I am done

rl/Lib/site-packages/sympy/vector/__init__.py

@@ -0,0 +1,47 @@
+from sympy.vector.coordsysrect import CoordSys3D
+from sympy.vector.vector import (Vector, VectorAdd, VectorMul,
+                                 BaseVector, VectorZero, Cross, Dot, cross, dot)
+from sympy.vector.dyadic import (Dyadic, DyadicAdd, DyadicMul,
+                                 BaseDyadic, DyadicZero)
+from sympy.vector.scalar import BaseScalar
+from sympy.vector.deloperator import Del
+from sympy.vector.functions import (express, matrix_to_vector,
+                                    laplacian, is_conservative,
+                                    is_solenoidal, scalar_potential,
+                                    directional_derivative,
+                                    scalar_potential_difference)
+from sympy.vector.point import Point
+from sympy.vector.orienters import (AxisOrienter, BodyOrienter,
+                                    SpaceOrienter, QuaternionOrienter)
+from sympy.vector.operators import Gradient, Divergence, Curl, Laplacian, gradient, curl, divergence
+from sympy.vector.implicitregion import ImplicitRegion
+from sympy.vector.parametricregion import (ParametricRegion, parametric_region_list)
+from sympy.vector.integrals import (ParametricIntegral, vector_integrate)
+
+__all__ = [
+    'Vector', 'VectorAdd', 'VectorMul', 'BaseVector', 'VectorZero', 'Cross',
+    'Dot', 'cross', 'dot',
+
+    'Dyadic', 'DyadicAdd', 'DyadicMul', 'BaseDyadic', 'DyadicZero',
+
+    'BaseScalar',
+
+    'Del',
+
+    'CoordSys3D',
+
+    'express', 'matrix_to_vector', 'laplacian', 'is_conservative',
+    'is_solenoidal', 'scalar_potential', 'directional_derivative',
+    'scalar_potential_difference',
+
+    'Point',
+
+    'AxisOrienter', 'BodyOrienter', 'SpaceOrienter', 'QuaternionOrienter',
+
+    'Gradient', 'Divergence', 'Curl', 'Laplacian', 'gradient', 'curl',
+    'divergence',
+
+    'ParametricRegion', 'parametric_region_list', 'ImplicitRegion',
+
+    'ParametricIntegral', 'vector_integrate',
+]

rl/Lib/site-packages/sympy/vector/basisdependent.py

@@ -0,0 +1,365 @@
+from __future__ import annotations
+from typing import TYPE_CHECKING
+
+from sympy.simplify import simplify as simp, trigsimp as tsimp  # type: ignore
+from sympy.core.decorators import call_highest_priority, _sympifyit
+from sympy.core.assumptions import StdFactKB
+from sympy.core.function import diff as df
+from sympy.integrals.integrals import Integral
+from sympy.polys.polytools import factor as fctr
+from sympy.core import S, Add, Mul
+from sympy.core.expr import Expr
+
+if TYPE_CHECKING:
+    from sympy.vector.vector import BaseVector
+
+
+class BasisDependent(Expr):
+    """
+    Super class containing functionality common to vectors and
+    dyadics.
+    Named so because the representation of these quantities in
+    sympy.vector is dependent on the basis they are expressed in.
+    """
+
+    zero: BasisDependentZero
+
+    @call_highest_priority('__radd__')
+    def __add__(self, other):
+        return self._add_func(self, other)
+
+    @call_highest_priority('__add__')
+    def __radd__(self, other):
+        return self._add_func(other, self)
+
+    @call_highest_priority('__rsub__')
+    def __sub__(self, other):
+        return self._add_func(self, -other)
+
+    @call_highest_priority('__sub__')
+    def __rsub__(self, other):
+        return self._add_func(other, -self)
+
+    @_sympifyit('other', NotImplemented)
+    @call_highest_priority('__rmul__')
+    def __mul__(self, other):
+        return self._mul_func(self, other)
+
+    @_sympifyit('other', NotImplemented)
+    @call_highest_priority('__mul__')
+    def __rmul__(self, other):
+        return self._mul_func(other, self)
+
+    def __neg__(self):
+        return self._mul_func(S.NegativeOne, self)
+
+    @_sympifyit('other', NotImplemented)
+    @call_highest_priority('__rtruediv__')
+    def __truediv__(self, other):
+        return self._div_helper(other)
+
+    @call_highest_priority('__truediv__')
+    def __rtruediv__(self, other):
+        return TypeError("Invalid divisor for division")
+
+    def evalf(self, n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False):
+        """
+        Implements the SymPy evalf routine for this quantity.
+
+        evalf's documentation
+        =====================
+
+        """
+        options = {'subs':subs, 'maxn':maxn, 'chop':chop, 'strict':strict,
+                'quad':quad, 'verbose':verbose}
+        vec = self.zero
+        for k, v in self.components.items():
+            vec += v.evalf(n, **options) * k
+        return vec
+
+    evalf.__doc__ += Expr.evalf.__doc__  # type: ignore
+
+    n = evalf
+
+    def simplify(self, **kwargs):
+        """
+        Implements the SymPy simplify routine for this quantity.
+
+        simplify's documentation
+        ========================
+
+        """
+        simp_components = [simp(v, **kwargs) * k for
+                           k, v in self.components.items()]
+        return self._add_func(*simp_components)
+
+    simplify.__doc__ += simp.__doc__  # type: ignore
+
+    def trigsimp(self, **opts):
+        """
+        Implements the SymPy trigsimp routine, for this quantity.
+
+        trigsimp's documentation
+        ========================
+
+        """
+        trig_components = [tsimp(v, **opts) * k for
+                           k, v in self.components.items()]
+        return self._add_func(*trig_components)
+
+    trigsimp.__doc__ += tsimp.__doc__  # type: ignore
+
+    def _eval_simplify(self, **kwargs):
+        return self.simplify(**kwargs)
+
+    def _eval_trigsimp(self, **opts):
+        return self.trigsimp(**opts)
+
+    def _eval_derivative(self, wrt):
+        return self.diff(wrt)
+
+    def _eval_Integral(self, *symbols, **assumptions):
+        integral_components = [Integral(v, *symbols, **assumptions) * k
+                               for k, v in self.components.items()]
+        return self._add_func(*integral_components)
+
+    def as_numer_denom(self):
+        """
+        Returns the expression as a tuple wrt the following
+        transformation -
+
+        expression -> a/b -> a, b
+
+        """
+        return self, S.One
+
+    def factor(self, *args, **kwargs):
+        """
+        Implements the SymPy factor routine, on the scalar parts
+        of a basis-dependent expression.
+
+        factor's documentation
+        ========================
+
+        """
+        fctr_components = [fctr(v, *args, **kwargs) * k for
+                           k, v in self.components.items()]
+        return self._add_func(*fctr_components)
+
+    factor.__doc__ += fctr.__doc__  # type: ignore
+
+    def as_coeff_Mul(self, rational=False):
+        """Efficiently extract the coefficient of a product."""
+        return (S.One, self)
+
+    def as_coeff_add(self, *deps):
+        """Efficiently extract the coefficient of a summation."""
+        l = [x * self.components[x] for x in self.components]
+        return 0, tuple(l)
+
+    def diff(self, *args, **kwargs):
+        """
+        Implements the SymPy diff routine, for vectors.
+
+        diff's documentation
+        ========================
+
+        """
+        for x in args:
+            if isinstance(x, BasisDependent):
+                raise TypeError("Invalid arg for differentiation")
+        diff_components = [df(v, *args, **kwargs) * k for
+                           k, v in self.components.items()]
+        return self._add_func(*diff_components)
+
+    diff.__doc__ += df.__doc__  # type: ignore
+
+    def doit(self, **hints):
+        """Calls .doit() on each term in the Dyadic"""
+        doit_components = [self.components[x].doit(**hints) * x
+                           for x in self.components]
+        return self._add_func(*doit_components)
+
+
+class BasisDependentAdd(BasisDependent, Add):
+    """
+    Denotes sum of basis dependent quantities such that they cannot
+    be expressed as base or Mul instances.
+    """
+
+    def __new__(cls, *args, **options):
+        components = {}
+
+        # Check each arg and simultaneously learn the components
+        for arg in args:
+            if not isinstance(arg, cls._expr_type):
+                if isinstance(arg, Mul):
+                    arg = cls._mul_func(*(arg.args))
+                elif isinstance(arg, Add):
+                    arg = cls._add_func(*(arg.args))
+                else:
+                    raise TypeError(str(arg) +
+                                    " cannot be interpreted correctly")
+            # If argument is zero, ignore
+            if arg == cls.zero:
+                continue
+            # Else, update components accordingly
+            if hasattr(arg, "components"):
+                for x in arg.components:
+                    components[x] = components.get(x, 0) + arg.components[x]
+
+        temp = list(components.keys())
+        for x in temp:
+            if components[x] == 0:
+                del components[x]
+
+        # Handle case of zero vector
+        if len(components) == 0:
+            return cls.zero
+
+        # Build object
+        newargs = [x * components[x] for x in components]
+        obj = super().__new__(cls, *newargs, **options)
+        if isinstance(obj, Mul):
+            return cls._mul_func(*obj.args)
+        assumptions = {'commutative': True}
+        obj._assumptions = StdFactKB(assumptions)
+        obj._components = components
+        obj._sys = (list(components.keys()))[0]._sys
+
+        return obj
+
+
+class BasisDependentMul(BasisDependent, Mul):
+    """
+    Denotes product of base- basis dependent quantity with a scalar.
+    """
+
+    def __new__(cls, *args, **options):
+        from sympy.vector import Cross, Dot, Curl, Gradient
+        count = 0
+        measure_number = S.One
+        zeroflag = False
+        extra_args = []
+
+        # Determine the component and check arguments
+        # Also keep a count to ensure two vectors aren't
+        # being multiplied
+        for arg in args:
+            if isinstance(arg, cls._zero_func):
+                count += 1
+                zeroflag = True
+            elif arg == S.Zero:
+                zeroflag = True
+            elif isinstance(arg, (cls._base_func, cls._mul_func)):
+                count += 1
+                expr = arg._base_instance
+                measure_number *= arg._measure_number
+            elif isinstance(arg, cls._add_func):
+                count += 1
+                expr = arg
+            elif isinstance(arg, (Cross, Dot, Curl, Gradient)):
+                extra_args.append(arg)
+            else:
+                measure_number *= arg
+        # Make sure incompatible types weren't multiplied
+        if count > 1:
+            raise ValueError("Invalid multiplication")
+        elif count == 0:
+            return Mul(*args, **options)
+        # Handle zero vector case
+        if zeroflag:
+            return cls.zero
+
+        # If one of the args was a VectorAdd, return an
+        # appropriate VectorAdd instance
+        if isinstance(expr, cls._add_func):
+            newargs = [cls._mul_func(measure_number, x) for
+                       x in expr.args]
+            return cls._add_func(*newargs)
+
+        obj = super().__new__(cls, measure_number,
+                              expr._base_instance,
+                              *extra_args,
+                              **options)
+        if isinstance(obj, Add):
+            return cls._add_func(*obj.args)
+        obj._base_instance = expr._base_instance
+        obj._measure_number = measure_number
+        assumptions = {'commutative': True}
+        obj._assumptions = StdFactKB(assumptions)
+        obj._components = {expr._base_instance: measure_number}
+        obj._sys = expr._base_instance._sys
+
+        return obj
+
+    def _sympystr(self, printer):
+        measure_str = printer._print(self._measure_number)
+        if ('(' in measure_str or '-' in measure_str or
+                '+' in measure_str):
+            measure_str = '(' + measure_str + ')'
+        return measure_str + '*' + printer._print(self._base_instance)
+
+
+class BasisDependentZero(BasisDependent):
+    """
+    Class to denote a zero basis dependent instance.
+    """
+    components: dict['BaseVector', Expr] = {}
+    _latex_form: str
+
+    def __new__(cls):
+        obj = super().__new__(cls)
+        # Pre-compute a specific hash value for the zero vector
+        # Use the same one always
+        obj._hash = (S.Zero, cls).__hash__()
+        return obj
+
+    def __hash__(self):
+        return self._hash
+
+    @call_highest_priority('__req__')
+    def __eq__(self, other):
+        return isinstance(other, self._zero_func)
+
+    __req__ = __eq__
+
+    @call_highest_priority('__radd__')
+    def __add__(self, other):
+        if isinstance(other, self._expr_type):
+            return other
+        else:
+            raise TypeError("Invalid argument types for addition")
+
+    @call_highest_priority('__add__')
+    def __radd__(self, other):
+        if isinstance(other, self._expr_type):
+            return other
+        else:
+            raise TypeError("Invalid argument types for addition")
+
+    @call_highest_priority('__rsub__')
+    def __sub__(self, other):
+        if isinstance(other, self._expr_type):
+            return -other
+        else:
+            raise TypeError("Invalid argument types for subtraction")
+
+    @call_highest_priority('__sub__')
+    def __rsub__(self, other):
+        if isinstance(other, self._expr_type):
+            return other
+        else:
+            raise TypeError("Invalid argument types for subtraction")
+
+    def __neg__(self):
+        return self
+
+    def normalize(self):
+        """
+        Returns the normalized version of this vector.
+        """
+        return self
+
+    def _sympystr(self, printer):
+        return '0'

rl/Lib/site-packages/sympy/vector/deloperator.py

@@ -0,0 +1,121 @@
+from sympy.core import Basic
+from sympy.vector.operators import gradient, divergence, curl
+
+
+class Del(Basic):
+    """
+    Represents the vector differential operator, usually represented in
+    mathematical expressions as the 'nabla' symbol.
+    """
+
+    def __new__(cls):
+        obj = super().__new__(cls)
+        obj._name = "delop"
+        return obj
+
+    def gradient(self, scalar_field, doit=False):
+        """
+        Returns the gradient of the given scalar field, as a
+        Vector instance.
+
+        Parameters
+        ==========
+
+        scalar_field : SymPy expression
+            The scalar field to calculate the gradient of.
+
+        doit : bool
+            If True, the result is returned after calling .doit() on
+            each component. Else, the returned expression contains
+            Derivative instances
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D, Del
+        >>> C = CoordSys3D('C')
+        >>> delop = Del()
+        >>> delop.gradient(9)
+        0
+        >>> delop(C.x*C.y*C.z).doit()
+        C.y*C.z*C.i + C.x*C.z*C.j + C.x*C.y*C.k
+
+        """
+
+        return gradient(scalar_field, doit=doit)
+
+    __call__ = gradient
+    __call__.__doc__ = gradient.__doc__
+
+    def dot(self, vect, doit=False):
+        """
+        Represents the dot product between this operator and a given
+        vector - equal to the divergence of the vector field.
+
+        Parameters
+        ==========
+
+        vect : Vector
+            The vector whose divergence is to be calculated.
+
+        doit : bool
+            If True, the result is returned after calling .doit() on
+            each component. Else, the returned expression contains
+            Derivative instances
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D, Del
+        >>> delop = Del()
+        >>> C = CoordSys3D('C')
+        >>> delop.dot(C.x*C.i)
+        Derivative(C.x, C.x)
+        >>> v = C.x*C.y*C.z * (C.i + C.j + C.k)
+        >>> (delop & v).doit()
+        C.x*C.y + C.x*C.z + C.y*C.z
+
+        """
+        return divergence(vect, doit=doit)
+
+    __and__ = dot
+    __and__.__doc__ = dot.__doc__
+
+    def cross(self, vect, doit=False):
+        """
+        Represents the cross product between this operator and a given
+        vector - equal to the curl of the vector field.
+
+        Parameters
+        ==========
+
+        vect : Vector
+            The vector whose curl is to be calculated.
+
+        doit : bool
+            If True, the result is returned after calling .doit() on
+            each component. Else, the returned expression contains
+            Derivative instances
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D, Del
+        >>> C = CoordSys3D('C')
+        >>> delop = Del()
+        >>> v = C.x*C.y*C.z * (C.i + C.j + C.k)
+        >>> delop.cross(v, doit = True)
+        (-C.x*C.y + C.x*C.z)*C.i + (C.x*C.y - C.y*C.z)*C.j +
+            (-C.x*C.z + C.y*C.z)*C.k
+        >>> (delop ^ C.i).doit()
+        0
+
+        """
+
+        return curl(vect, doit=doit)
+
+    __xor__ = cross
+    __xor__.__doc__ = cross.__doc__
+
+    def _sympystr(self, printer):
+        return self._name

rl/Lib/site-packages/sympy/vector/dyadic.py

@@ -0,0 +1,285 @@
+from __future__ import annotations
+
+from sympy.vector.basisdependent import (BasisDependent, BasisDependentAdd,
+                                         BasisDependentMul, BasisDependentZero)
+from sympy.core import S, Pow
+from sympy.core.expr import AtomicExpr
+from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix
+import sympy.vector
+
+
+class Dyadic(BasisDependent):
+    """
+    Super class for all Dyadic-classes.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Dyadic_tensor
+    .. [2] Kane, T., Levinson, D. Dynamics Theory and Applications. 1985
+           McGraw-Hill
+
+    """
+
+    _op_priority = 13.0
+
+    _expr_type: type[Dyadic]
+    _mul_func: type[Dyadic]
+    _add_func: type[Dyadic]
+    _zero_func: type[Dyadic]
+    _base_func: type[Dyadic]
+    zero: DyadicZero
+
+    @property
+    def components(self):
+        """
+        Returns the components of this dyadic in the form of a
+        Python dictionary mapping BaseDyadic instances to the
+        corresponding measure numbers.
+
+        """
+        # The '_components' attribute is defined according to the
+        # subclass of Dyadic the instance belongs to.
+        return self._components
+
+    def dot(self, other):
+        """
+        Returns the dot product(also called inner product) of this
+        Dyadic, with another Dyadic or Vector.
+        If 'other' is a Dyadic, this returns a Dyadic. Else, it returns
+        a Vector (unless an error is encountered).
+
+        Parameters
+        ==========
+
+        other : Dyadic/Vector
+            The other Dyadic or Vector to take the inner product with
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D
+        >>> N = CoordSys3D('N')
+        >>> D1 = N.i.outer(N.j)
+        >>> D2 = N.j.outer(N.j)
+        >>> D1.dot(D2)
+        (N.i|N.j)
+        >>> D1.dot(N.j)
+        N.i
+
+        """
+
+        Vector = sympy.vector.Vector
+        if isinstance(other, BasisDependentZero):
+            return Vector.zero
+        elif isinstance(other, Vector):
+            outvec = Vector.zero
+            for k, v in self.components.items():
+                vect_dot = k.args[1].dot(other)
+                outvec += vect_dot * v * k.args[0]
+            return outvec
+        elif isinstance(other, Dyadic):
+            outdyad = Dyadic.zero
+            for k1, v1 in self.components.items():
+                for k2, v2 in other.components.items():
+                    vect_dot = k1.args[1].dot(k2.args[0])
+                    outer_product = k1.args[0].outer(k2.args[1])
+                    outdyad += vect_dot * v1 * v2 * outer_product
+            return outdyad
+        else:
+            raise TypeError("Inner product is not defined for " +
+                            str(type(other)) + " and Dyadics.")
+
+    def __and__(self, other):
+        return self.dot(other)
+
+    __and__.__doc__ = dot.__doc__
+
+    def cross(self, other):
+        """
+        Returns the cross product between this Dyadic, and a Vector, as a
+        Vector instance.
+
+        Parameters
+        ==========
+
+        other : Vector
+            The Vector that we are crossing this Dyadic with
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D
+        >>> N = CoordSys3D('N')
+        >>> d = N.i.outer(N.i)
+        >>> d.cross(N.j)
+        (N.i|N.k)
+
+        """
+
+        Vector = sympy.vector.Vector
+        if other == Vector.zero:
+            return Dyadic.zero
+        elif isinstance(other, Vector):
+            outdyad = Dyadic.zero
+            for k, v in self.components.items():
+                cross_product = k.args[1].cross(other)
+                outer = k.args[0].outer(cross_product)
+                outdyad += v * outer
+            return outdyad
+        else:
+            raise TypeError(str(type(other)) + " not supported for " +
+                            "cross with dyadics")
+
+    def __xor__(self, other):
+        return self.cross(other)
+
+    __xor__.__doc__ = cross.__doc__
+
+    def to_matrix(self, system, second_system=None):
+        """
+        Returns the matrix form of the dyadic with respect to one or two
+        coordinate systems.
+
+        Parameters
+        ==========
+
+        system : CoordSys3D
+            The coordinate system that the rows and columns of the matrix
+            correspond to. If a second system is provided, this
+            only corresponds to the rows of the matrix.
+        second_system : CoordSys3D, optional, default=None
+            The coordinate system that the columns of the matrix correspond
+            to.
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D
+        >>> N = CoordSys3D('N')
+        >>> v = N.i + 2*N.j
+        >>> d = v.outer(N.i)
+        >>> d.to_matrix(N)
+        Matrix([
+        [1, 0, 0],
+        [2, 0, 0],
+        [0, 0, 0]])
+        >>> from sympy import Symbol
+        >>> q = Symbol('q')
+        >>> P = N.orient_new_axis('P', q, N.k)
+        >>> d.to_matrix(N, P)
+        Matrix([
+        [  cos(q),   -sin(q), 0],
+        [2*cos(q), -2*sin(q), 0],
+        [       0,         0, 0]])
+
+        """
+
+        if second_system is None:
+            second_system = system
+
+        return Matrix([i.dot(self).dot(j) for i in system for j in
+                       second_system]).reshape(3, 3)
+
+    def _div_helper(one, other):
+        """ Helper for division involving dyadics """
+        if isinstance(one, Dyadic) and isinstance(other, Dyadic):
+            raise TypeError("Cannot divide two dyadics")
+        elif isinstance(one, Dyadic):
+            return DyadicMul(one, Pow(other, S.NegativeOne))
+        else:
+            raise TypeError("Cannot divide by a dyadic")
+
+
+class BaseDyadic(Dyadic, AtomicExpr):
+    """
+    Class to denote a base dyadic tensor component.
+    """
+
+    def __new__(cls, vector1, vector2):
+        Vector = sympy.vector.Vector
+        BaseVector = sympy.vector.BaseVector
+        VectorZero = sympy.vector.VectorZero
+        # Verify arguments
+        if not isinstance(vector1, (BaseVector, VectorZero)) or \
+                not isinstance(vector2, (BaseVector, VectorZero)):
+            raise TypeError("BaseDyadic cannot be composed of non-base " +
+                            "vectors")
+        # Handle special case of zero vector
+        elif vector1 == Vector.zero or vector2 == Vector.zero:
+            return Dyadic.zero
+        # Initialize instance
+        obj = super().__new__(cls, vector1, vector2)
+        obj._base_instance = obj
+        obj._measure_number = 1
+        obj._components = {obj: S.One}
+        obj._sys = vector1._sys
+        obj._pretty_form = ('(' + vector1._pretty_form + '|' +
+                             vector2._pretty_form + ')')
+        obj._latex_form = (r'\left(' + vector1._latex_form + r"{\middle|}" +
+                           vector2._latex_form + r'\right)')
+
+        return obj
+
+    def _sympystr(self, printer):
+        return "({}|{})".format(
+            printer._print(self.args[0]), printer._print(self.args[1]))
+
+    def _sympyrepr(self, printer):
+        return "BaseDyadic({}, {})".format(
+            printer._print(self.args[0]), printer._print(self.args[1]))
+
+
+class DyadicMul(BasisDependentMul, Dyadic):
+    """ Products of scalars and BaseDyadics """
+
+    def __new__(cls, *args, **options):
+        obj = BasisDependentMul.__new__(cls, *args, **options)
+        return obj
+
+    @property
+    def base_dyadic(self):
+        """ The BaseDyadic involved in the product. """
+        return self._base_instance
+
+    @property
+    def measure_number(self):
+        """ The scalar expression involved in the definition of
+        this DyadicMul.
+        """
+        return self._measure_number
+
+
+class DyadicAdd(BasisDependentAdd, Dyadic):
+    """ Class to hold dyadic sums """
+
+    def __new__(cls, *args, **options):
+        obj = BasisDependentAdd.__new__(cls, *args, **options)
+        return obj
+
+    def _sympystr(self, printer):
+        items = list(self.components.items())
+        items.sort(key=lambda x: x[0].__str__())
+        return " + ".join(printer._print(k * v) for k, v in items)
+
+
+class DyadicZero(BasisDependentZero, Dyadic):
+    """
+    Class to denote a zero dyadic
+    """
+
+    _op_priority = 13.1
+    _pretty_form = '(0|0)'
+    _latex_form = r'(\mathbf{\hat{0}}|\mathbf{\hat{0}})'
+
+    def __new__(cls):
+        obj = BasisDependentZero.__new__(cls)
+        return obj
+
+
+Dyadic._expr_type = Dyadic
+Dyadic._mul_func = DyadicMul
+Dyadic._add_func = DyadicAdd
+Dyadic._zero_func = DyadicZero
+Dyadic._base_func = BaseDyadic
+Dyadic.zero = DyadicZero()

rl/Lib/site-packages/sympy/vector/functions.py

@@ -0,0 +1,517 @@
+from sympy.vector.coordsysrect import CoordSys3D
+from sympy.vector.deloperator import Del
+from sympy.vector.scalar import BaseScalar
+from sympy.vector.vector import Vector, BaseVector
+from sympy.vector.operators import gradient, curl, divergence
+from sympy.core.function import diff
+from sympy.core.singleton import S
+from sympy.integrals.integrals import integrate
+from sympy.simplify.simplify import simplify
+from sympy.core import sympify
+from sympy.vector.dyadic import Dyadic
+
+
+def express(expr, system, system2=None, variables=False):
+    """
+    Global function for 'express' functionality.
+
+    Re-expresses a Vector, Dyadic or scalar(sympyfiable) in the given
+    coordinate system.
+
+    If 'variables' is True, then the coordinate variables (base scalars)
+    of other coordinate systems present in the vector/scalar field or
+    dyadic are also substituted in terms of the base scalars of the
+    given system.
+
+    Parameters
+    ==========
+
+    expr : Vector/Dyadic/scalar(sympyfiable)
+        The expression to re-express in CoordSys3D 'system'
+
+    system: CoordSys3D
+        The coordinate system the expr is to be expressed in
+
+    system2: CoordSys3D
+        The other coordinate system required for re-expression
+        (only for a Dyadic Expr)
+
+    variables : boolean
+        Specifies whether to substitute the coordinate variables present
+        in expr, in terms of those of parameter system
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D
+    >>> from sympy import Symbol, cos, sin
+    >>> N = CoordSys3D('N')
+    >>> q = Symbol('q')
+    >>> B = N.orient_new_axis('B', q, N.k)
+    >>> from sympy.vector import express
+    >>> express(B.i, N)
+    (cos(q))*N.i + (sin(q))*N.j
+    >>> express(N.x, B, variables=True)
+    B.x*cos(q) - B.y*sin(q)
+    >>> d = N.i.outer(N.i)
+    >>> express(d, B, N) == (cos(q))*(B.i|N.i) + (-sin(q))*(B.j|N.i)
+    True
+
+    """
+
+    if expr in (0, Vector.zero):
+        return expr
+
+    if not isinstance(system, CoordSys3D):
+        raise TypeError("system should be a CoordSys3D \
+                        instance")
+
+    if isinstance(expr, Vector):
+        if system2 is not None:
+            raise ValueError("system2 should not be provided for \
+                                Vectors")
+        # Given expr is a Vector
+        if variables:
+            # If variables attribute is True, substitute
+            # the coordinate variables in the Vector
+            system_list = {x.system for x in expr.atoms(BaseScalar, BaseVector)} - {system}
+            subs_dict = {}
+            for f in system_list:
+                subs_dict.update(f.scalar_map(system))
+            expr = expr.subs(subs_dict)
+        # Re-express in this coordinate system
+        outvec = Vector.zero
+        parts = expr.separate()
+        for x in parts:
+            if x != system:
+                temp = system.rotation_matrix(x) * parts[x].to_matrix(x)
+                outvec += matrix_to_vector(temp, system)
+            else:
+                outvec += parts[x]
+        return outvec
+
+    elif isinstance(expr, Dyadic):
+        if system2 is None:
+            system2 = system
+        if not isinstance(system2, CoordSys3D):
+            raise TypeError("system2 should be a CoordSys3D \
+                            instance")
+        outdyad = Dyadic.zero
+        var = variables
+        for k, v in expr.components.items():
+            outdyad += (express(v, system, variables=var) *
+                        (express(k.args[0], system, variables=var) |
+                         express(k.args[1], system2, variables=var)))
+
+        return outdyad
+
+    else:
+        if system2 is not None:
+            raise ValueError("system2 should not be provided for \
+                                Vectors")
+        if variables:
+            # Given expr is a scalar field
+            system_set = set()
+            expr = sympify(expr)
+            # Substitute all the coordinate variables
+            for x in expr.atoms(BaseScalar):
+                if x.system != system:
+                    system_set.add(x.system)
+            subs_dict = {}
+            for f in system_set:
+                subs_dict.update(f.scalar_map(system))
+            return expr.subs(subs_dict)
+        return expr
+
+
+def directional_derivative(field, direction_vector):
+    """
+    Returns the directional derivative of a scalar or vector field computed
+    along a given vector in coordinate system which parameters are expressed.
+
+    Parameters
+    ==========
+
+    field : Vector or Scalar
+        The scalar or vector field to compute the directional derivative of
+
+    direction_vector : Vector
+        The vector to calculated directional derivative along them.
+
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, directional_derivative
+    >>> R = CoordSys3D('R')
+    >>> f1 = R.x*R.y*R.z
+    >>> v1 = 3*R.i + 4*R.j + R.k
+    >>> directional_derivative(f1, v1)
+    R.x*R.y + 4*R.x*R.z + 3*R.y*R.z
+    >>> f2 = 5*R.x**2*R.z
+    >>> directional_derivative(f2, v1)
+    5*R.x**2 + 30*R.x*R.z
+
+    """
+    from sympy.vector.operators import _get_coord_systems
+    coord_sys = _get_coord_systems(field)
+    if len(coord_sys) > 0:
+        # TODO: This gets a random coordinate system in case of multiple ones:
+        coord_sys = next(iter(coord_sys))
+        field = express(field, coord_sys, variables=True)
+        i, j, k = coord_sys.base_vectors()
+        x, y, z = coord_sys.base_scalars()
+        out = Vector.dot(direction_vector, i) * diff(field, x)
+        out += Vector.dot(direction_vector, j) * diff(field, y)
+        out += Vector.dot(direction_vector, k) * diff(field, z)
+        if out == 0 and isinstance(field, Vector):
+            out = Vector.zero
+        return out
+    elif isinstance(field, Vector):
+        return Vector.zero
+    else:
+        return S.Zero
+
+
+def laplacian(expr):
+    """
+    Return the laplacian of the given field computed in terms of
+    the base scalars of the given coordinate system.
+
+    Parameters
+    ==========
+
+    expr : SymPy Expr or Vector
+        expr denotes a scalar or vector field.
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, laplacian
+    >>> R = CoordSys3D('R')
+    >>> f = R.x**2*R.y**5*R.z
+    >>> laplacian(f)
+    20*R.x**2*R.y**3*R.z + 2*R.y**5*R.z
+    >>> f = R.x**2*R.i + R.y**3*R.j + R.z**4*R.k
+    >>> laplacian(f)
+    2*R.i + 6*R.y*R.j + 12*R.z**2*R.k
+
+    """
+
+    delop = Del()
+    if expr.is_Vector:
+        return (gradient(divergence(expr)) - curl(curl(expr))).doit()
+    return delop.dot(delop(expr)).doit()
+
+
+def is_conservative(field):
+    """
+    Checks if a field is conservative.
+
+    Parameters
+    ==========
+
+    field : Vector
+        The field to check for conservative property
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D
+    >>> from sympy.vector import is_conservative
+    >>> R = CoordSys3D('R')
+    >>> is_conservative(R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k)
+    True
+    >>> is_conservative(R.z*R.j)
+    False
+
+    """
+
+    # Field is conservative irrespective of system
+    # Take the first coordinate system in the result of the
+    # separate method of Vector
+    if not isinstance(field, Vector):
+        raise TypeError("field should be a Vector")
+    if field == Vector.zero:
+        return True
+    return curl(field).simplify() == Vector.zero
+
+
+def is_solenoidal(field):
+    """
+    Checks if a field is solenoidal.
+
+    Parameters
+    ==========
+
+    field : Vector
+        The field to check for solenoidal property
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D
+    >>> from sympy.vector import is_solenoidal
+    >>> R = CoordSys3D('R')
+    >>> is_solenoidal(R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k)
+    True
+    >>> is_solenoidal(R.y * R.j)
+    False
+
+    """
+
+    # Field is solenoidal irrespective of system
+    # Take the first coordinate system in the result of the
+    # separate method in Vector
+    if not isinstance(field, Vector):
+        raise TypeError("field should be a Vector")
+    if field == Vector.zero:
+        return True
+    return divergence(field).simplify() is S.Zero
+
+
+def scalar_potential(field, coord_sys):
+    """
+    Returns the scalar potential function of a field in a given
+    coordinate system (without the added integration constant).
+
+    Parameters
+    ==========
+
+    field : Vector
+        The vector field whose scalar potential function is to be
+        calculated
+
+    coord_sys : CoordSys3D
+        The coordinate system to do the calculation in
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D
+    >>> from sympy.vector import scalar_potential, gradient
+    >>> R = CoordSys3D('R')
+    >>> scalar_potential(R.k, R) == R.z
+    True
+    >>> scalar_field = 2*R.x**2*R.y*R.z
+    >>> grad_field = gradient(scalar_field)
+    >>> scalar_potential(grad_field, R)
+    2*R.x**2*R.y*R.z
+
+    """
+
+    # Check whether field is conservative
+    if not is_conservative(field):
+        raise ValueError("Field is not conservative")
+    if field == Vector.zero:
+        return S.Zero
+    # Express the field exntirely in coord_sys
+    # Substitute coordinate variables also
+    if not isinstance(coord_sys, CoordSys3D):
+        raise TypeError("coord_sys must be a CoordSys3D")
+    field = express(field, coord_sys, variables=True)
+    dimensions = coord_sys.base_vectors()
+    scalars = coord_sys.base_scalars()
+    # Calculate scalar potential function
+    temp_function = integrate(field.dot(dimensions[0]), scalars[0])
+    for i, dim in enumerate(dimensions[1:]):
+        partial_diff = diff(temp_function, scalars[i + 1])
+        partial_diff = field.dot(dim) - partial_diff
+        temp_function += integrate(partial_diff, scalars[i + 1])
+    return temp_function
+
+
+def scalar_potential_difference(field, coord_sys, point1, point2):
+    """
+    Returns the scalar potential difference between two points in a
+    certain coordinate system, wrt a given field.
+
+    If a scalar field is provided, its values at the two points are
+    considered. If a conservative vector field is provided, the values
+    of its scalar potential function at the two points are used.
+
+    Returns (potential at point2) - (potential at point1)
+
+    The position vectors of the two Points are calculated wrt the
+    origin of the coordinate system provided.
+
+    Parameters
+    ==========
+
+    field : Vector/Expr
+        The field to calculate wrt
+
+    coord_sys : CoordSys3D
+        The coordinate system to do the calculations in
+
+    point1 : Point
+        The initial Point in given coordinate system
+
+    position2 : Point
+        The second Point in the given coordinate system
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D
+    >>> from sympy.vector import scalar_potential_difference
+    >>> R = CoordSys3D('R')
+    >>> P = R.origin.locate_new('P', R.x*R.i + R.y*R.j + R.z*R.k)
+    >>> vectfield = 4*R.x*R.y*R.i + 2*R.x**2*R.j
+    >>> scalar_potential_difference(vectfield, R, R.origin, P)
+    2*R.x**2*R.y
+    >>> Q = R.origin.locate_new('O', 3*R.i + R.j + 2*R.k)
+    >>> scalar_potential_difference(vectfield, R, P, Q)
+    -2*R.x**2*R.y + 18
+
+    """
+
+    if not isinstance(coord_sys, CoordSys3D):
+        raise TypeError("coord_sys must be a CoordSys3D")
+    if isinstance(field, Vector):
+        # Get the scalar potential function
+        scalar_fn = scalar_potential(field, coord_sys)
+    else:
+        # Field is a scalar
+        scalar_fn = field
+    # Express positions in required coordinate system
+    origin = coord_sys.origin
+    position1 = express(point1.position_wrt(origin), coord_sys,
+                        variables=True)
+    position2 = express(point2.position_wrt(origin), coord_sys,
+                        variables=True)
+    # Get the two positions as substitution dicts for coordinate variables
+    subs_dict1 = {}
+    subs_dict2 = {}
+    scalars = coord_sys.base_scalars()
+    for i, x in enumerate(coord_sys.base_vectors()):
+        subs_dict1[scalars[i]] = x.dot(position1)
+        subs_dict2[scalars[i]] = x.dot(position2)
+    return scalar_fn.subs(subs_dict2) - scalar_fn.subs(subs_dict1)
+
+
+def matrix_to_vector(matrix, system):
+    """
+    Converts a vector in matrix form to a Vector instance.
+
+    It is assumed that the elements of the Matrix represent the
+    measure numbers of the components of the vector along basis
+    vectors of 'system'.
+
+    Parameters
+    ==========
+
+    matrix : SymPy Matrix, Dimensions: (3, 1)
+        The matrix to be converted to a vector
+
+    system : CoordSys3D
+        The coordinate system the vector is to be defined in
+
+    Examples
+    ========
+
+    >>> from sympy import ImmutableMatrix as Matrix
+    >>> m = Matrix([1, 2, 3])
+    >>> from sympy.vector import CoordSys3D, matrix_to_vector
+    >>> C = CoordSys3D('C')
+    >>> v = matrix_to_vector(m, C)
+    >>> v
+    C.i + 2*C.j + 3*C.k
+    >>> v.to_matrix(C) == m
+    True
+
+    """
+
+    outvec = Vector.zero
+    vects = system.base_vectors()
+    for i, x in enumerate(matrix):
+        outvec += x * vects[i]
+    return outvec
+
+
+def _path(from_object, to_object):
+    """
+    Calculates the 'path' of objects starting from 'from_object'
+    to 'to_object', along with the index of the first common
+    ancestor in the tree.
+
+    Returns (index, list) tuple.
+    """
+
+    if from_object._root != to_object._root:
+        raise ValueError("No connecting path found between " +
+                         str(from_object) + " and " + str(to_object))
+
+    other_path = []
+    obj = to_object
+    while obj._parent is not None:
+        other_path.append(obj)
+        obj = obj._parent
+    other_path.append(obj)
+    object_set = set(other_path)
+    from_path = []
+    obj = from_object
+    while obj not in object_set:
+        from_path.append(obj)
+        obj = obj._parent
+    index = len(from_path)
+    i = other_path.index(obj)
+    while i >= 0:
+        from_path.append(other_path[i])
+        i -= 1
+    return index, from_path
+
+
+def orthogonalize(*vlist, orthonormal=False):
+    """
+    Takes a sequence of independent vectors and orthogonalizes them
+    using the Gram - Schmidt process. Returns a list of
+    orthogonal or orthonormal vectors.
+
+    Parameters
+    ==========
+
+    vlist : sequence of independent vectors to be made orthogonal.
+
+    orthonormal : Optional parameter
+                  Set to True if the vectors returned should be
+                  orthonormal.
+                  Default: False
+
+    Examples
+    ========
+
+    >>> from sympy.vector.coordsysrect import CoordSys3D
+    >>> from sympy.vector.functions import orthogonalize
+    >>> C = CoordSys3D('C')
+    >>> i, j, k = C.base_vectors()
+    >>> v1 = i + 2*j
+    >>> v2 = 2*i + 3*j
+    >>> orthogonalize(v1, v2)
+    [C.i + 2*C.j, 2/5*C.i + (-1/5)*C.j]
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Gram-Schmidt_process
+
+    """
+
+    if not all(isinstance(vec, Vector) for vec in vlist):
+        raise TypeError('Each element must be of Type Vector')
+
+    ortho_vlist = []
+    for i, term in enumerate(vlist):
+        for j in range(i):
+            term -= ortho_vlist[j].projection(vlist[i])
+        # TODO : The following line introduces a performance issue
+        # and needs to be changed once a good solution for issue #10279 is
+        # found.
+        if simplify(term).equals(Vector.zero):
+            raise ValueError("Vector set not linearly independent")
+        ortho_vlist.append(term)
+
+    if orthonormal:
+        ortho_vlist = [vec.normalize() for vec in ortho_vlist]
+
+    return ortho_vlist

rl/Lib/site-packages/sympy/vector/implicitregion.py

@@ -0,0 +1,506 @@
+from sympy.core.numbers import Rational
+from sympy.core.singleton import S
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.complexes import sign
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.polys.polytools import gcd
+from sympy.sets.sets import Complement
+from sympy.core import Basic, Tuple, diff, expand, Eq, Integer
+from sympy.core.sorting import ordered
+from sympy.core.symbol import _symbol
+from sympy.solvers import solveset, nonlinsolve, diophantine
+from sympy.polys import total_degree
+from sympy.geometry import Point
+from sympy.ntheory.factor_ import core
+
+
+class ImplicitRegion(Basic):
+    """
+    Represents an implicit region in space.
+
+    Examples
+    ========
+
+    >>> from sympy import Eq
+    >>> from sympy.abc import x, y, z, t
+    >>> from sympy.vector import ImplicitRegion
+
+    >>> ImplicitRegion((x, y), x**2 + y**2 - 4)
+    ImplicitRegion((x, y), x**2 + y**2 - 4)
+    >>> ImplicitRegion((x, y), Eq(y*x, 1))
+    ImplicitRegion((x, y), x*y - 1)
+
+    >>> parabola = ImplicitRegion((x, y), y**2 - 4*x)
+    >>> parabola.degree
+    2
+    >>> parabola.equation
+    -4*x + y**2
+    >>> parabola.rational_parametrization(t)
+    (4/t**2, 4/t)
+
+    >>> r = ImplicitRegion((x, y, z), Eq(z, x**2 + y**2))
+    >>> r.variables
+    (x, y, z)
+    >>> r.singular_points()
+    EmptySet
+    >>> r.regular_point()
+    (-10, -10, 200)
+
+    Parameters
+    ==========
+
+    variables : tuple to map variables in implicit equation to base scalars.
+
+    equation : An expression or Eq denoting the implicit equation of the region.
+
+    """
+    def __new__(cls, variables, equation):
+        if not isinstance(variables, Tuple):
+            variables = Tuple(*variables)
+
+        if isinstance(equation, Eq):
+            equation = equation.lhs - equation.rhs
+
+        return super().__new__(cls, variables, equation)
+
+    @property
+    def variables(self):
+        return self.args[0]
+
+    @property
+    def equation(self):
+        return self.args[1]
+
+    @property
+    def degree(self):
+        return total_degree(self.equation)
+
+    def regular_point(self):
+        """
+        Returns a point on the implicit region.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x, y, z
+        >>> from sympy.vector import ImplicitRegion
+        >>> circle = ImplicitRegion((x, y), (x + 2)**2 + (y - 3)**2 - 16)
+        >>> circle.regular_point()
+        (-2, -1)
+        >>> parabola = ImplicitRegion((x, y), x**2 - 4*y)
+        >>> parabola.regular_point()
+        (0, 0)
+        >>> r = ImplicitRegion((x, y, z), (x + y + z)**4)
+        >>> r.regular_point()
+        (-10, -10, 20)
+
+        References
+        ==========
+
+        - Erik Hillgarter, "Rational Points on Conics", Diploma Thesis, RISC-Linz,
+          J. Kepler Universitat Linz, 1996. Available:
+          https://www3.risc.jku.at/publications/download/risc_1355/Rational%20Points%20on%20Conics.pdf
+
+        """
+        equation = self.equation
+
+        if len(self.variables) == 1:
+            return (list(solveset(equation, self.variables[0], domain=S.Reals))[0],)
+        elif len(self.variables) == 2:
+
+            if self.degree == 2:
+                coeffs = a, b, c, d, e, f = conic_coeff(self.variables, equation)
+
+                if b**2 == 4*a*c:
+                    x_reg, y_reg = self._regular_point_parabola(*coeffs)
+                else:
+                    x_reg, y_reg = self._regular_point_ellipse(*coeffs)
+                return x_reg, y_reg
+
+        if len(self.variables) == 3:
+            x, y, z = self.variables
+
+            for x_reg in range(-10, 10):
+                for y_reg in range(-10, 10):
+                    if not solveset(equation.subs({x: x_reg, y: y_reg}), self.variables[2], domain=S.Reals).is_empty:
+                        return (x_reg, y_reg, list(solveset(equation.subs({x: x_reg, y: y_reg})))[0])
+
+        if len(self.singular_points()) != 0:
+            return list[self.singular_points()][0]
+
+        raise NotImplementedError()
+
+    def _regular_point_parabola(self, a, b, c, d, e, f):
+            ok = (a, d) != (0, 0) and (c, e) != (0, 0) and b**2 == 4*a*c and (a, c) != (0, 0)
+
+            if not ok:
+                raise ValueError("Rational Point on the conic does not exist")
+
+            if a != 0:
+                d_dash, f_dash = (4*a*e - 2*b*d, 4*a*f - d**2)
+                if d_dash != 0:
+                    y_reg = -f_dash/d_dash
+                    x_reg = -(d + b*y_reg)/(2*a)
+                else:
+                    ok = False
+            elif c != 0:
+                d_dash, f_dash = (4*c*d - 2*b*e, 4*c*f - e**2)
+                if d_dash != 0:
+                    x_reg = -f_dash/d_dash
+                    y_reg = -(e + b*x_reg)/(2*c)
+                else:
+                    ok = False
+
+            if ok:
+                return x_reg, y_reg
+            else:
+                raise ValueError("Rational Point on the conic does not exist")
+
+    def _regular_point_ellipse(self, a, b, c, d, e, f):
+            D = 4*a*c - b**2
+            ok = D
+
+            if not ok:
+                raise ValueError("Rational Point on the conic does not exist")
+
+            if a == 0 and c == 0:
+                K = -1
+                L = 4*(d*e - b*f)
+            elif c != 0:
+                K = D
+                L = 4*c**2*d**2 - 4*b*c*d*e + 4*a*c*e**2 + 4*b**2*c*f - 16*a*c**2*f
+            else:
+                K = D
+                L = 4*a**2*e**2 - 4*b*a*d*e + 4*b**2*a*f
+
+            ok = L != 0 and not(K > 0 and L < 0)
+            if not ok:
+                raise ValueError("Rational Point on the conic does not exist")
+
+            K = Rational(K).limit_denominator(10**12)
+            L = Rational(L).limit_denominator(10**12)
+
+            k1, k2 = K.p, K.q
+            l1, l2 = L.p, L.q
+            g = gcd(k2, l2)
+
+            a1 = (l2*k2)/g
+            b1 = (k1*l2)/g
+            c1 = -(l1*k2)/g
+            a2 = sign(a1)*core(abs(a1), 2)
+            r1 = sqrt(a1/a2)
+            b2 = sign(b1)*core(abs(b1), 2)
+            r2 = sqrt(b1/b2)
+            c2 = sign(c1)*core(abs(c1), 2)
+            r3 = sqrt(c1/c2)
+
+            g = gcd(gcd(a2, b2), c2)
+            a2 = a2/g
+            b2 = b2/g
+            c2 = c2/g
+
+            g1 = gcd(a2, b2)
+            a2 = a2/g1
+            b2 = b2/g1
+            c2 = c2*g1
+
+            g2 = gcd(a2,c2)
+            a2 = a2/g2
+            b2 = b2*g2
+            c2 = c2/g2
+
+            g3 = gcd(b2, c2)
+            a2 = a2*g3
+            b2 = b2/g3
+            c2 = c2/g3
+
+            x, y, z = symbols("x y z")
+            eq = a2*x**2 + b2*y**2 + c2*z**2
+
+            solutions = diophantine(eq)
+
+            if len(solutions) == 0:
+                raise ValueError("Rational Point on the conic does not exist")
+
+            flag = False
+            for sol in solutions:
+                syms = Tuple(*sol).free_symbols
+                rep = dict.fromkeys(syms, 3)
+                sol_z = sol[2]
+
+                if sol_z == 0:
+                    flag = True
+                    continue
+
+                if not isinstance(sol_z, (int, Integer)):
+                    syms_z = sol_z.free_symbols
+
+                    if len(syms_z) == 1:
+                        p = next(iter(syms_z))
+                        p_values = Complement(S.Integers, solveset(Eq(sol_z, 0), p, S.Integers))
+                        rep[p] = next(iter(p_values))
+
+                    if len(syms_z) == 2:
+                        p, q = list(ordered(syms_z))
+
+                        for i in S.Integers:
+                            subs_sol_z = sol_z.subs(p, i)
+                            q_values = Complement(S.Integers, solveset(Eq(subs_sol_z, 0), q, S.Integers))
+
+                            if not q_values.is_empty:
+                                rep[p] = i
+                                rep[q] = next(iter(q_values))
+                                break
+
+                    if len(syms) != 0:
+                        x, y, z = tuple(s.subs(rep) for s in sol)
+                    else:
+                        x, y, z =   sol
+                    flag = False
+                    break
+
+            if flag:
+                raise ValueError("Rational Point on the conic does not exist")
+
+            x = (x*g3)/r1
+            y = (y*g2)/r2
+            z = (z*g1)/r3
+            x = x/z
+            y = y/z
+
+            if a == 0 and c == 0:
+                x_reg = (x + y - 2*e)/(2*b)
+                y_reg = (x - y - 2*d)/(2*b)
+            elif c != 0:
+                x_reg = (x - 2*d*c + b*e)/K
+                y_reg = (y - b*x_reg - e)/(2*c)
+            else:
+                y_reg = (x - 2*e*a + b*d)/K
+                x_reg = (y - b*y_reg - d)/(2*a)
+
+            return x_reg, y_reg
+
+    def singular_points(self):
+        """
+        Returns a set of singular points of the region.
+
+        The singular points are those points on the region
+        where all partial derivatives vanish.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x, y
+        >>> from sympy.vector import ImplicitRegion
+        >>> I = ImplicitRegion((x, y), (y-1)**2 -x**3 + 2*x**2 -x)
+        >>> I.singular_points()
+        {(1, 1)}
+
+        """
+        eq_list = [self.equation]
+        for var in self.variables:
+            eq_list += [diff(self.equation, var)]
+
+        return nonlinsolve(eq_list, list(self.variables))
+
+    def multiplicity(self, point):
+        """
+        Returns the multiplicity of a singular point on the region.
+
+        A singular point (x,y) of region is said to be of multiplicity m
+        if all the partial derivatives off to order m - 1 vanish there.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x, y, z
+        >>> from sympy.vector import ImplicitRegion
+        >>> I = ImplicitRegion((x, y, z), x**2 + y**3 - z**4)
+        >>> I.singular_points()
+        {(0, 0, 0)}
+        >>> I.multiplicity((0, 0, 0))
+        2
+
+        """
+        if isinstance(point, Point):
+            point = point.args
+
+        modified_eq = self.equation
+
+        for i, var in enumerate(self.variables):
+            modified_eq = modified_eq.subs(var, var + point[i])
+        modified_eq = expand(modified_eq)
+
+        if len(modified_eq.args) != 0:
+            terms = modified_eq.args
+            m = min(total_degree(term) for term in terms)
+        else:
+            terms = modified_eq
+            m = total_degree(terms)
+
+        return m
+
+    def rational_parametrization(self, parameters=('t', 's'), reg_point=None):
+        """
+        Returns the rational parametrization of implicit region.
+
+        Examples
+        ========
+
+        >>> from sympy import Eq
+        >>> from sympy.abc import x, y, z, s, t
+        >>> from sympy.vector import ImplicitRegion
+
+        >>> parabola = ImplicitRegion((x, y), y**2 - 4*x)
+        >>> parabola.rational_parametrization()
+        (4/t**2, 4/t)
+
+        >>> circle = ImplicitRegion((x, y), Eq(x**2 + y**2, 4))
+        >>> circle.rational_parametrization()
+        (4*t/(t**2 + 1), 4*t**2/(t**2 + 1) - 2)
+
+        >>> I = ImplicitRegion((x, y), x**3 + x**2 - y**2)
+        >>> I.rational_parametrization()
+        (t**2 - 1, t*(t**2 - 1))
+
+        >>> cubic_curve = ImplicitRegion((x, y), x**3 + x**2 - y**2)
+        >>> cubic_curve.rational_parametrization(parameters=(t))
+        (t**2 - 1, t*(t**2 - 1))
+
+        >>> sphere = ImplicitRegion((x, y, z), x**2 + y**2 + z**2 - 4)
+        >>> sphere.rational_parametrization(parameters=(t, s))
+        (-2 + 4/(s**2 + t**2 + 1), 4*s/(s**2 + t**2 + 1), 4*t/(s**2 + t**2 + 1))
+
+        For some conics, regular_points() is unable to find a point on curve.
+        To calulcate the parametric representation in such cases, user need
+        to determine a point on the region and pass it using reg_point.
+
+        >>> c = ImplicitRegion((x, y), (x  - 1/2)**2 + (y)**2 - (1/4)**2)
+        >>> c.rational_parametrization(reg_point=(3/4, 0))
+        (0.75 - 0.5/(t**2 + 1), -0.5*t/(t**2 + 1))
+
+        References
+        ==========
+
+        - Christoph M. Hoffmann, "Conversion Methods between Parametric and
+          Implicit Curves and Surfaces", Purdue e-Pubs, 1990. Available:
+          https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1827&context=cstech
+
+        """
+        equation = self.equation
+        degree = self.degree
+
+        if degree == 1:
+            if len(self.variables) == 1:
+                return (equation,)
+            elif len(self.variables) == 2:
+                x, y = self.variables
+                y_par = list(solveset(equation, y))[0]
+                return x, y_par
+            else:
+                raise NotImplementedError()
+
+        point = ()
+
+        # Finding the (n - 1) fold point of the monoid of degree
+        if degree == 2:
+            # For degree 2 curves, either a regular point or a singular point can be used.
+            if reg_point is not None:
+                # Using point provided by the user as regular point
+                point = reg_point
+            else:
+                if len(self.singular_points()) != 0:
+                    point = list(self.singular_points())[0]
+                else:
+                    point = self.regular_point()
+
+        if len(self.singular_points()) != 0:
+            singular_points = self.singular_points()
+            for spoint in singular_points:
+                syms = Tuple(*spoint).free_symbols
+                rep = dict.fromkeys(syms, 2)
+
+                if len(syms) != 0:
+                   spoint = tuple(s.subs(rep) for s in spoint)
+
+                if self.multiplicity(spoint) == degree - 1:
+                    point = spoint
+                    break
+
+        if len(point) == 0:
+            # The region in not a monoid
+            raise NotImplementedError()
+
+        modified_eq = equation
+
+        # Shifting the region such that fold point moves to origin
+        for i, var in enumerate(self.variables):
+            modified_eq = modified_eq.subs(var, var + point[i])
+        modified_eq = expand(modified_eq)
+
+        hn = hn_1 = 0
+        for term in modified_eq.args:
+            if total_degree(term) == degree:
+                hn += term
+            else:
+                hn_1 += term
+
+        hn_1 = -1*hn_1
+
+        if not isinstance(parameters, tuple):
+            parameters = (parameters,)
+
+        if len(self.variables) == 2:
+
+            parameter1 = parameters[0]
+            if parameter1 == 's':
+                # To avoid name conflict between parameters
+                s = _symbol('s_', real=True)
+            else:
+                s = _symbol('s', real=True)
+            t = _symbol(parameter1, real=True)
+
+            hn = hn.subs({self.variables[0]: s, self.variables[1]: t})
+            hn_1 = hn_1.subs({self.variables[0]: s, self.variables[1]: t})
+
+            x_par = (s*(hn_1/hn)).subs(s, 1) + point[0]
+            y_par = (t*(hn_1/hn)).subs(s, 1) + point[1]
+
+            return x_par, y_par
+
+        elif len(self.variables) == 3:
+
+            parameter1, parameter2 = parameters
+            if 'r' in parameters:
+                # To avoid name conflict between parameters
+                r = _symbol('r_', real=True)
+            else:
+                r = _symbol('r', real=True)
+            s = _symbol(parameter2, real=True)
+            t = _symbol(parameter1, real=True)
+
+            hn = hn.subs({self.variables[0]: r, self.variables[1]: s, self.variables[2]: t})
+            hn_1 = hn_1.subs({self.variables[0]: r, self.variables[1]: s, self.variables[2]: t})
+
+            x_par = (r*(hn_1/hn)).subs(r, 1) + point[0]
+            y_par = (s*(hn_1/hn)).subs(r, 1) + point[1]
+            z_par = (t*(hn_1/hn)).subs(r, 1) + point[2]
+
+            return x_par, y_par, z_par
+
+        raise NotImplementedError()
+
+def conic_coeff(variables, equation):
+    if total_degree(equation) != 2:
+        raise ValueError()
+    x = variables[0]
+    y = variables[1]
+
+    equation = expand(equation)
+    a = equation.coeff(x**2)
+    b = equation.coeff(x*y)
+    c = equation.coeff(y**2)
+    d = equation.coeff(x, 1).coeff(y, 0)
+    e = equation.coeff(y, 1).coeff(x, 0)
+    f = equation.coeff(x, 0).coeff(y, 0)
+    return a, b, c, d, e, f

rl/Lib/site-packages/sympy/vector/integrals.py

@@ -0,0 +1,206 @@
+from sympy.core import Basic, diff
+from sympy.core.singleton import S
+from sympy.core.sorting import default_sort_key
+from sympy.matrices import Matrix
+from sympy.integrals import Integral, integrate
+from sympy.geometry.entity import GeometryEntity
+from sympy.simplify.simplify import simplify
+from sympy.utilities.iterables import topological_sort
+from sympy.vector import (CoordSys3D, Vector, ParametricRegion,
+                        parametric_region_list, ImplicitRegion)
+from sympy.vector.operators import _get_coord_systems
+
+
+class ParametricIntegral(Basic):
+    """
+    Represents integral of a scalar or vector field
+    over a Parametric Region
+
+    Examples
+    ========
+
+    >>> from sympy import cos, sin, pi
+    >>> from sympy.vector import CoordSys3D, ParametricRegion, ParametricIntegral
+    >>> from sympy.abc import r, t, theta, phi
+
+    >>> C = CoordSys3D('C')
+    >>> curve = ParametricRegion((3*t - 2, t + 1), (t, 1, 2))
+    >>> ParametricIntegral(C.x, curve)
+    5*sqrt(10)/2
+    >>> length = ParametricIntegral(1, curve)
+    >>> length
+    sqrt(10)
+    >>> semisphere = ParametricRegion((2*sin(phi)*cos(theta), 2*sin(phi)*sin(theta), 2*cos(phi)),\
+                            (theta, 0, 2*pi), (phi, 0, pi/2))
+    >>> ParametricIntegral(C.z, semisphere)
+    8*pi
+
+    >>> ParametricIntegral(C.j + C.k, ParametricRegion((r*cos(theta), r*sin(theta)), r, theta))
+    0
+
+    """
+
+    def __new__(cls, field, parametricregion):
+
+        coord_set = _get_coord_systems(field)
+
+        if len(coord_set) == 0:
+            coord_sys = CoordSys3D('C')
+        elif len(coord_set) > 1:
+            raise ValueError
+        else:
+            coord_sys = next(iter(coord_set))
+
+        if parametricregion.dimensions == 0:
+            return S.Zero
+
+        base_vectors = coord_sys.base_vectors()
+        base_scalars = coord_sys.base_scalars()
+
+        parametricfield = field
+
+        r = Vector.zero
+        for i in range(len(parametricregion.definition)):
+            r += base_vectors[i]*parametricregion.definition[i]
+
+        if len(coord_set) != 0:
+            for i in range(len(parametricregion.definition)):
+                parametricfield = parametricfield.subs(base_scalars[i], parametricregion.definition[i])
+
+        if parametricregion.dimensions == 1:
+            parameter = parametricregion.parameters[0]
+
+            r_diff = diff(r, parameter)
+            lower, upper = parametricregion.limits[parameter][0], parametricregion.limits[parameter][1]
+
+            if isinstance(parametricfield, Vector):
+                integrand = simplify(r_diff.dot(parametricfield))
+            else:
+                integrand = simplify(r_diff.magnitude()*parametricfield)
+
+            result = integrate(integrand, (parameter, lower, upper))
+
+        elif parametricregion.dimensions == 2:
+            u, v = cls._bounds_case(parametricregion.parameters, parametricregion.limits)
+
+            r_u = diff(r, u)
+            r_v = diff(r, v)
+            normal_vector = simplify(r_u.cross(r_v))
+
+            if isinstance(parametricfield, Vector):
+                integrand = parametricfield.dot(normal_vector)
+            else:
+                integrand = parametricfield*normal_vector.magnitude()
+
+            integrand = simplify(integrand)
+
+            lower_u, upper_u = parametricregion.limits[u][0], parametricregion.limits[u][1]
+            lower_v, upper_v = parametricregion.limits[v][0], parametricregion.limits[v][1]
+
+            result = integrate(integrand, (u, lower_u, upper_u), (v, lower_v, upper_v))
+
+        else:
+            variables = cls._bounds_case(parametricregion.parameters, parametricregion.limits)
+            coeff = Matrix(parametricregion.definition).jacobian(variables).det()
+            integrand = simplify(parametricfield*coeff)
+
+            l = [(var, parametricregion.limits[var][0], parametricregion.limits[var][1]) for var in variables]
+            result = integrate(integrand, *l)
+
+        if not isinstance(result, Integral):
+            return result
+        else:
+            return super().__new__(cls, field, parametricregion)
+
+    @classmethod
+    def _bounds_case(cls, parameters, limits):
+
+        V = list(limits.keys())
+        E = []
+
+        for p in V:
+            lower_p = limits[p][0]
+            upper_p = limits[p][1]
+
+            lower_p = lower_p.atoms()
+            upper_p = upper_p.atoms()
+            E.extend((p, q) for q in V if p != q and
+                     (lower_p.issuperset({q}) or upper_p.issuperset({q})))
+
+        if not E:
+            return parameters
+        else:
+            return topological_sort((V, E), key=default_sort_key)
+
+    @property
+    def field(self):
+        return self.args[0]
+
+    @property
+    def parametricregion(self):
+        return self.args[1]
+
+
+def vector_integrate(field, *region):
+    """
+    Compute the integral of a vector/scalar field
+    over a a region or a set of parameters.
+
+    Examples
+    ========
+    >>> from sympy.vector import CoordSys3D, ParametricRegion, vector_integrate
+    >>> from sympy.abc import x, y, t
+    >>> C = CoordSys3D('C')
+
+    >>> region = ParametricRegion((t, t**2), (t, 1, 5))
+    >>> vector_integrate(C.x*C.i, region)
+    12
+
+    Integrals over some objects of geometry module can also be calculated.
+
+    >>> from sympy.geometry import Point, Circle, Triangle
+    >>> c = Circle(Point(0, 2), 5)
+    >>> vector_integrate(C.x**2 + C.y**2, c)
+    290*pi
+    >>> triangle = Triangle(Point(-2, 3), Point(2, 3), Point(0, 5))
+    >>> vector_integrate(3*C.x**2*C.y*C.i + C.j, triangle)
+    -8
+
+    Integrals over some simple implicit regions can be computed. But in most cases,
+    it takes too long to compute over them. This is due to the expressions of parametric
+    representation becoming large.
+
+    >>> from sympy.vector import ImplicitRegion
+    >>> c2 = ImplicitRegion((x, y), (x - 2)**2 + (y - 1)**2 - 9)
+    >>> vector_integrate(1, c2)
+    6*pi
+
+    Integral of fields with respect to base scalars:
+
+    >>> vector_integrate(12*C.y**3, (C.y, 1, 3))
+    240
+    >>> vector_integrate(C.x**2*C.z, C.x)
+    C.x**3*C.z/3
+    >>> vector_integrate(C.x*C.i - C.y*C.k, C.x)
+    (Integral(C.x, C.x))*C.i + (Integral(-C.y, C.x))*C.k
+    >>> _.doit()
+    C.x**2/2*C.i + (-C.x*C.y)*C.k
+
+    """
+    if len(region) == 1:
+        if isinstance(region[0], ParametricRegion):
+            return ParametricIntegral(field, region[0])
+
+        if isinstance(region[0], ImplicitRegion):
+            region = parametric_region_list(region[0])[0]
+            return vector_integrate(field, region)
+
+        if isinstance(region[0], GeometryEntity):
+            regions_list = parametric_region_list(region[0])
+
+            result = 0
+            for reg in regions_list:
+                result += vector_integrate(field, reg)
+            return result
+
+    return integrate(field, *region)

rl/Lib/site-packages/sympy/vector/operators.py

@@ -0,0 +1,335 @@
+import collections
+from sympy.core.expr import Expr
+from sympy.core import sympify, S, preorder_traversal
+from sympy.vector.coordsysrect import CoordSys3D
+from sympy.vector.vector import Vector, VectorMul, VectorAdd, Cross, Dot
+from sympy.core.function import Derivative
+from sympy.core.add import Add
+from sympy.core.mul import Mul
+
+
+def _get_coord_systems(expr):
+    g = preorder_traversal(expr)
+    ret = set()
+    for i in g:
+        if isinstance(i, CoordSys3D):
+            ret.add(i)
+            g.skip()
+    return frozenset(ret)
+
+
+def _split_mul_args_wrt_coordsys(expr):
+    d = collections.defaultdict(lambda: S.One)
+    for i in expr.args:
+        d[_get_coord_systems(i)] *= i
+    return list(d.values())
+
+
+class Gradient(Expr):
+    """
+    Represents unevaluated Gradient.
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, Gradient
+    >>> R = CoordSys3D('R')
+    >>> s = R.x*R.y*R.z
+    >>> Gradient(s)
+    Gradient(R.x*R.y*R.z)
+
+    """
+
+    def __new__(cls, expr):
+        expr = sympify(expr)
+        obj = Expr.__new__(cls, expr)
+        obj._expr = expr
+        return obj
+
+    def doit(self, **hints):
+        return gradient(self._expr, doit=True)
+
+
+class Divergence(Expr):
+    """
+    Represents unevaluated Divergence.
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, Divergence
+    >>> R = CoordSys3D('R')
+    >>> v = R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k
+    >>> Divergence(v)
+    Divergence(R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k)
+
+    """
+
+    def __new__(cls, expr):
+        expr = sympify(expr)
+        obj = Expr.__new__(cls, expr)
+        obj._expr = expr
+        return obj
+
+    def doit(self, **hints):
+        return divergence(self._expr, doit=True)
+
+
+class Curl(Expr):
+    """
+    Represents unevaluated Curl.
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, Curl
+    >>> R = CoordSys3D('R')
+    >>> v = R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k
+    >>> Curl(v)
+    Curl(R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k)
+
+    """
+
+    def __new__(cls, expr):
+        expr = sympify(expr)
+        obj = Expr.__new__(cls, expr)
+        obj._expr = expr
+        return obj
+
+    def doit(self, **hints):
+        return curl(self._expr, doit=True)
+
+
+def curl(vect, doit=True):
+    """
+    Returns the curl of a vector field computed wrt the base scalars
+    of the given coordinate system.
+
+    Parameters
+    ==========
+
+    vect : Vector
+        The vector operand
+
+    doit : bool
+        If True, the result is returned after calling .doit() on
+        each component. Else, the returned expression contains
+        Derivative instances
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, curl
+    >>> R = CoordSys3D('R')
+    >>> v1 = R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k
+    >>> curl(v1)
+    0
+    >>> v2 = R.x*R.y*R.z*R.i
+    >>> curl(v2)
+    R.x*R.y*R.j + (-R.x*R.z)*R.k
+
+    """
+
+    coord_sys = _get_coord_systems(vect)
+
+    if len(coord_sys) == 0:
+        return Vector.zero
+    elif len(coord_sys) == 1:
+        coord_sys = next(iter(coord_sys))
+        i, j, k = coord_sys.base_vectors()
+        x, y, z = coord_sys.base_scalars()
+        h1, h2, h3 = coord_sys.lame_coefficients()
+        vectx = vect.dot(i)
+        vecty = vect.dot(j)
+        vectz = vect.dot(k)
+        outvec = Vector.zero
+        outvec += (Derivative(vectz * h3, y) -
+                   Derivative(vecty * h2, z)) * i / (h2 * h3)
+        outvec += (Derivative(vectx * h1, z) -
+                   Derivative(vectz * h3, x)) * j / (h1 * h3)
+        outvec += (Derivative(vecty * h2, x) -
+                   Derivative(vectx * h1, y)) * k / (h2 * h1)
+
+        if doit:
+            return outvec.doit()
+        return outvec
+    else:
+        if isinstance(vect, (Add, VectorAdd)):
+            from sympy.vector import express
+            try:
+                cs = next(iter(coord_sys))
+                args = [express(i, cs, variables=True) for i in vect.args]
+            except ValueError:
+                args = vect.args
+            return VectorAdd.fromiter(curl(i, doit=doit) for i in args)
+        elif isinstance(vect, (Mul, VectorMul)):
+            vector = [i for i in vect.args if isinstance(i, (Vector, Cross, Gradient))][0]
+            scalar = Mul.fromiter(i for i in vect.args if not isinstance(i, (Vector, Cross, Gradient)))
+            res = Cross(gradient(scalar), vector).doit() + scalar*curl(vector, doit=doit)
+            if doit:
+                return res.doit()
+            return res
+        elif isinstance(vect, (Cross, Curl, Gradient)):
+            return Curl(vect)
+        else:
+            raise Curl(vect)
+
+
+def divergence(vect, doit=True):
+    """
+    Returns the divergence of a vector field computed wrt the base
+    scalars of the given coordinate system.
+
+    Parameters
+    ==========
+
+    vector : Vector
+        The vector operand
+
+    doit : bool
+        If True, the result is returned after calling .doit() on
+        each component. Else, the returned expression contains
+        Derivative instances
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, divergence
+    >>> R = CoordSys3D('R')
+    >>> v1 = R.x*R.y*R.z * (R.i+R.j+R.k)
+
+    >>> divergence(v1)
+    R.x*R.y + R.x*R.z + R.y*R.z
+    >>> v2 = 2*R.y*R.z*R.j
+    >>> divergence(v2)
+    2*R.z
+
+    """
+    coord_sys = _get_coord_systems(vect)
+    if len(coord_sys) == 0:
+        return S.Zero
+    elif len(coord_sys) == 1:
+        if isinstance(vect, (Cross, Curl, Gradient)):
+            return Divergence(vect)
+        # TODO: is case of many coord systems, this gets a random one:
+        coord_sys = next(iter(coord_sys))
+        i, j, k = coord_sys.base_vectors()
+        x, y, z = coord_sys.base_scalars()
+        h1, h2, h3 = coord_sys.lame_coefficients()
+        vx = _diff_conditional(vect.dot(i), x, h2, h3) \
+             / (h1 * h2 * h3)
+        vy = _diff_conditional(vect.dot(j), y, h3, h1) \
+             / (h1 * h2 * h3)
+        vz = _diff_conditional(vect.dot(k), z, h1, h2) \
+             / (h1 * h2 * h3)
+        res = vx + vy + vz
+        if doit:
+            return res.doit()
+        return res
+    else:
+        if isinstance(vect, (Add, VectorAdd)):
+            return Add.fromiter(divergence(i, doit=doit) for i in vect.args)
+        elif isinstance(vect, (Mul, VectorMul)):
+            vector = [i for i in vect.args if isinstance(i, (Vector, Cross, Gradient))][0]
+            scalar = Mul.fromiter(i for i in vect.args if not isinstance(i, (Vector, Cross, Gradient)))
+            res = Dot(vector, gradient(scalar)) + scalar*divergence(vector, doit=doit)
+            if doit:
+                return res.doit()
+            return res
+        elif isinstance(vect, (Cross, Curl, Gradient)):
+            return Divergence(vect)
+        else:
+            raise Divergence(vect)
+
+
+def gradient(scalar_field, doit=True):
+    """
+    Returns the vector gradient of a scalar field computed wrt the
+    base scalars of the given coordinate system.
+
+    Parameters
+    ==========
+
+    scalar_field : SymPy Expr
+        The scalar field to compute the gradient of
+
+    doit : bool
+        If True, the result is returned after calling .doit() on
+        each component. Else, the returned expression contains
+        Derivative instances
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, gradient
+    >>> R = CoordSys3D('R')
+    >>> s1 = R.x*R.y*R.z
+    >>> gradient(s1)
+    R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k
+    >>> s2 = 5*R.x**2*R.z
+    >>> gradient(s2)
+    10*R.x*R.z*R.i + 5*R.x**2*R.k
+
+    """
+    coord_sys = _get_coord_systems(scalar_field)
+
+    if len(coord_sys) == 0:
+        return Vector.zero
+    elif len(coord_sys) == 1:
+        coord_sys = next(iter(coord_sys))
+        h1, h2, h3 = coord_sys.lame_coefficients()
+        i, j, k = coord_sys.base_vectors()
+        x, y, z = coord_sys.base_scalars()
+        vx = Derivative(scalar_field, x) / h1
+        vy = Derivative(scalar_field, y) / h2
+        vz = Derivative(scalar_field, z) / h3
+
+        if doit:
+            return (vx * i + vy * j + vz * k).doit()
+        return vx * i + vy * j + vz * k
+    else:
+        if isinstance(scalar_field, (Add, VectorAdd)):
+            return VectorAdd.fromiter(gradient(i) for i in scalar_field.args)
+        if isinstance(scalar_field, (Mul, VectorMul)):
+            s = _split_mul_args_wrt_coordsys(scalar_field)
+            return VectorAdd.fromiter(scalar_field / i * gradient(i) for i in s)
+        return Gradient(scalar_field)
+
+
+class Laplacian(Expr):
+    """
+    Represents unevaluated Laplacian.
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, Laplacian
+    >>> R = CoordSys3D('R')
+    >>> v = 3*R.x**3*R.y**2*R.z**3
+    >>> Laplacian(v)
+    Laplacian(3*R.x**3*R.y**2*R.z**3)
+
+    """
+
+    def __new__(cls, expr):
+        expr = sympify(expr)
+        obj = Expr.__new__(cls, expr)
+        obj._expr = expr
+        return obj
+
+    def doit(self, **hints):
+        from sympy.vector.functions import laplacian
+        return laplacian(self._expr)
+
+
+def _diff_conditional(expr, base_scalar, coeff_1, coeff_2):
+    """
+    First re-expresses expr in the system that base_scalar belongs to.
+    If base_scalar appears in the re-expressed form, differentiates
+    it wrt base_scalar.
+    Else, returns 0
+    """
+    from sympy.vector.functions import express
+    new_expr = express(expr, base_scalar.system, variables=True)
+    arg = coeff_1 * coeff_2 * new_expr
+    return Derivative(arg, base_scalar) if arg else S.Zero

rl/Lib/site-packages/sympy/vector/orienters.py

@@ -0,0 +1,398 @@
+from sympy.core.basic import Basic
+from sympy.core.sympify import sympify
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.matrices.dense import (eye, rot_axis1, rot_axis2, rot_axis3)
+from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix
+from sympy.core.cache import cacheit
+from sympy.core.symbol import Str
+import sympy.vector
+
+
+class Orienter(Basic):
+    """
+    Super-class for all orienter classes.
+    """
+
+    def rotation_matrix(self):
+        """
+        The rotation matrix corresponding to this orienter
+        instance.
+        """
+        return self._parent_orient
+
+
+class AxisOrienter(Orienter):
+    """
+    Class to denote an axis orienter.
+    """
+
+    def __new__(cls, angle, axis):
+        if not isinstance(axis, sympy.vector.Vector):
+            raise TypeError("axis should be a Vector")
+        angle = sympify(angle)
+
+        obj = super().__new__(cls, angle, axis)
+        obj._angle = angle
+        obj._axis = axis
+
+        return obj
+
+    def __init__(self, angle, axis):
+        """
+        Axis rotation is a rotation about an arbitrary axis by
+        some angle. The angle is supplied as a SymPy expr scalar, and
+        the axis is supplied as a Vector.
+
+        Parameters
+        ==========
+
+        angle : Expr
+            The angle by which the new system is to be rotated
+
+        axis : Vector
+            The axis around which the rotation has to be performed
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D
+        >>> from sympy import symbols
+        >>> q1 = symbols('q1')
+        >>> N = CoordSys3D('N')
+        >>> from sympy.vector import AxisOrienter
+        >>> orienter = AxisOrienter(q1, N.i + 2 * N.j)
+        >>> B = N.orient_new('B', (orienter, ))
+
+        """
+        # Dummy initializer for docstrings
+        pass
+
+    @cacheit
+    def rotation_matrix(self, system):
+        """
+        The rotation matrix corresponding to this orienter
+        instance.
+
+        Parameters
+        ==========
+
+        system : CoordSys3D
+            The coordinate system wrt which the rotation matrix
+            is to be computed
+        """
+
+        axis = sympy.vector.express(self.axis, system).normalize()
+        axis = axis.to_matrix(system)
+        theta = self.angle
+        parent_orient = ((eye(3) - axis * axis.T) * cos(theta) +
+                         Matrix([[0, -axis[2], axis[1]],
+                                 [axis[2], 0, -axis[0]],
+                                 [-axis[1], axis[0], 0]]) * sin(theta) +
+                         axis * axis.T)
+        parent_orient = parent_orient.T
+        return parent_orient
+
+    @property
+    def angle(self):
+        return self._angle
+
+    @property
+    def axis(self):
+        return self._axis
+
+
+class ThreeAngleOrienter(Orienter):
+    """
+    Super-class for Body and Space orienters.
+    """
+
+    def __new__(cls, angle1, angle2, angle3, rot_order):
+        if isinstance(rot_order, Str):
+            rot_order = rot_order.name
+
+        approved_orders = ('123', '231', '312', '132', '213',
+                           '321', '121', '131', '212', '232',
+                           '313', '323', '')
+        original_rot_order = rot_order
+        rot_order = str(rot_order).upper()
+        if not (len(rot_order) == 3):
+            raise TypeError('rot_order should be a str of length 3')
+        rot_order = [i.replace('X', '1') for i in rot_order]
+        rot_order = [i.replace('Y', '2') for i in rot_order]
+        rot_order = [i.replace('Z', '3') for i in rot_order]
+        rot_order = ''.join(rot_order)
+        if rot_order not in approved_orders:
+            raise TypeError('Invalid rot_type parameter')
+        a1 = int(rot_order[0])
+        a2 = int(rot_order[1])
+        a3 = int(rot_order[2])
+        angle1 = sympify(angle1)
+        angle2 = sympify(angle2)
+        angle3 = sympify(angle3)
+        if cls._in_order:
+            parent_orient = (_rot(a1, angle1) *
+                             _rot(a2, angle2) *
+                             _rot(a3, angle3))
+        else:
+            parent_orient = (_rot(a3, angle3) *
+                             _rot(a2, angle2) *
+                             _rot(a1, angle1))
+        parent_orient = parent_orient.T
+
+        obj = super().__new__(
+            cls, angle1, angle2, angle3, Str(rot_order))
+        obj._angle1 = angle1
+        obj._angle2 = angle2
+        obj._angle3 = angle3
+        obj._rot_order = original_rot_order
+        obj._parent_orient = parent_orient
+
+        return obj
+
+    @property
+    def angle1(self):
+        return self._angle1
+
+    @property
+    def angle2(self):
+        return self._angle2
+
+    @property
+    def angle3(self):
+        return self._angle3
+
+    @property
+    def rot_order(self):
+        return self._rot_order
+
+
+class BodyOrienter(ThreeAngleOrienter):
+    """
+    Class to denote a body-orienter.
+    """
+
+    _in_order = True
+
+    def __new__(cls, angle1, angle2, angle3, rot_order):
+        obj = ThreeAngleOrienter.__new__(cls, angle1, angle2, angle3,
+                                         rot_order)
+        return obj
+
+    def __init__(self, angle1, angle2, angle3, rot_order):
+        """
+        Body orientation takes this coordinate system through three
+        successive simple rotations.
+
+        Body fixed rotations include both Euler Angles and
+        Tait-Bryan Angles, see https://en.wikipedia.org/wiki/Euler_angles.
+
+        Parameters
+        ==========
+
+        angle1, angle2, angle3 : Expr
+            Three successive angles to rotate the coordinate system by
+
+        rotation_order : string
+            String defining the order of axes for rotation
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D, BodyOrienter
+        >>> from sympy import symbols
+        >>> q1, q2, q3 = symbols('q1 q2 q3')
+        >>> N = CoordSys3D('N')
+
+        A 'Body' fixed rotation is described by three angles and
+        three body-fixed rotation axes. To orient a coordinate system D
+        with respect to N, each sequential rotation is always about
+        the orthogonal unit vectors fixed to D. For example, a '123'
+        rotation will specify rotations about N.i, then D.j, then
+        D.k. (Initially, D.i is same as N.i)
+        Therefore,
+
+        >>> body_orienter = BodyOrienter(q1, q2, q3, '123')
+        >>> D = N.orient_new('D', (body_orienter, ))
+
+        is same as
+
+        >>> from sympy.vector import AxisOrienter
+        >>> axis_orienter1 = AxisOrienter(q1, N.i)
+        >>> D = N.orient_new('D', (axis_orienter1, ))
+        >>> axis_orienter2 = AxisOrienter(q2, D.j)
+        >>> D = D.orient_new('D', (axis_orienter2, ))
+        >>> axis_orienter3 = AxisOrienter(q3, D.k)
+        >>> D = D.orient_new('D', (axis_orienter3, ))
+
+        Acceptable rotation orders are of length 3, expressed in XYZ or
+        123, and cannot have a rotation about about an axis twice in a row.
+
+        >>> body_orienter1 = BodyOrienter(q1, q2, q3, '123')
+        >>> body_orienter2 = BodyOrienter(q1, q2, 0, 'ZXZ')
+        >>> body_orienter3 = BodyOrienter(0, 0, 0, 'XYX')
+
+        """
+        # Dummy initializer for docstrings
+        pass
+
+
+class SpaceOrienter(ThreeAngleOrienter):
+    """
+    Class to denote a space-orienter.
+    """
+
+    _in_order = False
+
+    def __new__(cls, angle1, angle2, angle3, rot_order):
+        obj = ThreeAngleOrienter.__new__(cls, angle1, angle2, angle3,
+                                         rot_order)
+        return obj
+
+    def __init__(self, angle1, angle2, angle3, rot_order):
+        """
+        Space rotation is similar to Body rotation, but the rotations
+        are applied in the opposite order.
+
+        Parameters
+        ==========
+
+        angle1, angle2, angle3 : Expr
+            Three successive angles to rotate the coordinate system by
+
+        rotation_order : string
+            String defining the order of axes for rotation
+
+        See Also
+        ========
+
+        BodyOrienter : Orienter to orient systems wrt Euler angles.
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D, SpaceOrienter
+        >>> from sympy import symbols
+        >>> q1, q2, q3 = symbols('q1 q2 q3')
+        >>> N = CoordSys3D('N')
+
+        To orient a coordinate system D with respect to N, each
+        sequential rotation is always about N's orthogonal unit vectors.
+        For example, a '123' rotation will specify rotations about
+        N.i, then N.j, then N.k.
+        Therefore,
+
+        >>> space_orienter = SpaceOrienter(q1, q2, q3, '312')
+        >>> D = N.orient_new('D', (space_orienter, ))
+
+        is same as
+
+        >>> from sympy.vector import AxisOrienter
+        >>> axis_orienter1 = AxisOrienter(q1, N.i)
+        >>> B = N.orient_new('B', (axis_orienter1, ))
+        >>> axis_orienter2 = AxisOrienter(q2, N.j)
+        >>> C = B.orient_new('C', (axis_orienter2, ))
+        >>> axis_orienter3 = AxisOrienter(q3, N.k)
+        >>> D = C.orient_new('C', (axis_orienter3, ))
+
+        """
+        # Dummy initializer for docstrings
+        pass
+
+
+class QuaternionOrienter(Orienter):
+    """
+    Class to denote a quaternion-orienter.
+    """
+
+    def __new__(cls, q0, q1, q2, q3):
+        q0 = sympify(q0)
+        q1 = sympify(q1)
+        q2 = sympify(q2)
+        q3 = sympify(q3)
+        parent_orient = (Matrix([[q0 ** 2 + q1 ** 2 - q2 ** 2 -
+                                  q3 ** 2,
+                                  2 * (q1 * q2 - q0 * q3),
+                                  2 * (q0 * q2 + q1 * q3)],
+                                 [2 * (q1 * q2 + q0 * q3),
+                                  q0 ** 2 - q1 ** 2 +
+                                  q2 ** 2 - q3 ** 2,
+                                  2 * (q2 * q3 - q0 * q1)],
+                                 [2 * (q1 * q3 - q0 * q2),
+                                  2 * (q0 * q1 + q2 * q3),
+                                  q0 ** 2 - q1 ** 2 -
+                                  q2 ** 2 + q3 ** 2]]))
+        parent_orient = parent_orient.T
+
+        obj = super().__new__(cls, q0, q1, q2, q3)
+        obj._q0 = q0
+        obj._q1 = q1
+        obj._q2 = q2
+        obj._q3 = q3
+        obj._parent_orient = parent_orient
+
+        return obj
+
+    def __init__(self, angle1, angle2, angle3, rot_order):
+        """
+        Quaternion orientation orients the new CoordSys3D with
+        Quaternions, defined as a finite rotation about lambda, a unit
+        vector, by some amount theta.
+
+        This orientation is described by four parameters:
+
+        q0 = cos(theta/2)
+
+        q1 = lambda_x sin(theta/2)
+
+        q2 = lambda_y sin(theta/2)
+
+        q3 = lambda_z sin(theta/2)
+
+        Quaternion does not take in a rotation order.
+
+        Parameters
+        ==========
+
+        q0, q1, q2, q3 : Expr
+            The quaternions to rotate the coordinate system by
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D
+        >>> from sympy import symbols
+        >>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3')
+        >>> N = CoordSys3D('N')
+        >>> from sympy.vector import QuaternionOrienter
+        >>> q_orienter = QuaternionOrienter(q0, q1, q2, q3)
+        >>> B = N.orient_new('B', (q_orienter, ))
+
+        """
+        # Dummy initializer for docstrings
+        pass
+
+    @property
+    def q0(self):
+        return self._q0
+
+    @property
+    def q1(self):
+        return self._q1
+
+    @property
+    def q2(self):
+        return self._q2
+
+    @property
+    def q3(self):
+        return self._q3
+
+
+def _rot(axis, angle):
+    """DCM for simple axis 1, 2 or 3 rotations. """
+    if axis == 1:
+        return Matrix(rot_axis1(angle).T)
+    elif axis == 2:
+        return Matrix(rot_axis2(angle).T)
+    elif axis == 3:
+        return Matrix(rot_axis3(angle).T)

rl/Lib/site-packages/sympy/vector/parametricregion.py

@@ -0,0 +1,189 @@
+from functools import singledispatch
+from sympy.core.numbers import pi
+from sympy.functions.elementary.trigonometric import tan
+from sympy.simplify import trigsimp
+from sympy.core import Basic, Tuple
+from sympy.core.symbol import _symbol
+from sympy.solvers import solve
+from sympy.geometry import Point, Segment, Curve, Ellipse, Polygon
+from sympy.vector import ImplicitRegion
+
+
+class ParametricRegion(Basic):
+    """
+    Represents a parametric region in space.
+
+    Examples
+    ========
+
+    >>> from sympy import cos, sin, pi
+    >>> from sympy.abc import r, theta, t, a, b, x, y
+    >>> from sympy.vector import ParametricRegion
+
+    >>> ParametricRegion((t, t**2), (t, -1, 2))
+    ParametricRegion((t, t**2), (t, -1, 2))
+    >>> ParametricRegion((x, y), (x, 3, 4), (y, 5, 6))
+    ParametricRegion((x, y), (x, 3, 4), (y, 5, 6))
+    >>> ParametricRegion((r*cos(theta), r*sin(theta)), (r, -2, 2), (theta, 0, pi))
+    ParametricRegion((r*cos(theta), r*sin(theta)), (r, -2, 2), (theta, 0, pi))
+    >>> ParametricRegion((a*cos(t), b*sin(t)), t)
+    ParametricRegion((a*cos(t), b*sin(t)), t)
+
+    >>> circle = ParametricRegion((r*cos(theta), r*sin(theta)), r, (theta, 0, pi))
+    >>> circle.parameters
+    (r, theta)
+    >>> circle.definition
+    (r*cos(theta), r*sin(theta))
+    >>> circle.limits
+    {theta: (0, pi)}
+
+    Dimension of a parametric region determines whether a region is a curve, surface
+    or volume region. It does not represent its dimensions in space.
+
+    >>> circle.dimensions
+    1
+
+    Parameters
+    ==========
+
+    definition : tuple to define base scalars in terms of parameters.
+
+    bounds : Parameter or a tuple of length 3 to define parameter and corresponding lower and upper bound.
+
+    """
+    def __new__(cls, definition, *bounds):
+        parameters = ()
+        limits = {}
+
+        if not isinstance(bounds, Tuple):
+            bounds = Tuple(*bounds)
+
+        for bound in bounds:
+            if isinstance(bound, (tuple, Tuple)):
+                if len(bound) != 3:
+                    raise ValueError("Tuple should be in the form (parameter, lowerbound, upperbound)")
+                parameters += (bound[0],)
+                limits[bound[0]] = (bound[1], bound[2])
+            else:
+                parameters += (bound,)
+
+        if not isinstance(definition, (tuple, Tuple)):
+            definition = (definition,)
+
+        obj = super().__new__(cls, Tuple(*definition), *bounds)
+        obj._parameters = parameters
+        obj._limits = limits
+
+        return obj
+
+    @property
+    def definition(self):
+        return self.args[0]
+
+    @property
+    def limits(self):
+        return self._limits
+
+    @property
+    def parameters(self):
+        return self._parameters
+
+    @property
+    def dimensions(self):
+        return len(self.limits)
+
+
+@singledispatch
+def parametric_region_list(reg):
+    """
+    Returns a list of ParametricRegion objects representing the geometric region.
+
+    Examples
+    ========
+
+    >>> from sympy.abc import t
+    >>> from sympy.vector import parametric_region_list
+    >>> from sympy.geometry import Point, Curve, Ellipse, Segment, Polygon
+
+    >>> p = Point(2, 5)
+    >>> parametric_region_list(p)
+    [ParametricRegion((2, 5))]
+
+    >>> c = Curve((t**3, 4*t), (t, -3, 4))
+    >>> parametric_region_list(c)
+    [ParametricRegion((t**3, 4*t), (t, -3, 4))]
+
+    >>> e = Ellipse(Point(1, 3), 2, 3)
+    >>> parametric_region_list(e)
+    [ParametricRegion((2*cos(t) + 1, 3*sin(t) + 3), (t, 0, 2*pi))]
+
+    >>> s = Segment(Point(1, 3), Point(2, 6))
+    >>> parametric_region_list(s)
+    [ParametricRegion((t + 1, 3*t + 3), (t, 0, 1))]
+
+    >>> p1, p2, p3, p4 = [(0, 1), (2, -3), (5, 3), (-2, 3)]
+    >>> poly = Polygon(p1, p2, p3, p4)
+    >>> parametric_region_list(poly)
+    [ParametricRegion((2*t, 1 - 4*t), (t, 0, 1)), ParametricRegion((3*t + 2, 6*t - 3), (t, 0, 1)),\
+     ParametricRegion((5 - 7*t, 3), (t, 0, 1)), ParametricRegion((2*t - 2, 3 - 2*t),  (t, 0, 1))]
+
+    """
+    raise ValueError("SymPy cannot determine parametric representation of the region.")
+
+
+@parametric_region_list.register(Point)
+def _(obj):
+    return [ParametricRegion(obj.args)]
+
+
+@parametric_region_list.register(Curve)  # type: ignore
+def _(obj):
+    definition = obj.arbitrary_point(obj.parameter).args
+    bounds = obj.limits
+    return [ParametricRegion(definition, bounds)]
+
+
+@parametric_region_list.register(Ellipse) # type: ignore
+def _(obj, parameter='t'):
+    definition = obj.arbitrary_point(parameter).args
+    t = _symbol(parameter, real=True)
+    bounds = (t, 0, 2*pi)
+    return [ParametricRegion(definition, bounds)]
+
+
+@parametric_region_list.register(Segment) # type: ignore
+def _(obj, parameter='t'):
+    t = _symbol(parameter, real=True)
+    definition = obj.arbitrary_point(t).args
+
+    for i in range(0, 3):
+        lower_bound = solve(definition[i] - obj.points[0].args[i], t)
+        upper_bound = solve(definition[i] - obj.points[1].args[i], t)
+
+        if len(lower_bound) == 1 and len(upper_bound) == 1:
+            bounds = t, lower_bound[0], upper_bound[0]
+            break
+
+    definition_tuple = obj.arbitrary_point(parameter).args
+    return [ParametricRegion(definition_tuple, bounds)]
+
+
+@parametric_region_list.register(Polygon) # type: ignore
+def _(obj, parameter='t'):
+    l = [parametric_region_list(side, parameter)[0] for side in obj.sides]
+    return l
+
+
+@parametric_region_list.register(ImplicitRegion) # type: ignore
+def _(obj, parameters=('t', 's')):
+    definition = obj.rational_parametrization(parameters)
+    bounds = []
+
+    for i in range(len(obj.variables) - 1):
+        # Each parameter is replaced by its tangent to simplify intergation
+        parameter = _symbol(parameters[i], real=True)
+        definition = [trigsimp(elem.subs(parameter, tan(parameter/2))) for elem in definition]
+        bounds.append((parameter, 0, 2*pi),)
+
+    definition = Tuple(*definition)
+    return [ParametricRegion(definition, *bounds)]

rl/Lib/site-packages/sympy/vector/point.py

@@ -0,0 +1,151 @@
+from sympy.core.basic import Basic
+from sympy.core.symbol import Str
+from sympy.vector.vector import Vector
+from sympy.vector.coordsysrect import CoordSys3D
+from sympy.vector.functions import _path
+from sympy.core.cache import cacheit
+
+
+class Point(Basic):
+    """
+    Represents a point in 3-D space.
+    """
+
+    def __new__(cls, name, position=Vector.zero, parent_point=None):
+        name = str(name)
+        # Check the args first
+        if not isinstance(position, Vector):
+            raise TypeError(
+                "position should be an instance of Vector, not %s" % type(
+                    position))
+        if (not isinstance(parent_point, Point) and
+                parent_point is not None):
+            raise TypeError(
+                "parent_point should be an instance of Point, not %s" % type(
+                    parent_point))
+        # Super class construction
+        if parent_point is None:
+            obj = super().__new__(cls, Str(name), position)
+        else:
+            obj = super().__new__(cls, Str(name), position, parent_point)
+        # Decide the object parameters
+        obj._name = name
+        obj._pos = position
+        if parent_point is None:
+            obj._parent = None
+            obj._root = obj
+        else:
+            obj._parent = parent_point
+            obj._root = parent_point._root
+        # Return object
+        return obj
+
+    @cacheit
+    def position_wrt(self, other):
+        """
+        Returns the position vector of this Point with respect to
+        another Point/CoordSys3D.
+
+        Parameters
+        ==========
+
+        other : Point/CoordSys3D
+            If other is a Point, the position of this Point wrt it is
+            returned. If its an instance of CoordSyRect, the position
+            wrt its origin is returned.
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D
+        >>> N = CoordSys3D('N')
+        >>> p1 = N.origin.locate_new('p1', 10 * N.i)
+        >>> N.origin.position_wrt(p1)
+        (-10)*N.i
+
+        """
+
+        if (not isinstance(other, Point) and
+                not isinstance(other, CoordSys3D)):
+            raise TypeError(str(other) +
+                            "is not a Point or CoordSys3D")
+        if isinstance(other, CoordSys3D):
+            other = other.origin
+        # Handle special cases
+        if other == self:
+            return Vector.zero
+        elif other == self._parent:
+            return self._pos
+        elif other._parent == self:
+            return -1 * other._pos
+        # Else, use point tree to calculate position
+        rootindex, path = _path(self, other)
+        result = Vector.zero
+        i = -1
+        for i in range(rootindex):
+            result += path[i]._pos
+        i += 2
+        while i < len(path):
+            result -= path[i]._pos
+            i += 1
+        return result
+
+    def locate_new(self, name, position):
+        """
+        Returns a new Point located at the given position wrt this
+        Point.
+        Thus, the position vector of the new Point wrt this one will
+        be equal to the given 'position' parameter.
+
+        Parameters
+        ==========
+
+        name : str
+            Name of the new point
+
+        position : Vector
+            The position vector of the new Point wrt this one
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D
+        >>> N = CoordSys3D('N')
+        >>> p1 = N.origin.locate_new('p1', 10 * N.i)
+        >>> p1.position_wrt(N.origin)
+        10*N.i
+
+        """
+        return Point(name, position, self)
+
+    def express_coordinates(self, coordinate_system):
+        """
+        Returns the Cartesian/rectangular coordinates of this point
+        wrt the origin of the given CoordSys3D instance.
+
+        Parameters
+        ==========
+
+        coordinate_system : CoordSys3D
+            The coordinate system to express the coordinates of this
+            Point in.
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D
+        >>> N = CoordSys3D('N')
+        >>> p1 = N.origin.locate_new('p1', 10 * N.i)
+        >>> p2 = p1.locate_new('p2', 5 * N.j)
+        >>> p2.express_coordinates(N)
+        (10, 5, 0)
+
+        """
+
+        # Determine the position vector
+        pos_vect = self.position_wrt(coordinate_system.origin)
+        # Express it in the given coordinate system
+        return tuple(pos_vect.to_matrix(coordinate_system))
+
+    def _sympystr(self, printer):
+        return self._name

rl/Lib/site-packages/sympy/vector/scalar.py

@@ -0,0 +1,69 @@
+from sympy.core import AtomicExpr, Symbol, S
+from sympy.core.sympify import _sympify
+from sympy.printing.pretty.stringpict import prettyForm
+from sympy.printing.precedence import PRECEDENCE
+
+
+class BaseScalar(AtomicExpr):
+    """
+    A coordinate symbol/base scalar.
+
+    Ideally, users should not instantiate this class.
+
+    """
+
+    def __new__(cls, index, system, pretty_str=None, latex_str=None):
+        from sympy.vector.coordsysrect import CoordSys3D
+        if pretty_str is None:
+            pretty_str = "x{}".format(index)
+        elif isinstance(pretty_str, Symbol):
+            pretty_str = pretty_str.name
+        if latex_str is None:
+            latex_str = "x_{}".format(index)
+        elif isinstance(latex_str, Symbol):
+            latex_str = latex_str.name
+
+        index = _sympify(index)
+        system = _sympify(system)
+        obj = super().__new__(cls, index, system)
+        if not isinstance(system, CoordSys3D):
+            raise TypeError("system should be a CoordSys3D")
+        if index not in range(0, 3):
+            raise ValueError("Invalid index specified.")
+        # The _id is used for equating purposes, and for hashing
+        obj._id = (index, system)
+        obj._name = obj.name = system._name + '.' + system._variable_names[index]
+        obj._pretty_form = '' + pretty_str
+        obj._latex_form = latex_str
+        obj._system = system
+
+        return obj
+
+    is_commutative = True
+    is_symbol = True
+
+    @property
+    def free_symbols(self):
+        return {self}
+
+    _diff_wrt = True
+
+    def _eval_derivative(self, s):
+        if self == s:
+            return S.One
+        return S.Zero
+
+    def _latex(self, printer=None):
+        return self._latex_form
+
+    def _pretty(self, printer=None):
+        return prettyForm(self._pretty_form)
+
+    precedence = PRECEDENCE['Atom']
+
+    @property
+    def system(self):
+        return self._system
+
+    def _sympystr(self, printer):
+        return self._name

rl/Lib/site-packages/sympy/vector/tests/test_coordsysrect.py

@@ -0,0 +1,464 @@
+from sympy.testing.pytest import raises
+from sympy.vector.coordsysrect import CoordSys3D
+from sympy.vector.scalar import BaseScalar
+from sympy.core.function import expand
+from sympy.core.numbers import pi
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.hyperbolic import (cosh, sinh)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (acos, atan2, cos, sin)
+from sympy.matrices.dense import zeros
+from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix
+from sympy.simplify.simplify import simplify
+from sympy.vector.functions import express
+from sympy.vector.point import Point
+from sympy.vector.vector import Vector
+from sympy.vector.orienters import (AxisOrienter, BodyOrienter,
+                                    SpaceOrienter, QuaternionOrienter)
+
+
+x, y, z = symbols('x y z')
+a, b, c, q = symbols('a b c q')
+q1, q2, q3, q4 = symbols('q1 q2 q3 q4')
+
+
+def test_func_args():
+    A = CoordSys3D('A')
+    assert A.x.func(*A.x.args) == A.x
+    expr = 3*A.x + 4*A.y
+    assert expr.func(*expr.args) == expr
+    assert A.i.func(*A.i.args) == A.i
+    v = A.x*A.i + A.y*A.j + A.z*A.k
+    assert v.func(*v.args) == v
+    assert A.origin.func(*A.origin.args) == A.origin
+
+
+def test_coordsys3d_equivalence():
+    A = CoordSys3D('A')
+    A1 = CoordSys3D('A')
+    assert A1 == A
+    B = CoordSys3D('B')
+    assert A != B
+
+
+def test_orienters():
+    A = CoordSys3D('A')
+    axis_orienter = AxisOrienter(a, A.k)
+    body_orienter = BodyOrienter(a, b, c, '123')
+    space_orienter = SpaceOrienter(a, b, c, '123')
+    q_orienter = QuaternionOrienter(q1, q2, q3, q4)
+    assert axis_orienter.rotation_matrix(A) == Matrix([
+        [ cos(a), sin(a), 0],
+        [-sin(a), cos(a), 0],
+        [      0,      0, 1]])
+    assert body_orienter.rotation_matrix() == Matrix([
+        [ cos(b)*cos(c),  sin(a)*sin(b)*cos(c) + sin(c)*cos(a),
+          sin(a)*sin(c) - sin(b)*cos(a)*cos(c)],
+        [-sin(c)*cos(b), -sin(a)*sin(b)*sin(c) + cos(a)*cos(c),
+         sin(a)*cos(c) + sin(b)*sin(c)*cos(a)],
+        [        sin(b),                        -sin(a)*cos(b),
+                 cos(a)*cos(b)]])
+    assert space_orienter.rotation_matrix() == Matrix([
+        [cos(b)*cos(c), sin(c)*cos(b),       -sin(b)],
+        [sin(a)*sin(b)*cos(c) - sin(c)*cos(a),
+         sin(a)*sin(b)*sin(c) + cos(a)*cos(c), sin(a)*cos(b)],
+        [sin(a)*sin(c) + sin(b)*cos(a)*cos(c), -sin(a)*cos(c) +
+         sin(b)*sin(c)*cos(a), cos(a)*cos(b)]])
+    assert q_orienter.rotation_matrix() == Matrix([
+        [q1**2 + q2**2 - q3**2 - q4**2, 2*q1*q4 + 2*q2*q3,
+         -2*q1*q3 + 2*q2*q4],
+        [-2*q1*q4 + 2*q2*q3, q1**2 - q2**2 + q3**2 - q4**2,
+         2*q1*q2 + 2*q3*q4],
+        [2*q1*q3 + 2*q2*q4,
+         -2*q1*q2 + 2*q3*q4, q1**2 - q2**2 - q3**2 + q4**2]])
+
+
+def test_coordinate_vars():
+    """
+    Tests the coordinate variables functionality with respect to
+    reorientation of coordinate systems.
+    """
+    A = CoordSys3D('A')
+    # Note that the name given on the lhs is different from A.x._name
+    assert BaseScalar(0, A, 'A_x', r'\mathbf{{x}_{A}}') == A.x
+    assert BaseScalar(1, A, 'A_y', r'\mathbf{{y}_{A}}') == A.y
+    assert BaseScalar(2, A, 'A_z', r'\mathbf{{z}_{A}}') == A.z
+    assert BaseScalar(0, A, 'A_x', r'\mathbf{{x}_{A}}').__hash__() == A.x.__hash__()
+    assert isinstance(A.x, BaseScalar) and \
+           isinstance(A.y, BaseScalar) and \
+           isinstance(A.z, BaseScalar)
+    assert A.x*A.y == A.y*A.x
+    assert A.scalar_map(A) == {A.x: A.x, A.y: A.y, A.z: A.z}
+    assert A.x.system == A
+    assert A.x.diff(A.x) == 1
+    B = A.orient_new_axis('B', q, A.k)
+    assert B.scalar_map(A) == {B.z: A.z, B.y: -A.x*sin(q) + A.y*cos(q),
+                                 B.x: A.x*cos(q) + A.y*sin(q)}
+    assert A.scalar_map(B) == {A.x: B.x*cos(q) - B.y*sin(q),
+                                 A.y: B.x*sin(q) + B.y*cos(q), A.z: B.z}
+    assert express(B.x, A, variables=True) == A.x*cos(q) + A.y*sin(q)
+    assert express(B.y, A, variables=True) == -A.x*sin(q) + A.y*cos(q)
+    assert express(B.z, A, variables=True) == A.z
+    assert expand(express(B.x*B.y*B.z, A, variables=True)) == \
+           expand(A.z*(-A.x*sin(q) + A.y*cos(q))*(A.x*cos(q) + A.y*sin(q)))
+    assert express(B.x*B.i + B.y*B.j + B.z*B.k, A) == \
+           (B.x*cos(q) - B.y*sin(q))*A.i + (B.x*sin(q) + \
+           B.y*cos(q))*A.j + B.z*A.k
+    assert simplify(express(B.x*B.i + B.y*B.j + B.z*B.k, A, \
+                            variables=True)) == \
+           A.x*A.i + A.y*A.j + A.z*A.k
+    assert express(A.x*A.i + A.y*A.j + A.z*A.k, B) == \
+           (A.x*cos(q) + A.y*sin(q))*B.i + \
+           (-A.x*sin(q) + A.y*cos(q))*B.j + A.z*B.k
+    assert simplify(express(A.x*A.i + A.y*A.j + A.z*A.k, B, \
+                            variables=True)) == \
+           B.x*B.i + B.y*B.j + B.z*B.k
+    N = B.orient_new_axis('N', -q, B.k)
+    assert N.scalar_map(A) == \
+           {N.x: A.x, N.z: A.z, N.y: A.y}
+    C = A.orient_new_axis('C', q, A.i + A.j + A.k)
+    mapping = A.scalar_map(C)
+    assert mapping[A.x].equals(C.x*(2*cos(q) + 1)/3 +
+                            C.y*(-2*sin(q + pi/6) + 1)/3 +
+                            C.z*(-2*cos(q + pi/3) + 1)/3)
+    assert mapping[A.y].equals(C.x*(-2*cos(q + pi/3) + 1)/3 +
+                            C.y*(2*cos(q) + 1)/3 +
+                            C.z*(-2*sin(q + pi/6) + 1)/3)
+    assert mapping[A.z].equals(C.x*(-2*sin(q + pi/6) + 1)/3 +
+                            C.y*(-2*cos(q + pi/3) + 1)/3 +
+                            C.z*(2*cos(q) + 1)/3)
+    D = A.locate_new('D', a*A.i + b*A.j + c*A.k)
+    assert D.scalar_map(A) == {D.z: A.z - c, D.x: A.x - a, D.y: A.y - b}
+    E = A.orient_new_axis('E', a, A.k, a*A.i + b*A.j + c*A.k)
+    assert A.scalar_map(E) == {A.z: E.z + c,
+                               A.x: E.x*cos(a) - E.y*sin(a) + a,
+                               A.y: E.x*sin(a) + E.y*cos(a) + b}
+    assert E.scalar_map(A) == {E.x: (A.x - a)*cos(a) + (A.y - b)*sin(a),
+                               E.y: (-A.x + a)*sin(a) + (A.y - b)*cos(a),
+                               E.z: A.z - c}
+    F = A.locate_new('F', Vector.zero)
+    assert A.scalar_map(F) == {A.z: F.z, A.x: F.x, A.y: F.y}
+
+
+def test_rotation_matrix():
+    N = CoordSys3D('N')
+    A = N.orient_new_axis('A', q1, N.k)
+    B = A.orient_new_axis('B', q2, A.i)
+    C = B.orient_new_axis('C', q3, B.j)
+    D = N.orient_new_axis('D', q4, N.j)
+    E = N.orient_new_space('E', q1, q2, q3, '123')
+    F = N.orient_new_quaternion('F', q1, q2, q3, q4)
+    G = N.orient_new_body('G', q1, q2, q3, '123')
+    assert N.rotation_matrix(C) == Matrix([
+        [- sin(q1) * sin(q2) * sin(q3) + cos(q1) * cos(q3), - sin(q1) *
+        cos(q2), sin(q1) * sin(q2) * cos(q3) + sin(q3) * cos(q1)], \
+        [sin(q1) * cos(q3) + sin(q2) * sin(q3) * cos(q1), \
+         cos(q1) * cos(q2), sin(q1) * sin(q3) - sin(q2) * cos(q1) * \
+         cos(q3)], [- sin(q3) * cos(q2), sin(q2), cos(q2) * cos(q3)]])
+    test_mat = D.rotation_matrix(C) - Matrix(
+        [[cos(q1) * cos(q3) * cos(q4) - sin(q3) * (- sin(q4) * cos(q2) +
+        sin(q1) * sin(q2) * cos(q4)), - sin(q2) * sin(q4) - sin(q1) *
+            cos(q2) * cos(q4), sin(q3) * cos(q1) * cos(q4) + cos(q3) * \
+          (- sin(q4) * cos(q2) + sin(q1) * sin(q2) * cos(q4))], \
+         [sin(q1) * cos(q3) + sin(q2) * sin(q3) * cos(q1), cos(q1) * \
+          cos(q2), sin(q1) * sin(q3) - sin(q2) * cos(q1) * cos(q3)], \
+         [sin(q4) * cos(q1) * cos(q3) - sin(q3) * (cos(q2) * cos(q4) + \
+                                                   sin(q1) * sin(q2) * \
+                                                   sin(q4)), sin(q2) *
+                cos(q4) - sin(q1) * sin(q4) * cos(q2), sin(q3) * \
+          sin(q4) * cos(q1) + cos(q3) * (cos(q2) * cos(q4) + \
+                                         sin(q1) * sin(q2) * sin(q4))]])
+    assert test_mat.expand() == zeros(3, 3)
+    assert E.rotation_matrix(N) == Matrix(
+        [[cos(q2)*cos(q3), sin(q3)*cos(q2), -sin(q2)],
+        [sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1), \
+         sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), sin(q1)*cos(q2)], \
+         [sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3), - \
+          sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1), cos(q1)*cos(q2)]])
+    assert F.rotation_matrix(N) == Matrix([[
+        q1**2 + q2**2 - q3**2 - q4**2,
+        2*q1*q4 + 2*q2*q3, -2*q1*q3 + 2*q2*q4],[ -2*q1*q4 + 2*q2*q3,
+            q1**2 - q2**2 + q3**2 - q4**2, 2*q1*q2 + 2*q3*q4],
+                                           [2*q1*q3 + 2*q2*q4,
+                                            -2*q1*q2 + 2*q3*q4,
+                                q1**2 - q2**2 - q3**2 + q4**2]])
+    assert G.rotation_matrix(N) == Matrix([[
+        cos(q2)*cos(q3),  sin(q1)*sin(q2)*cos(q3) + sin(q3)*cos(q1),
+        sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3)], [
+            -sin(q3)*cos(q2), -sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3),
+            sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1)],[
+                sin(q2), -sin(q1)*cos(q2), cos(q1)*cos(q2)]])
+
+
+def test_vector_with_orientation():
+    """
+    Tests the effects of orientation of coordinate systems on
+    basic vector operations.
+    """
+    N = CoordSys3D('N')
+    A = N.orient_new_axis('A', q1, N.k)
+    B = A.orient_new_axis('B', q2, A.i)
+    C = B.orient_new_axis('C', q3, B.j)
+
+    # Test to_matrix
+    v1 = a*N.i + b*N.j + c*N.k
+    assert v1.to_matrix(A) == Matrix([[ a*cos(q1) + b*sin(q1)],
+                                      [-a*sin(q1) + b*cos(q1)],
+                                      [                     c]])
+
+    # Test dot
+    assert N.i.dot(A.i) == cos(q1)
+    assert N.i.dot(A.j) == -sin(q1)
+    assert N.i.dot(A.k) == 0
+    assert N.j.dot(A.i) == sin(q1)
+    assert N.j.dot(A.j) == cos(q1)
+    assert N.j.dot(A.k) == 0
+    assert N.k.dot(A.i) == 0
+    assert N.k.dot(A.j) == 0
+    assert N.k.dot(A.k) == 1
+
+    assert N.i.dot(A.i + A.j) == -sin(q1) + cos(q1) == \
+           (A.i + A.j).dot(N.i)
+
+    assert A.i.dot(C.i) == cos(q3)
+    assert A.i.dot(C.j) == 0
+    assert A.i.dot(C.k) == sin(q3)
+    assert A.j.dot(C.i) == sin(q2)*sin(q3)
+    assert A.j.dot(C.j) == cos(q2)
+    assert A.j.dot(C.k) == -sin(q2)*cos(q3)
+    assert A.k.dot(C.i) == -cos(q2)*sin(q3)
+    assert A.k.dot(C.j) == sin(q2)
+    assert A.k.dot(C.k) == cos(q2)*cos(q3)
+
+    # Test cross
+    assert N.i.cross(A.i) == sin(q1)*A.k
+    assert N.i.cross(A.j) == cos(q1)*A.k
+    assert N.i.cross(A.k) == -sin(q1)*A.i - cos(q1)*A.j
+    assert N.j.cross(A.i) == -cos(q1)*A.k
+    assert N.j.cross(A.j) == sin(q1)*A.k
+    assert N.j.cross(A.k) == cos(q1)*A.i - sin(q1)*A.j
+    assert N.k.cross(A.i) == A.j
+    assert N.k.cross(A.j) == -A.i
+    assert N.k.cross(A.k) == Vector.zero
+
+    assert N.i.cross(A.i) == sin(q1)*A.k
+    assert N.i.cross(A.j) == cos(q1)*A.k
+    assert N.i.cross(A.i + A.j) == sin(q1)*A.k + cos(q1)*A.k
+    assert (A.i + A.j).cross(N.i) == (-sin(q1) - cos(q1))*N.k
+
+    assert A.i.cross(C.i) == sin(q3)*C.j
+    assert A.i.cross(C.j) == -sin(q3)*C.i + cos(q3)*C.k
+    assert A.i.cross(C.k) == -cos(q3)*C.j
+    assert C.i.cross(A.i) == (-sin(q3)*cos(q2))*A.j + \
+           (-sin(q2)*sin(q3))*A.k
+    assert C.j.cross(A.i) == (sin(q2))*A.j + (-cos(q2))*A.k
+    assert express(C.k.cross(A.i), C).trigsimp() == cos(q3)*C.j
+
+
+def test_orient_new_methods():
+    N = CoordSys3D('N')
+    orienter1 = AxisOrienter(q4, N.j)
+    orienter2 = SpaceOrienter(q1, q2, q3, '123')
+    orienter3 = QuaternionOrienter(q1, q2, q3, q4)
+    orienter4 = BodyOrienter(q1, q2, q3, '123')
+    D = N.orient_new('D', (orienter1, ))
+    E = N.orient_new('E', (orienter2, ))
+    F = N.orient_new('F', (orienter3, ))
+    G = N.orient_new('G', (orienter4, ))
+    assert D == N.orient_new_axis('D', q4, N.j)
+    assert E == N.orient_new_space('E', q1, q2, q3, '123')
+    assert F == N.orient_new_quaternion('F', q1, q2, q3, q4)
+    assert G == N.orient_new_body('G', q1, q2, q3, '123')
+
+
+def test_locatenew_point():
+    """
+    Tests Point class, and locate_new method in CoordSys3D.
+    """
+    A = CoordSys3D('A')
+    assert isinstance(A.origin, Point)
+    v = a*A.i + b*A.j + c*A.k
+    C = A.locate_new('C', v)
+    assert C.origin.position_wrt(A) == \
+           C.position_wrt(A) == \
+           C.origin.position_wrt(A.origin) == v
+    assert A.origin.position_wrt(C) == \
+           A.position_wrt(C) == \
+           A.origin.position_wrt(C.origin) == -v
+    assert A.origin.express_coordinates(C) == (-a, -b, -c)
+    p = A.origin.locate_new('p', -v)
+    assert p.express_coordinates(A) == (-a, -b, -c)
+    assert p.position_wrt(C.origin) == p.position_wrt(C) == \
+           -2 * v
+    p1 = p.locate_new('p1', 2*v)
+    assert p1.position_wrt(C.origin) == Vector.zero
+    assert p1.express_coordinates(C) == (0, 0, 0)
+    p2 = p.locate_new('p2', A.i)
+    assert p1.position_wrt(p2) == 2*v - A.i
+    assert p2.express_coordinates(C) == (-2*a + 1, -2*b, -2*c)
+
+
+def test_create_new():
+    a = CoordSys3D('a')
+    c = a.create_new('c', transformation='spherical')
+    assert c._parent == a
+    assert c.transformation_to_parent() == \
+           (c.r*sin(c.theta)*cos(c.phi), c.r*sin(c.theta)*sin(c.phi), c.r*cos(c.theta))
+    assert c.transformation_from_parent() == \
+           (sqrt(a.x**2 + a.y**2 + a.z**2), acos(a.z/sqrt(a.x**2 + a.y**2 + a.z**2)), atan2(a.y, a.x))
+
+
+def test_evalf():
+    A = CoordSys3D('A')
+    v = 3*A.i + 4*A.j + a*A.k
+    assert v.n() == v.evalf()
+    assert v.evalf(subs={a:1}) == v.subs(a, 1).evalf()
+
+
+def test_lame_coefficients():
+    a = CoordSys3D('a', 'spherical')
+    assert a.lame_coefficients() == (1, a.r, sin(a.theta)*a.r)
+    a = CoordSys3D('a')
+    assert a.lame_coefficients() == (1, 1, 1)
+    a = CoordSys3D('a', 'cartesian')
+    assert a.lame_coefficients() == (1, 1, 1)
+    a = CoordSys3D('a', 'cylindrical')
+    assert a.lame_coefficients() == (1, a.r, 1)
+
+
+def test_transformation_equations():
+
+    x, y, z = symbols('x y z')
+    # Str
+    a = CoordSys3D('a', transformation='spherical',
+                   variable_names=["r", "theta", "phi"])
+    r, theta, phi = a.base_scalars()
+
+    assert r == a.r
+    assert theta == a.theta
+    assert phi == a.phi
+
+    raises(AttributeError, lambda: a.x)
+    raises(AttributeError, lambda: a.y)
+    raises(AttributeError, lambda: a.z)
+
+    assert a.transformation_to_parent() == (
+        r*sin(theta)*cos(phi),
+        r*sin(theta)*sin(phi),
+        r*cos(theta)
+    )
+    assert a.lame_coefficients() == (1, r, r*sin(theta))
+    assert a.transformation_from_parent_function()(x, y, z) == (
+        sqrt(x ** 2 + y ** 2 + z ** 2),
+        acos((z) / sqrt(x**2 + y**2 + z**2)),
+        atan2(y, x)
+    )
+    a = CoordSys3D('a', transformation='cylindrical',
+                   variable_names=["r", "theta", "z"])
+    r, theta, z = a.base_scalars()
+    assert a.transformation_to_parent() == (
+        r*cos(theta),
+        r*sin(theta),
+        z
+    )
+    assert a.lame_coefficients() == (1, a.r, 1)
+    assert a.transformation_from_parent_function()(x, y, z) == (sqrt(x**2 + y**2),
+                            atan2(y, x), z)
+
+    a = CoordSys3D('a', 'cartesian')
+    assert a.transformation_to_parent() == (a.x, a.y, a.z)
+    assert a.lame_coefficients() == (1, 1, 1)
+    assert a.transformation_from_parent_function()(x, y, z) == (x, y, z)
+
+    # Variables and expressions
+
+    # Cartesian with equation tuple:
+    x, y, z = symbols('x y z')
+    a = CoordSys3D('a', ((x, y, z), (x, y, z)))
+    a._calculate_inv_trans_equations()
+    assert a.transformation_to_parent() == (a.x1, a.x2, a.x3)
+    assert a.lame_coefficients() == (1, 1, 1)
+    assert a.transformation_from_parent_function()(x, y, z) == (x, y, z)
+    r, theta, z = symbols("r theta z")
+
+    # Cylindrical with equation tuple:
+    a = CoordSys3D('a', [(r, theta, z), (r*cos(theta), r*sin(theta), z)],
+                   variable_names=["r", "theta", "z"])
+    r, theta, z = a.base_scalars()
+    assert a.transformation_to_parent() == (
+        r*cos(theta), r*sin(theta), z
+    )
+    assert a.lame_coefficients() == (
+        sqrt(sin(theta)**2 + cos(theta)**2),
+        sqrt(r**2*sin(theta)**2 + r**2*cos(theta)**2),
+        1
+    )  # ==> this should simplify to (1, r, 1), tests are too slow with `simplify`.
+
+    # Definitions with `lambda`:
+
+    # Cartesian with `lambda`
+    a = CoordSys3D('a', lambda x, y, z: (x, y, z))
+    assert a.transformation_to_parent() == (a.x1, a.x2, a.x3)
+    assert a.lame_coefficients() == (1, 1, 1)
+    a._calculate_inv_trans_equations()
+    assert a.transformation_from_parent_function()(x, y, z) == (x, y, z)
+
+    # Spherical with `lambda`
+    a = CoordSys3D('a', lambda r, theta, phi: (r*sin(theta)*cos(phi), r*sin(theta)*sin(phi), r*cos(theta)),
+                   variable_names=["r", "theta", "phi"])
+    r, theta, phi = a.base_scalars()
+    assert a.transformation_to_parent() == (
+        r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta)
+    )
+    assert a.lame_coefficients() == (
+        sqrt(sin(phi)**2*sin(theta)**2 + sin(theta)**2*cos(phi)**2 + cos(theta)**2),
+        sqrt(r**2*sin(phi)**2*cos(theta)**2 + r**2*sin(theta)**2 + r**2*cos(phi)**2*cos(theta)**2),
+        sqrt(r**2*sin(phi)**2*sin(theta)**2 + r**2*sin(theta)**2*cos(phi)**2)
+    )  # ==> this should simplify to (1, r, sin(theta)*r), `simplify` is too slow.
+
+    # Cylindrical with `lambda`
+    a = CoordSys3D('a', lambda r, theta, z:
+        (r*cos(theta), r*sin(theta), z),
+        variable_names=["r", "theta", "z"]
+    )
+    r, theta, z = a.base_scalars()
+    assert a.transformation_to_parent() == (r*cos(theta), r*sin(theta), z)
+    assert a.lame_coefficients() == (
+        sqrt(sin(theta)**2 + cos(theta)**2),
+        sqrt(r**2*sin(theta)**2 + r**2*cos(theta)**2),
+        1
+    )  # ==> this should simplify to (1, a.x, 1)
+
+    raises(TypeError, lambda: CoordSys3D('a', transformation={
+        x: x*sin(y)*cos(z), y:x*sin(y)*sin(z), z:  x*cos(y)}))
+
+
+def test_check_orthogonality():
+    x, y, z = symbols('x y z')
+    u,v = symbols('u, v')
+    a = CoordSys3D('a', transformation=((x, y, z), (x*sin(y)*cos(z), x*sin(y)*sin(z), x*cos(y))))
+    assert a._check_orthogonality(a._transformation) is True
+    a = CoordSys3D('a', transformation=((x, y, z), (x * cos(y), x * sin(y), z)))
+    assert a._check_orthogonality(a._transformation) is True
+    a = CoordSys3D('a', transformation=((u, v, z), (cosh(u) * cos(v), sinh(u) * sin(v), z)))
+    assert a._check_orthogonality(a._transformation) is True
+
+    raises(ValueError, lambda: CoordSys3D('a', transformation=((x, y, z), (x, x, z))))
+    raises(ValueError, lambda: CoordSys3D('a', transformation=(
+        (x, y, z), (x*sin(y/2)*cos(z), x*sin(y)*sin(z), x*cos(y)))))
+
+
+def test_rotation_trans_equations():
+    a = CoordSys3D('a')
+    from sympy.core.symbol import symbols
+    q0 = symbols('q0')
+    assert a._rotation_trans_equations(a._parent_rotation_matrix, a.base_scalars()) == (a.x, a.y, a.z)
+    assert a._rotation_trans_equations(a._inverse_rotation_matrix(), a.base_scalars()) == (a.x, a.y, a.z)
+    b = a.orient_new_axis('b', 0, -a.k)
+    assert b._rotation_trans_equations(b._parent_rotation_matrix, b.base_scalars()) == (b.x, b.y, b.z)
+    assert b._rotation_trans_equations(b._inverse_rotation_matrix(), b.base_scalars()) == (b.x, b.y, b.z)
+    c = a.orient_new_axis('c', q0, -a.k)
+    assert c._rotation_trans_equations(c._parent_rotation_matrix, c.base_scalars()) == \
+           (-sin(q0) * c.y + cos(q0) * c.x, sin(q0) * c.x + cos(q0) * c.y, c.z)
+    assert c._rotation_trans_equations(c._inverse_rotation_matrix(), c.base_scalars()) == \
+           (sin(q0) * c.y + cos(q0) * c.x, -sin(q0) * c.x + cos(q0) * c.y, c.z)

rl/Lib/site-packages/sympy/vector/tests/test_dyadic.py

@@ -0,0 +1,134 @@
+from sympy.core.numbers import pi
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix
+from sympy.simplify.simplify import simplify
+from sympy.vector import (CoordSys3D, Vector, Dyadic,
+                          DyadicAdd, DyadicMul, DyadicZero,
+                          BaseDyadic, express)
+
+
+A = CoordSys3D('A')
+
+
+def test_dyadic():
+    a, b = symbols('a, b')
+    assert Dyadic.zero != 0
+    assert isinstance(Dyadic.zero, DyadicZero)
+    assert BaseDyadic(A.i, A.j) != BaseDyadic(A.j, A.i)
+    assert (BaseDyadic(Vector.zero, A.i) ==
+            BaseDyadic(A.i, Vector.zero) == Dyadic.zero)
+
+    d1 = A.i | A.i
+    d2 = A.j | A.j
+    d3 = A.i | A.j
+
+    assert isinstance(d1, BaseDyadic)
+    d_mul = a*d1
+    assert isinstance(d_mul, DyadicMul)
+    assert d_mul.base_dyadic == d1
+    assert d_mul.measure_number == a
+    assert isinstance(a*d1 + b*d3, DyadicAdd)
+    assert d1 == A.i.outer(A.i)
+    assert d3 == A.i.outer(A.j)
+    v1 = a*A.i - A.k
+    v2 = A.i + b*A.j
+    assert v1 | v2 == v1.outer(v2) == a * (A.i|A.i) + (a*b) * (A.i|A.j) +\
+           - (A.k|A.i) - b * (A.k|A.j)
+    assert d1 * 0 == Dyadic.zero
+    assert d1 != Dyadic.zero
+    assert d1 * 2 == 2 * (A.i | A.i)
+    assert d1 / 2. == 0.5 * d1
+
+    assert d1.dot(0 * d1) == Vector.zero
+    assert d1 & d2 == Dyadic.zero
+    assert d1.dot(A.i) == A.i == d1 & A.i
+
+    assert d1.cross(Vector.zero) == Dyadic.zero
+    assert d1.cross(A.i) == Dyadic.zero
+    assert d1 ^ A.j == d1.cross(A.j)
+    assert d1.cross(A.k) == - A.i | A.j
+    assert d2.cross(A.i) == - A.j | A.k == d2 ^ A.i
+
+    assert A.i ^ d1 == Dyadic.zero
+    assert A.j.cross(d1) == - A.k | A.i == A.j ^ d1
+    assert Vector.zero.cross(d1) == Dyadic.zero
+    assert A.k ^ d1 == A.j | A.i
+    assert A.i.dot(d1) == A.i & d1 == A.i
+    assert A.j.dot(d1) == Vector.zero
+    assert Vector.zero.dot(d1) == Vector.zero
+    assert A.j & d2 == A.j
+
+    assert d1.dot(d3) == d1 & d3 == A.i | A.j == d3
+    assert d3 & d1 == Dyadic.zero
+
+    q = symbols('q')
+    B = A.orient_new_axis('B', q, A.k)
+    assert express(d1, B) == express(d1, B, B)
+
+    expr1 = ((cos(q)**2) * (B.i | B.i) + (-sin(q) * cos(q)) *
+            (B.i | B.j) + (-sin(q) * cos(q)) * (B.j | B.i) + (sin(q)**2) *
+            (B.j | B.j))
+    assert (express(d1, B) - expr1).simplify() == Dyadic.zero
+
+    expr2 = (cos(q)) * (B.i | A.i) + (-sin(q)) * (B.j | A.i)
+    assert (express(d1, B, A) - expr2).simplify() == Dyadic.zero
+
+    expr3 = (cos(q)) * (A.i | B.i) + (-sin(q)) * (A.i | B.j)
+    assert (express(d1, A, B) - expr3).simplify() == Dyadic.zero
+
+    assert d1.to_matrix(A) == Matrix([[1, 0, 0], [0, 0, 0], [0, 0, 0]])
+    assert d1.to_matrix(A, B) == Matrix([[cos(q), -sin(q), 0],
+                                         [0, 0, 0],
+                                         [0, 0, 0]])
+    assert d3.to_matrix(A) == Matrix([[0, 1, 0], [0, 0, 0], [0, 0, 0]])
+    a, b, c, d, e, f = symbols('a, b, c, d, e, f')
+    v1 = a * A.i + b * A.j + c * A.k
+    v2 = d * A.i + e * A.j + f * A.k
+    d4 = v1.outer(v2)
+    assert d4.to_matrix(A) == Matrix([[a * d, a * e, a * f],
+                                      [b * d, b * e, b * f],
+                                      [c * d, c * e, c * f]])
+    d5 = v1.outer(v1)
+    C = A.orient_new_axis('C', q, A.i)
+    for expected, actual in zip(C.rotation_matrix(A) * d5.to_matrix(A) * \
+                               C.rotation_matrix(A).T, d5.to_matrix(C)):
+        assert (expected - actual).simplify() == 0
+
+
+def test_dyadic_simplify():
+    x, y, z, k, n, m, w, f, s, A = symbols('x, y, z, k, n, m, w, f, s, A')
+    N = CoordSys3D('N')
+
+    dy = N.i | N.i
+    test1 = (1 / x + 1 / y) * dy
+    assert (N.i & test1 & N.i) != (x + y) / (x * y)
+    test1 = test1.simplify()
+    assert test1.simplify() == simplify(test1)
+    assert (N.i & test1 & N.i) == (x + y) / (x * y)
+
+    test2 = (A**2 * s**4 / (4 * pi * k * m**3)) * dy
+    test2 = test2.simplify()
+    assert (N.i & test2 & N.i) == (A**2 * s**4 / (4 * pi * k * m**3))
+
+    test3 = ((4 + 4 * x - 2 * (2 + 2 * x)) / (2 + 2 * x)) * dy
+    test3 = test3.simplify()
+    assert (N.i & test3 & N.i) == 0
+
+    test4 = ((-4 * x * y**2 - 2 * y**3 - 2 * x**2 * y) / (x + y)**2) * dy
+    test4 = test4.simplify()
+    assert (N.i & test4 & N.i) == -2 * y
+
+
+def test_dyadic_srepr():
+    from sympy.printing.repr import srepr
+    N = CoordSys3D('N')
+
+    dy = N.i | N.j
+    res = "BaseDyadic(CoordSys3D(Str('N'), Tuple(ImmutableDenseMatrix([["\
+        "Integer(1), Integer(0), Integer(0)], [Integer(0), Integer(1), "\
+        "Integer(0)], [Integer(0), Integer(0), Integer(1)]]), "\
+        "VectorZero())).i, CoordSys3D(Str('N'), Tuple(ImmutableDenseMatrix("\
+        "[[Integer(1), Integer(0), Integer(0)], [Integer(0), Integer(1), "\
+        "Integer(0)], [Integer(0), Integer(0), Integer(1)]]), VectorZero())).j)"
+    assert srepr(dy) == res

rl/Lib/site-packages/sympy/vector/tests/test_field_functions.py

@@ -0,0 +1,321 @@
+from sympy.core.function import Derivative
+from sympy.vector.vector import Vector
+from sympy.vector.coordsysrect import CoordSys3D
+from sympy.simplify import simplify
+from sympy.core.symbol import symbols
+from sympy.core import S
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.vector.vector import Dot
+from sympy.vector.operators import curl, divergence, gradient, Gradient, Divergence, Cross
+from sympy.vector.deloperator import Del
+from sympy.vector.functions import (is_conservative, is_solenoidal,
+                                    scalar_potential, directional_derivative,
+                                    laplacian, scalar_potential_difference)
+from sympy.testing.pytest import raises
+
+C = CoordSys3D('C')
+i, j, k = C.base_vectors()
+x, y, z = C.base_scalars()
+delop = Del()
+a, b, c, q = symbols('a b c q')
+
+
+def test_del_operator():
+    # Tests for curl
+
+    assert delop ^ Vector.zero == Vector.zero
+    assert ((delop ^ Vector.zero).doit() == Vector.zero ==
+            curl(Vector.zero))
+    assert delop.cross(Vector.zero) == delop ^ Vector.zero
+    assert (delop ^ i).doit() == Vector.zero
+    assert delop.cross(2*y**2*j, doit=True) == Vector.zero
+    assert delop.cross(2*y**2*j) == delop ^ 2*y**2*j
+    v = x*y*z * (i + j + k)
+    assert ((delop ^ v).doit() ==
+            (-x*y + x*z)*i + (x*y - y*z)*j + (-x*z + y*z)*k ==
+            curl(v))
+    assert delop ^ v == delop.cross(v)
+    assert (delop.cross(2*x**2*j) ==
+            (Derivative(0, C.y) - Derivative(2*C.x**2, C.z))*C.i +
+            (-Derivative(0, C.x) + Derivative(0, C.z))*C.j +
+            (-Derivative(0, C.y) + Derivative(2*C.x**2, C.x))*C.k)
+    assert (delop.cross(2*x**2*j, doit=True) == 4*x*k ==
+            curl(2*x**2*j))
+
+    #Tests for divergence
+    assert delop & Vector.zero is S.Zero == divergence(Vector.zero)
+    assert (delop & Vector.zero).doit() is S.Zero
+    assert delop.dot(Vector.zero) == delop & Vector.zero
+    assert (delop & i).doit() is S.Zero
+    assert (delop & x**2*i).doit() == 2*x == divergence(x**2*i)
+    assert (delop.dot(v, doit=True) == x*y + y*z + z*x ==
+            divergence(v))
+    assert delop & v == delop.dot(v)
+    assert delop.dot(1/(x*y*z) * (i + j + k), doit=True) == \
+           - 1 / (x*y*z**2) - 1 / (x*y**2*z) - 1 / (x**2*y*z)
+    v = x*i + y*j + z*k
+    assert (delop & v == Derivative(C.x, C.x) +
+            Derivative(C.y, C.y) + Derivative(C.z, C.z))
+    assert delop.dot(v, doit=True) == 3 == divergence(v)
+    assert delop & v == delop.dot(v)
+    assert simplify((delop & v).doit()) == 3
+
+    #Tests for gradient
+    assert (delop.gradient(0, doit=True) == Vector.zero ==
+            gradient(0))
+    assert delop.gradient(0) == delop(0)
+    assert (delop(S.Zero)).doit() == Vector.zero
+    assert (delop(x) == (Derivative(C.x, C.x))*C.i +
+            (Derivative(C.x, C.y))*C.j + (Derivative(C.x, C.z))*C.k)
+    assert (delop(x)).doit() == i == gradient(x)
+    assert (delop(x*y*z) ==
+            (Derivative(C.x*C.y*C.z, C.x))*C.i +
+            (Derivative(C.x*C.y*C.z, C.y))*C.j +
+            (Derivative(C.x*C.y*C.z, C.z))*C.k)
+    assert (delop.gradient(x*y*z, doit=True) ==
+            y*z*i + z*x*j + x*y*k ==
+            gradient(x*y*z))
+    assert delop(x*y*z) == delop.gradient(x*y*z)
+    assert (delop(2*x**2)).doit() == 4*x*i
+    assert ((delop(a*sin(y) / x)).doit() ==
+            -a*sin(y)/x**2 * i + a*cos(y)/x * j)
+
+    #Tests for directional derivative
+    assert (Vector.zero & delop)(a) is S.Zero
+    assert ((Vector.zero & delop)(a)).doit() is S.Zero
+    assert ((v & delop)(Vector.zero)).doit() == Vector.zero
+    assert ((v & delop)(S.Zero)).doit() is S.Zero
+    assert ((i & delop)(x)).doit() == 1
+    assert ((j & delop)(y)).doit() == 1
+    assert ((k & delop)(z)).doit() == 1
+    assert ((i & delop)(x*y*z)).doit() == y*z
+    assert ((v & delop)(x)).doit() == x
+    assert ((v & delop)(x*y*z)).doit() == 3*x*y*z
+    assert (v & delop)(x + y + z) == C.x + C.y + C.z
+    assert ((v & delop)(x + y + z)).doit() == x + y + z
+    assert ((v & delop)(v)).doit() == v
+    assert ((i & delop)(v)).doit() == i
+    assert ((j & delop)(v)).doit() == j
+    assert ((k & delop)(v)).doit() == k
+    assert ((v & delop)(Vector.zero)).doit() == Vector.zero
+
+    # Tests for laplacian on scalar fields
+    assert laplacian(x*y*z) is S.Zero
+    assert laplacian(x**2) == S(2)
+    assert laplacian(x**2*y**2*z**2) == \
+                    2*y**2*z**2 + 2*x**2*z**2 + 2*x**2*y**2
+    A = CoordSys3D('A', transformation="spherical", variable_names=["r", "theta", "phi"])
+    B = CoordSys3D('B', transformation='cylindrical', variable_names=["r", "theta", "z"])
+    assert laplacian(A.r + A.theta + A.phi) == 2/A.r + cos(A.theta)/(A.r**2*sin(A.theta))
+    assert laplacian(B.r + B.theta + B.z) == 1/B.r
+
+    # Tests for laplacian on vector fields
+    assert laplacian(x*y*z*(i + j + k)) == Vector.zero
+    assert laplacian(x*y**2*z*(i + j + k)) == \
+                            2*x*z*i + 2*x*z*j + 2*x*z*k
+
+
+def test_product_rules():
+    """
+    Tests the six product rules defined with respect to the Del
+    operator
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Del
+
+    """
+
+    #Define the scalar and vector functions
+    f = 2*x*y*z
+    g = x*y + y*z + z*x
+    u = x**2*i + 4*j - y**2*z*k
+    v = 4*i + x*y*z*k
+
+    # First product rule
+    lhs = delop(f * g, doit=True)
+    rhs = (f * delop(g) + g * delop(f)).doit()
+    assert simplify(lhs) == simplify(rhs)
+
+    # Second product rule
+    lhs = delop(u & v).doit()
+    rhs = ((u ^ (delop ^ v)) + (v ^ (delop ^ u)) + \
+          ((u & delop)(v)) + ((v & delop)(u))).doit()
+    assert simplify(lhs) == simplify(rhs)
+
+    # Third product rule
+    lhs = (delop & (f*v)).doit()
+    rhs = ((f * (delop & v)) + (v & (delop(f)))).doit()
+    assert simplify(lhs) == simplify(rhs)
+
+    # Fourth product rule
+    lhs = (delop & (u ^ v)).doit()
+    rhs = ((v & (delop ^ u)) - (u & (delop ^ v))).doit()
+    assert simplify(lhs) == simplify(rhs)
+
+    # Fifth product rule
+    lhs = (delop ^ (f * v)).doit()
+    rhs = (((delop(f)) ^ v) + (f * (delop ^ v))).doit()
+    assert simplify(lhs) == simplify(rhs)
+
+    # Sixth product rule
+    lhs = (delop ^ (u ^ v)).doit()
+    rhs = (u * (delop & v) - v * (delop & u) +
+           (v & delop)(u) - (u & delop)(v)).doit()
+    assert simplify(lhs) == simplify(rhs)
+
+
+P = C.orient_new_axis('P', q, C.k)  # type: ignore
+scalar_field = 2*x**2*y*z
+grad_field = gradient(scalar_field)
+vector_field = y**2*i + 3*x*j + 5*y*z*k
+curl_field = curl(vector_field)
+
+
+def test_conservative():
+    assert is_conservative(Vector.zero) is True
+    assert is_conservative(i) is True
+    assert is_conservative(2 * i + 3 * j + 4 * k) is True
+    assert (is_conservative(y*z*i + x*z*j + x*y*k) is
+            True)
+    assert is_conservative(x * j) is False
+    assert is_conservative(grad_field) is True
+    assert is_conservative(curl_field) is False
+    assert (is_conservative(4*x*y*z*i + 2*x**2*z*j) is
+            False)
+    assert is_conservative(z*P.i + P.x*k) is True
+
+
+def test_solenoidal():
+    assert is_solenoidal(Vector.zero) is True
+    assert is_solenoidal(i) is True
+    assert is_solenoidal(2 * i + 3 * j + 4 * k) is True
+    assert (is_solenoidal(y*z*i + x*z*j + x*y*k) is
+            True)
+    assert is_solenoidal(y * j) is False
+    assert is_solenoidal(grad_field) is False
+    assert is_solenoidal(curl_field) is True
+    assert is_solenoidal((-2*y + 3)*k) is True
+    assert is_solenoidal(cos(q)*i + sin(q)*j + cos(q)*P.k) is True
+    assert is_solenoidal(z*P.i + P.x*k) is True
+
+
+def test_directional_derivative():
+    assert directional_derivative(C.x*C.y*C.z, 3*C.i + 4*C.j + C.k) == C.x*C.y + 4*C.x*C.z + 3*C.y*C.z
+    assert directional_derivative(5*C.x**2*C.z, 3*C.i + 4*C.j + C.k) == 5*C.x**2 + 30*C.x*C.z
+    assert directional_derivative(5*C.x**2*C.z, 4*C.j) is S.Zero
+
+    D = CoordSys3D("D", "spherical", variable_names=["r", "theta", "phi"],
+                   vector_names=["e_r", "e_theta", "e_phi"])
+    r, theta, phi = D.base_scalars()
+    e_r, e_theta, e_phi = D.base_vectors()
+    assert directional_derivative(r**2*e_r, e_r) == 2*r*e_r
+    assert directional_derivative(5*r**2*phi, 3*e_r + 4*e_theta + e_phi) == 5*r**2 + 30*r*phi
+
+
+def test_scalar_potential():
+    assert scalar_potential(Vector.zero, C) == 0
+    assert scalar_potential(i, C) == x
+    assert scalar_potential(j, C) == y
+    assert scalar_potential(k, C) == z
+    assert scalar_potential(y*z*i + x*z*j + x*y*k, C) == x*y*z
+    assert scalar_potential(grad_field, C) == scalar_field
+    assert scalar_potential(z*P.i + P.x*k, C) == x*z*cos(q) + y*z*sin(q)
+    assert scalar_potential(z*P.i + P.x*k, P) == P.x*P.z
+    raises(ValueError, lambda: scalar_potential(x*j, C))
+
+
+def test_scalar_potential_difference():
+    point1 = C.origin.locate_new('P1', 1*i + 2*j + 3*k)
+    point2 = C.origin.locate_new('P2', 4*i + 5*j + 6*k)
+    genericpointC = C.origin.locate_new('RP', x*i + y*j + z*k)
+    genericpointP = P.origin.locate_new('PP', P.x*P.i + P.y*P.j + P.z*P.k)
+    assert scalar_potential_difference(S.Zero, C, point1, point2) == 0
+    assert (scalar_potential_difference(scalar_field, C, C.origin,
+                                        genericpointC) ==
+            scalar_field)
+    assert (scalar_potential_difference(grad_field, C, C.origin,
+                                        genericpointC) ==
+            scalar_field)
+    assert scalar_potential_difference(grad_field, C, point1, point2) == 948
+    assert (scalar_potential_difference(y*z*i + x*z*j +
+                                        x*y*k, C, point1,
+                                        genericpointC) ==
+            x*y*z - 6)
+    potential_diff_P = (2*P.z*(P.x*sin(q) + P.y*cos(q))*
+                        (P.x*cos(q) - P.y*sin(q))**2)
+    assert (scalar_potential_difference(grad_field, P, P.origin,
+                                        genericpointP).simplify() ==
+            potential_diff_P.simplify())
+
+
+def test_differential_operators_curvilinear_system():
+    A = CoordSys3D('A', transformation="spherical", variable_names=["r", "theta", "phi"])
+    B = CoordSys3D('B', transformation='cylindrical', variable_names=["r", "theta", "z"])
+    # Test for spherical coordinate system and gradient
+    assert gradient(3*A.r + 4*A.theta) == 3*A.i + 4/A.r*A.j
+    assert gradient(3*A.r*A.phi + 4*A.theta) == 3*A.phi*A.i + 4/A.r*A.j + (3/sin(A.theta))*A.k
+    assert gradient(0*A.r + 0*A.theta+0*A.phi) == Vector.zero
+    assert gradient(A.r*A.theta*A.phi) == A.theta*A.phi*A.i + A.phi*A.j + (A.theta/sin(A.theta))*A.k
+    # Test for spherical coordinate system and divergence
+    assert divergence(A.r * A.i + A.theta * A.j + A.phi * A.k) == \
+           (sin(A.theta)*A.r + cos(A.theta)*A.r*A.theta)/(sin(A.theta)*A.r**2) + 3 + 1/(sin(A.theta)*A.r)
+    assert divergence(3*A.r*A.phi*A.i + A.theta*A.j + A.r*A.theta*A.phi*A.k) == \
+           (sin(A.theta)*A.r + cos(A.theta)*A.r*A.theta)/(sin(A.theta)*A.r**2) + 9*A.phi + A.theta/sin(A.theta)
+    assert divergence(Vector.zero) == 0
+    assert divergence(0*A.i + 0*A.j + 0*A.k) == 0
+    # Test for spherical coordinate system and curl
+    assert curl(A.r*A.i + A.theta*A.j + A.phi*A.k) == \
+           (cos(A.theta)*A.phi/(sin(A.theta)*A.r))*A.i + (-A.phi/A.r)*A.j + A.theta/A.r*A.k
+    assert curl(A.r*A.j + A.phi*A.k) == (cos(A.theta)*A.phi/(sin(A.theta)*A.r))*A.i + (-A.phi/A.r)*A.j + 2*A.k
+
+    # Test for cylindrical coordinate system and gradient
+    assert gradient(0*B.r + 0*B.theta+0*B.z) == Vector.zero
+    assert gradient(B.r*B.theta*B.z) == B.theta*B.z*B.i + B.z*B.j + B.r*B.theta*B.k
+    assert gradient(3*B.r) == 3*B.i
+    assert gradient(2*B.theta) == 2/B.r * B.j
+    assert gradient(4*B.z) == 4*B.k
+    # Test for cylindrical coordinate system and divergence
+    assert divergence(B.r*B.i + B.theta*B.j + B.z*B.k) == 3 + 1/B.r
+    assert divergence(B.r*B.j + B.z*B.k) == 1
+    # Test for cylindrical coordinate system and curl
+    assert curl(B.r*B.j + B.z*B.k) == 2*B.k
+    assert curl(3*B.i + 2/B.r*B.j + 4*B.k) == Vector.zero
+
+def test_mixed_coordinates():
+    # gradient
+    a = CoordSys3D('a')
+    b = CoordSys3D('b')
+    c = CoordSys3D('c')
+    assert gradient(a.x*b.y) == b.y*a.i + a.x*b.j
+    assert gradient(3*cos(q)*a.x*b.x+a.y*(a.x+(cos(q)+b.x))) ==\
+           (a.y + 3*b.x*cos(q))*a.i + (a.x + b.x + cos(q))*a.j + (3*a.x*cos(q) + a.y)*b.i
+    # Some tests need further work:
+    # assert gradient(a.x*(cos(a.x+b.x))) == (cos(a.x + b.x))*a.i + a.x*Gradient(cos(a.x + b.x))
+    # assert gradient(cos(a.x + b.x)*cos(a.x + b.z)) == Gradient(cos(a.x + b.x)*cos(a.x + b.z))
+    assert gradient(a.x**b.y) == Gradient(a.x**b.y)
+    # assert gradient(cos(a.x+b.y)*a.z) == None
+    assert gradient(cos(a.x*b.y)) == Gradient(cos(a.x*b.y))
+    assert gradient(3*cos(q)*a.x*b.x*a.z*a.y+ b.y*b.z + cos(a.x+a.y)*b.z) == \
+           (3*a.y*a.z*b.x*cos(q) - b.z*sin(a.x + a.y))*a.i + \
+           (3*a.x*a.z*b.x*cos(q) - b.z*sin(a.x + a.y))*a.j + (3*a.x*a.y*b.x*cos(q))*a.k + \
+           (3*a.x*a.y*a.z*cos(q))*b.i + b.z*b.j + (b.y + cos(a.x + a.y))*b.k
+    # divergence
+    assert divergence(a.i*a.x+a.j*a.y+a.z*a.k + b.i*b.x+b.j*b.y+b.z*b.k + c.i*c.x+c.j*c.y+c.z*c.k) == S(9)
+    # assert divergence(3*a.i*a.x*cos(a.x+b.z) + a.j*b.x*c.z) == None
+    assert divergence(3*a.i*a.x*a.z + b.j*b.x*c.z + 3*a.j*a.z*a.y) == \
+            6*a.z + b.x*Dot(b.j, c.k)
+    assert divergence(3*cos(q)*a.x*b.x*b.i*c.x) == \
+        3*a.x*b.x*cos(q)*Dot(b.i, c.i) + 3*a.x*c.x*cos(q) + 3*b.x*c.x*cos(q)*Dot(b.i, a.i)
+    assert divergence(a.x*b.x*c.x*Cross(a.x*a.i, a.y*b.j)) ==\
+           a.x*b.x*c.x*Divergence(Cross(a.x*a.i, a.y*b.j)) + \
+           b.x*c.x*Dot(Cross(a.x*a.i, a.y*b.j), a.i) + \
+           a.x*c.x*Dot(Cross(a.x*a.i, a.y*b.j), b.i) + \
+           a.x*b.x*Dot(Cross(a.x*a.i, a.y*b.j), c.i)
+    assert divergence(a.x*b.x*c.x*(a.x*a.i + b.x*b.i)) == \
+                4*a.x*b.x*c.x +\
+                a.x**2*c.x*Dot(a.i, b.i) +\
+                a.x**2*b.x*Dot(a.i, c.i) +\
+                b.x**2*c.x*Dot(b.i, a.i) +\
+                a.x*b.x**2*Dot(b.i, c.i)

rl/Lib/site-packages/sympy/vector/tests/test_functions.py

@@ -0,0 +1,184 @@
+from sympy.vector.vector import Vector
+from sympy.vector.coordsysrect import CoordSys3D
+from sympy.vector.functions import express, matrix_to_vector, orthogonalize
+from sympy.core.numbers import Rational
+from sympy.core.singleton import S
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix
+from sympy.testing.pytest import raises
+
+N = CoordSys3D('N')
+q1, q2, q3, q4, q5 = symbols('q1 q2 q3 q4 q5')
+A = N.orient_new_axis('A', q1, N.k)  # type: ignore
+B = A.orient_new_axis('B', q2, A.i)
+C = B.orient_new_axis('C', q3, B.j)
+
+
+def test_express():
+    assert express(Vector.zero, N) == Vector.zero
+    assert express(S.Zero, N) is S.Zero
+    assert express(A.i, C) == cos(q3)*C.i + sin(q3)*C.k
+    assert express(A.j, C) == sin(q2)*sin(q3)*C.i + cos(q2)*C.j - \
+        sin(q2)*cos(q3)*C.k
+    assert express(A.k, C) == -sin(q3)*cos(q2)*C.i + sin(q2)*C.j + \
+        cos(q2)*cos(q3)*C.k
+    assert express(A.i, N) == cos(q1)*N.i + sin(q1)*N.j
+    assert express(A.j, N) == -sin(q1)*N.i + cos(q1)*N.j
+    assert express(A.k, N) == N.k
+    assert express(A.i, A) == A.i
+    assert express(A.j, A) == A.j
+    assert express(A.k, A) == A.k
+    assert express(A.i, B) == B.i
+    assert express(A.j, B) == cos(q2)*B.j - sin(q2)*B.k
+    assert express(A.k, B) == sin(q2)*B.j + cos(q2)*B.k
+    assert express(A.i, C) == cos(q3)*C.i + sin(q3)*C.k
+    assert express(A.j, C) == sin(q2)*sin(q3)*C.i + cos(q2)*C.j - \
+        sin(q2)*cos(q3)*C.k
+    assert express(A.k, C) == -sin(q3)*cos(q2)*C.i + sin(q2)*C.j + \
+        cos(q2)*cos(q3)*C.k
+    # Check to make sure UnitVectors get converted properly
+    assert express(N.i, N) == N.i
+    assert express(N.j, N) == N.j
+    assert express(N.k, N) == N.k
+    assert express(N.i, A) == (cos(q1)*A.i - sin(q1)*A.j)
+    assert express(N.j, A) == (sin(q1)*A.i + cos(q1)*A.j)
+    assert express(N.k, A) == A.k
+    assert express(N.i, B) == (cos(q1)*B.i - sin(q1)*cos(q2)*B.j +
+            sin(q1)*sin(q2)*B.k)
+    assert express(N.j, B) == (sin(q1)*B.i + cos(q1)*cos(q2)*B.j -
+            sin(q2)*cos(q1)*B.k)
+    assert express(N.k, B) == (sin(q2)*B.j + cos(q2)*B.k)
+    assert express(N.i, C) == (
+        (cos(q1)*cos(q3) - sin(q1)*sin(q2)*sin(q3))*C.i -
+        sin(q1)*cos(q2)*C.j +
+        (sin(q3)*cos(q1) + sin(q1)*sin(q2)*cos(q3))*C.k)
+    assert express(N.j, C) == (
+        (sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*C.i +
+        cos(q1)*cos(q2)*C.j +
+        (sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*C.k)
+    assert express(N.k, C) == (-sin(q3)*cos(q2)*C.i + sin(q2)*C.j +
+            cos(q2)*cos(q3)*C.k)
+
+    assert express(A.i, N) == (cos(q1)*N.i + sin(q1)*N.j)
+    assert express(A.j, N) == (-sin(q1)*N.i + cos(q1)*N.j)
+    assert express(A.k, N) == N.k
+    assert express(A.i, A) == A.i
+    assert express(A.j, A) == A.j
+    assert express(A.k, A) == A.k
+    assert express(A.i, B) == B.i
+    assert express(A.j, B) == (cos(q2)*B.j - sin(q2)*B.k)
+    assert express(A.k, B) == (sin(q2)*B.j + cos(q2)*B.k)
+    assert express(A.i, C) == (cos(q3)*C.i + sin(q3)*C.k)
+    assert express(A.j, C) == (sin(q2)*sin(q3)*C.i + cos(q2)*C.j -
+            sin(q2)*cos(q3)*C.k)
+    assert express(A.k, C) == (-sin(q3)*cos(q2)*C.i + sin(q2)*C.j +
+            cos(q2)*cos(q3)*C.k)
+
+    assert express(B.i, N) == (cos(q1)*N.i + sin(q1)*N.j)
+    assert express(B.j, N) == (-sin(q1)*cos(q2)*N.i +
+            cos(q1)*cos(q2)*N.j + sin(q2)*N.k)
+    assert express(B.k, N) == (sin(q1)*sin(q2)*N.i -
+            sin(q2)*cos(q1)*N.j + cos(q2)*N.k)
+    assert express(B.i, A) == A.i
+    assert express(B.j, A) == (cos(q2)*A.j + sin(q2)*A.k)
+    assert express(B.k, A) == (-sin(q2)*A.j + cos(q2)*A.k)
+    assert express(B.i, B) == B.i
+    assert express(B.j, B) == B.j
+    assert express(B.k, B) == B.k
+    assert express(B.i, C) == (cos(q3)*C.i + sin(q3)*C.k)
+    assert express(B.j, C) == C.j
+    assert express(B.k, C) == (-sin(q3)*C.i + cos(q3)*C.k)
+
+    assert express(C.i, N) == (
+        (cos(q1)*cos(q3) - sin(q1)*sin(q2)*sin(q3))*N.i +
+        (sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*N.j -
+        sin(q3)*cos(q2)*N.k)
+    assert express(C.j, N) == (
+        -sin(q1)*cos(q2)*N.i + cos(q1)*cos(q2)*N.j + sin(q2)*N.k)
+    assert express(C.k, N) == (
+        (sin(q3)*cos(q1) + sin(q1)*sin(q2)*cos(q3))*N.i +
+        (sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*N.j +
+        cos(q2)*cos(q3)*N.k)
+    assert express(C.i, A) == (cos(q3)*A.i + sin(q2)*sin(q3)*A.j -
+            sin(q3)*cos(q2)*A.k)
+    assert express(C.j, A) == (cos(q2)*A.j + sin(q2)*A.k)
+    assert express(C.k, A) == (sin(q3)*A.i - sin(q2)*cos(q3)*A.j +
+            cos(q2)*cos(q3)*A.k)
+    assert express(C.i, B) == (cos(q3)*B.i - sin(q3)*B.k)
+    assert express(C.j, B) == B.j
+    assert express(C.k, B) == (sin(q3)*B.i + cos(q3)*B.k)
+    assert express(C.i, C) == C.i
+    assert express(C.j, C) == C.j
+    assert express(C.k, C) == C.k == (C.k)
+
+    #  Check to make sure Vectors get converted back to UnitVectors
+    assert N.i == express((cos(q1)*A.i - sin(q1)*A.j), N).simplify()
+    assert N.j == express((sin(q1)*A.i + cos(q1)*A.j), N).simplify()
+    assert N.i == express((cos(q1)*B.i - sin(q1)*cos(q2)*B.j +
+            sin(q1)*sin(q2)*B.k), N).simplify()
+    assert N.j == express((sin(q1)*B.i + cos(q1)*cos(q2)*B.j -
+        sin(q2)*cos(q1)*B.k), N).simplify()
+    assert N.k == express((sin(q2)*B.j + cos(q2)*B.k), N).simplify()
+
+
+    assert A.i == express((cos(q1)*N.i + sin(q1)*N.j), A).simplify()
+    assert A.j == express((-sin(q1)*N.i + cos(q1)*N.j), A).simplify()
+
+    assert A.j == express((cos(q2)*B.j - sin(q2)*B.k), A).simplify()
+    assert A.k == express((sin(q2)*B.j + cos(q2)*B.k), A).simplify()
+
+    assert A.i == express((cos(q3)*C.i + sin(q3)*C.k), A).simplify()
+    assert A.j == express((sin(q2)*sin(q3)*C.i + cos(q2)*C.j -
+            sin(q2)*cos(q3)*C.k), A).simplify()
+
+    assert A.k == express((-sin(q3)*cos(q2)*C.i + sin(q2)*C.j +
+            cos(q2)*cos(q3)*C.k), A).simplify()
+    assert B.i == express((cos(q1)*N.i + sin(q1)*N.j), B).simplify()
+    assert B.j == express((-sin(q1)*cos(q2)*N.i +
+            cos(q1)*cos(q2)*N.j + sin(q2)*N.k), B).simplify()
+
+    assert B.k == express((sin(q1)*sin(q2)*N.i -
+            sin(q2)*cos(q1)*N.j + cos(q2)*N.k), B).simplify()
+
+    assert B.j == express((cos(q2)*A.j + sin(q2)*A.k), B).simplify()
+    assert B.k == express((-sin(q2)*A.j + cos(q2)*A.k), B).simplify()
+    assert B.i == express((cos(q3)*C.i + sin(q3)*C.k), B).simplify()
+    assert B.k == express((-sin(q3)*C.i + cos(q3)*C.k), B).simplify()
+    assert C.i == express((cos(q3)*A.i + sin(q2)*sin(q3)*A.j -
+            sin(q3)*cos(q2)*A.k), C).simplify()
+    assert C.j == express((cos(q2)*A.j + sin(q2)*A.k), C).simplify()
+    assert C.k == express((sin(q3)*A.i - sin(q2)*cos(q3)*A.j +
+            cos(q2)*cos(q3)*A.k), C).simplify()
+    assert C.i == express((cos(q3)*B.i - sin(q3)*B.k), C).simplify()
+    assert C.k == express((sin(q3)*B.i + cos(q3)*B.k), C).simplify()
+
+
+def test_matrix_to_vector():
+    m = Matrix([[1], [2], [3]])
+    assert matrix_to_vector(m, C) == C.i + 2*C.j + 3*C.k
+    m = Matrix([[0], [0], [0]])
+    assert matrix_to_vector(m, N) == matrix_to_vector(m, C) == \
+           Vector.zero
+    m = Matrix([[q1], [q2], [q3]])
+    assert matrix_to_vector(m, N) == q1*N.i + q2*N.j + q3*N.k
+
+
+def test_orthogonalize():
+    C = CoordSys3D('C')
+    a, b = symbols('a b', integer=True)
+    i, j, k = C.base_vectors()
+    v1 = i + 2*j
+    v2 = 2*i + 3*j
+    v3 = 3*i + 5*j
+    v4 = 3*i + j
+    v5 = 2*i + 2*j
+    v6 = a*i + b*j
+    v7 = 4*a*i + 4*b*j
+    assert orthogonalize(v1, v2) == [C.i + 2*C.j, C.i*Rational(2, 5) + -C.j/5]
+    # from wikipedia
+    assert orthogonalize(v4, v5, orthonormal=True) == \
+        [(3*sqrt(10))*C.i/10 + (sqrt(10))*C.j/10, (-sqrt(10))*C.i/10 + (3*sqrt(10))*C.j/10]
+    raises(ValueError, lambda: orthogonalize(v1, v2, v3))
+    raises(ValueError, lambda: orthogonalize(v6, v7))

rl/Lib/site-packages/sympy/vector/tests/test_implicitregion.py

@@ -0,0 +1,90 @@
+from sympy.core.relational import Eq
+from sympy.core.singleton import S
+from sympy.abc import x, y, z, s, t
+from sympy.sets import FiniteSet, EmptySet
+from sympy.geometry import Point
+from sympy.vector import ImplicitRegion
+from sympy.testing.pytest import raises
+
+
+def test_ImplicitRegion():
+    ellipse = ImplicitRegion((x, y), (x**2/4 + y**2/16 - 1))
+    assert ellipse.equation == x**2/4 + y**2/16 - 1
+    assert ellipse.variables == (x, y)
+    assert ellipse.degree == 2
+    r = ImplicitRegion((x, y, z), Eq(x**4 + y**2 - x*y, 6))
+    assert r.equation == x**4 + y**2 - x*y - 6
+    assert r.variables == (x, y, z)
+    assert r.degree == 4
+
+
+def test_regular_point():
+    r1 = ImplicitRegion((x,), x**2 - 16)
+    assert r1.regular_point() == (-4,)
+    c1 = ImplicitRegion((x, y), x**2 + y**2 - 4)
+    assert c1.regular_point() == (0, -2)
+    c2 = ImplicitRegion((x, y), (x - S(5)/2)**2 + y**2 - (S(1)/4)**2)
+    assert c2.regular_point() == (S(5)/2, -S(1)/4)
+    c3 = ImplicitRegion((x, y), (y - 5)**2  - 16*(x - 5))
+    assert c3.regular_point() == (5, 5)
+    r2 = ImplicitRegion((x, y), x**2 - 4*x*y - 3*y**2 + 4*x + 8*y - 5)
+    assert r2.regular_point() == (S(4)/7, S(9)/7)
+    r3 = ImplicitRegion((x, y), x**2 - 2*x*y + 3*y**2 - 2*x - 5*y + 3/2)
+    raises(ValueError, lambda: r3.regular_point())
+
+
+def test_singular_points_and_multiplicty():
+    r1 = ImplicitRegion((x, y, z), Eq(x + y + z, 0))
+    assert r1.singular_points() == EmptySet
+    r2 = ImplicitRegion((x, y, z), x*y*z + y**4 -x**2*z**2)
+    assert r2.singular_points() == FiniteSet((0, 0, z), (x, 0, 0))
+    assert r2.multiplicity((0, 0, 0)) == 3
+    assert r2.multiplicity((0, 0, 6)) == 2
+    r3 = ImplicitRegion((x, y, z), z**2 - x**2 - y**2)
+    assert r3.singular_points() == FiniteSet((0, 0, 0))
+    assert r3.multiplicity((0, 0, 0)) == 2
+    r4 = ImplicitRegion((x, y), x**2 + y**2 - 2*x)
+    assert r4.singular_points() == EmptySet
+    assert r4.multiplicity(Point(1, 3)) == 0
+
+
+def test_rational_parametrization():
+    p = ImplicitRegion((x,), x - 2)
+    assert p.rational_parametrization() == (x - 2,)
+
+    line = ImplicitRegion((x, y), Eq(y, 3*x + 2))
+    assert line.rational_parametrization() == (x, 3*x + 2)
+
+    circle1 = ImplicitRegion((x, y), (x-2)**2 + (y+3)**2 - 4)
+    assert circle1.rational_parametrization(parameters=t) == (4*t/(t**2 + 1) + 2, 4*t**2/(t**2 + 1) - 5)
+    circle2 = ImplicitRegion((x, y), (x - S.Half)**2 + y**2 - (S(1)/2)**2)
+
+    assert circle2.rational_parametrization(parameters=t) == (t/(t**2 + 1) + S(1)/2, t**2/(t**2 + 1) - S(1)/2)
+    circle3 = ImplicitRegion((x, y), Eq(x**2 + y**2, 2*x))
+    assert circle3.rational_parametrization(parameters=(t,)) == (2*t/(t**2 + 1) + 1, 2*t**2/(t**2 + 1) - 1)
+
+    parabola = ImplicitRegion((x, y), (y - 3)**2 - 4*(x + 6))
+    assert parabola.rational_parametrization(t) == (-6 + 4/t**2, 3 + 4/t)
+
+    rect_hyperbola = ImplicitRegion((x, y), x*y - 1)
+    assert rect_hyperbola.rational_parametrization(t) == (-1 + (t + 1)/t, t)
+
+    cubic_curve = ImplicitRegion((x, y), x**3 + x**2 - y**2)
+    assert cubic_curve.rational_parametrization(parameters=(t)) == (t**2 - 1, t*(t**2 - 1))
+    cuspidal = ImplicitRegion((x, y), (x**3 - y**2))
+    assert cuspidal.rational_parametrization(t) == (t**2, t**3)
+
+    I = ImplicitRegion((x, y), x**3 + x**2 - y**2)
+    assert I.rational_parametrization(t) == (t**2 - 1, t*(t**2 - 1))
+
+    sphere = ImplicitRegion((x, y, z), Eq(x**2 + y**2 + z**2, 2*x))
+    assert sphere.rational_parametrization(parameters=(s, t)) == (2/(s**2 + t**2 + 1), 2*t/(s**2 + t**2 + 1), 2*s/(s**2 + t**2 + 1))
+
+    conic = ImplicitRegion((x, y), Eq(x**2 + 4*x*y + 3*y**2 + x - y + 10, 0))
+    assert conic.rational_parametrization(t) == (
+        S(17)/2 + 4/(3*t**2 + 4*t + 1), 4*t/(3*t**2 + 4*t + 1) - S(11)/2)
+
+    r1 = ImplicitRegion((x, y), y**2 - x**3 + x)
+    raises(NotImplementedError, lambda: r1.rational_parametrization())
+    r2 = ImplicitRegion((x, y), y**2 - x**3 - x**2 + 1)
+    raises(NotImplementedError, lambda: r2.rational_parametrization())

rl/Lib/site-packages/sympy/vector/tests/test_integrals.py

@@ -0,0 +1,106 @@
+from sympy.core.numbers import pi
+from sympy.core.singleton import S
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.testing.pytest import raises
+from sympy.vector.coordsysrect import CoordSys3D
+from sympy.vector.integrals import ParametricIntegral, vector_integrate
+from sympy.vector.parametricregion import ParametricRegion
+from sympy.vector.implicitregion import ImplicitRegion
+from sympy.abc import x, y, z, u, v, r, t, theta, phi
+from sympy.geometry import Point, Segment, Curve, Circle, Polygon, Plane
+
+C = CoordSys3D('C')
+
+def test_parametric_lineintegrals():
+    halfcircle = ParametricRegion((4*cos(theta), 4*sin(theta)), (theta, -pi/2, pi/2))
+    assert ParametricIntegral(C.x*C.y**4, halfcircle) == S(8192)/5
+
+    curve = ParametricRegion((t, t**2, t**3), (t, 0, 1))
+    field1 = 8*C.x**2*C.y*C.z*C.i + 5*C.z*C.j - 4*C.x*C.y*C.k
+    assert ParametricIntegral(field1, curve) == 1
+    line = ParametricRegion((4*t - 1, 2 - 2*t, t), (t, 0, 1))
+    assert ParametricIntegral(C.x*C.z*C.i - C.y*C.z*C.k, line) == 3
+
+    assert ParametricIntegral(4*C.x**3, ParametricRegion((1, t), (t, 0, 2))) == 8
+
+    helix = ParametricRegion((cos(t), sin(t), 3*t), (t, 0, 4*pi))
+    assert ParametricIntegral(C.x*C.y*C.z, helix) == -3*sqrt(10)*pi
+
+    field2 = C.y*C.i + C.z*C.j + C.z*C.k
+    assert ParametricIntegral(field2, ParametricRegion((cos(t), sin(t), t**2), (t, 0, pi))) == -5*pi/2 + pi**4/2
+
+def test_parametric_surfaceintegrals():
+
+    semisphere = ParametricRegion((2*sin(phi)*cos(theta), 2*sin(phi)*sin(theta), 2*cos(phi)),\
+                            (theta, 0, 2*pi), (phi, 0, pi/2))
+    assert ParametricIntegral(C.z, semisphere) == 8*pi
+
+    cylinder = ParametricRegion((sqrt(3)*cos(theta), sqrt(3)*sin(theta), z), (z, 0, 6), (theta, 0, 2*pi))
+    assert ParametricIntegral(C.y, cylinder) == 0
+
+    cone = ParametricRegion((v*cos(u), v*sin(u), v), (u, 0, 2*pi), (v, 0, 1))
+    assert ParametricIntegral(C.x*C.i + C.y*C.j + C.z**4*C.k, cone) == pi/3
+
+    triangle1 = ParametricRegion((x, y), (x, 0, 2), (y, 0, 10 - 5*x))
+    triangle2 = ParametricRegion((x, y), (y, 0, 10 - 5*x), (x, 0, 2))
+    assert ParametricIntegral(-15.6*C.y*C.k, triangle1) == ParametricIntegral(-15.6*C.y*C.k, triangle2)
+    assert ParametricIntegral(C.z, triangle1) == 10*C.z
+
+def test_parametric_volumeintegrals():
+
+    cube = ParametricRegion((x, y, z), (x, 0, 1), (y, 0, 1), (z, 0, 1))
+    assert ParametricIntegral(1, cube) == 1
+
+    solidsphere1 = ParametricRegion((r*sin(phi)*cos(theta), r*sin(phi)*sin(theta), r*cos(phi)),\
+                            (r, 0, 2), (theta, 0, 2*pi), (phi, 0, pi))
+    solidsphere2 = ParametricRegion((r*sin(phi)*cos(theta), r*sin(phi)*sin(theta), r*cos(phi)),\
+                            (r, 0, 2), (phi, 0, pi), (theta, 0, 2*pi))
+    assert ParametricIntegral(C.x**2 + C.y**2, solidsphere1) == -256*pi/15
+    assert ParametricIntegral(C.x**2 + C.y**2, solidsphere2) == 256*pi/15
+
+    region_under_plane1 = ParametricRegion((x, y, z), (x, 0, 3), (y, 0, -2*x/3 + 2),\
+                                    (z, 0, 6 - 2*x - 3*y))
+    region_under_plane2 = ParametricRegion((x, y, z), (x, 0, 3), (z, 0, 6 - 2*x - 3*y),\
+                                    (y, 0, -2*x/3 + 2))
+
+    assert ParametricIntegral(C.x*C.i + C.j - 100*C.k, region_under_plane1) == \
+        ParametricIntegral(C.x*C.i + C.j - 100*C.k, region_under_plane2)
+    assert ParametricIntegral(2*C.x, region_under_plane2) == -9
+
+def test_vector_integrate():
+    halfdisc = ParametricRegion((r*cos(theta), r* sin(theta)), (r, -2, 2), (theta, 0, pi))
+    assert vector_integrate(C.x**2, halfdisc) == 4*pi
+    assert vector_integrate(C.x, ParametricRegion((t, t**2), (t, 2, 3))) == -17*sqrt(17)/12 + 37*sqrt(37)/12
+
+    assert  vector_integrate(C.y**3*C.z, (C.x, 0, 3), (C.y, -1, 4)) == 765*C.z/4
+
+    s1 = Segment(Point(0, 0), Point(0, 1))
+    assert vector_integrate(-15*C.y, s1) == S(-15)/2
+    s2 = Segment(Point(4, 3, 9), Point(1, 1, 7))
+    assert vector_integrate(C.y*C.i, s2) == -6
+
+    curve = Curve((sin(t), cos(t)), (t, 0, 2))
+    assert vector_integrate(5*C.z, curve) == 10*C.z
+
+    c1 = Circle(Point(2, 3), 6)
+    assert vector_integrate(C.x*C.y, c1) == 72*pi
+    c2 = Circle(Point(0, 0), Point(1, 1), Point(1, 0))
+    assert vector_integrate(1, c2) == c2.circumference
+
+    triangle = Polygon((0, 0), (1, 0), (1, 1))
+    assert vector_integrate(C.x*C.i - 14*C.y*C.j, triangle) == 0
+    p1, p2, p3, p4 = [(0, 0), (1, 0), (5, 1), (0, 1)]
+    poly = Polygon(p1, p2, p3, p4)
+    assert vector_integrate(-23*C.z, poly) == -161*C.z - 23*sqrt(17)*C.z
+
+    point = Point(2, 3)
+    assert vector_integrate(C.i*C.y - C.z, point) == ParametricIntegral(C.y*C.i, ParametricRegion((2, 3)))
+
+    c3 = ImplicitRegion((x, y), x**2 + y**2 - 4)
+    assert vector_integrate(45, c3) == 180*pi
+    c4 = ImplicitRegion((x, y), (x - 3)**2 + (y - 4)**2 - 9)
+    assert vector_integrate(1, c4) == 6*pi
+
+    pl = Plane(Point(1, 1, 1), Point(2, 3, 4), Point(2, 2, 2))
+    raises(ValueError, lambda: vector_integrate(C.x*C.z*C.i + C.k, pl))

rl/Lib/site-packages/sympy/vector/tests/test_operators.py

@@ -0,0 +1,43 @@
+from sympy.vector import CoordSys3D, Gradient, Divergence, Curl, VectorZero, Laplacian
+from sympy.printing.repr import srepr
+
+R = CoordSys3D('R')
+s1 = R.x*R.y*R.z  # type: ignore
+s2 = R.x + 3*R.y**2  # type: ignore
+s3 = R.x**2 + R.y**2 + R.z**2  # type: ignore
+v1 = R.x*R.i + R.z*R.z*R.j  # type: ignore
+v2 = R.x*R.i + R.y*R.j + R.z*R.k  # type: ignore
+v3 = R.x**2*R.i + R.y**2*R.j + R.z**2*R.k  # type: ignore
+
+
+def test_Gradient():
+    assert Gradient(s1) == Gradient(R.x*R.y*R.z)
+    assert Gradient(s2) == Gradient(R.x + 3*R.y**2)
+    assert Gradient(s1).doit() == R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k
+    assert Gradient(s2).doit() == R.i + 6*R.y*R.j
+
+
+def test_Divergence():
+    assert Divergence(v1) == Divergence(R.x*R.i + R.z*R.z*R.j)
+    assert Divergence(v2) == Divergence(R.x*R.i + R.y*R.j + R.z*R.k)
+    assert Divergence(v1).doit() == 1
+    assert Divergence(v2).doit() == 3
+    # issue 22384
+    Rc = CoordSys3D('R', transformation='cylindrical')
+    assert Divergence(Rc.i).doit() == 1/Rc.r
+
+
+def test_Curl():
+    assert Curl(v1) == Curl(R.x*R.i + R.z*R.z*R.j)
+    assert Curl(v2) == Curl(R.x*R.i + R.y*R.j + R.z*R.k)
+    assert Curl(v1).doit() == (-2*R.z)*R.i
+    assert Curl(v2).doit() == VectorZero()
+
+
+def test_Laplacian():
+    assert Laplacian(s3) == Laplacian(R.x**2 + R.y**2 + R.z**2)
+    assert Laplacian(v3) == Laplacian(R.x**2*R.i + R.y**2*R.j + R.z**2*R.k)
+    assert Laplacian(s3).doit() == 6
+    assert Laplacian(v3).doit() == 2*R.i + 2*R.j + 2*R.k
+    assert srepr(Laplacian(s3)) == \
+            'Laplacian(Add(Pow(R.x, Integer(2)), Pow(R.y, Integer(2)), Pow(R.z, Integer(2))))'

rl/Lib/site-packages/sympy/vector/tests/test_parametricregion.py

@@ -0,0 +1,97 @@
+from sympy.core.numbers import pi
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.vector.coordsysrect import CoordSys3D
+from sympy.vector.parametricregion import ParametricRegion, parametric_region_list
+from sympy.geometry import Point, Segment, Curve, Ellipse, Line, Parabola, Polygon
+from sympy.testing.pytest import raises
+from sympy.abc import a, b, r, t, x, y, z, theta, phi
+
+
+C = CoordSys3D('C')
+
+def test_ParametricRegion():
+
+    point = ParametricRegion((3, 4))
+    assert point.definition == (3, 4)
+    assert point.parameters == ()
+    assert point.limits == {}
+    assert point.dimensions == 0
+
+    # line x = y
+    line_xy = ParametricRegion((y, y), (y, 1, 5))
+    assert line_xy .definition == (y, y)
+    assert line_xy.parameters == (y,)
+    assert line_xy.dimensions == 1
+
+    # line y = z
+    line_yz = ParametricRegion((x,t,t), x, (t, 1, 2))
+    assert line_yz.definition == (x,t,t)
+    assert line_yz.parameters == (x, t)
+    assert line_yz.limits == {t: (1, 2)}
+    assert line_yz.dimensions == 1
+
+    p1 = ParametricRegion((9*a, -16*b), (a, 0, 2), (b, -1, 5))
+    assert p1.definition == (9*a, -16*b)
+    assert p1.parameters == (a, b)
+    assert p1.limits == {a: (0, 2), b: (-1, 5)}
+    assert p1.dimensions == 2
+
+    p2 = ParametricRegion((t, t**3), t)
+    assert p2.parameters == (t,)
+    assert p2.limits == {}
+    assert p2.dimensions == 0
+
+    circle = ParametricRegion((r*cos(theta), r*sin(theta)), r, (theta, 0, 2*pi))
+    assert circle.definition == (r*cos(theta), r*sin(theta))
+    assert circle.dimensions == 1
+
+    halfdisc = ParametricRegion((r*cos(theta), r*sin(theta)), (r, -2, 2), (theta, 0, pi))
+    assert halfdisc.definition == (r*cos(theta), r*sin(theta))
+    assert halfdisc.parameters == (r, theta)
+    assert halfdisc.limits == {r: (-2, 2), theta: (0, pi)}
+    assert halfdisc.dimensions == 2
+
+    ellipse = ParametricRegion((a*cos(t), b*sin(t)), (t, 0, 8))
+    assert ellipse.parameters == (t,)
+    assert ellipse.limits == {t: (0, 8)}
+    assert ellipse.dimensions == 1
+
+    cylinder = ParametricRegion((r*cos(theta), r*sin(theta), z), (r, 0, 1), (theta, 0, 2*pi), (z, 0, 4))
+    assert cylinder.parameters == (r, theta, z)
+    assert cylinder.dimensions == 3
+
+    sphere = ParametricRegion((r*sin(phi)*cos(theta),r*sin(phi)*sin(theta), r*cos(phi)),
+                                r, (theta, 0, 2*pi), (phi, 0, pi))
+    assert sphere.definition == (r*sin(phi)*cos(theta),r*sin(phi)*sin(theta), r*cos(phi))
+    assert sphere.parameters == (r, theta, phi)
+    assert sphere.dimensions == 2
+
+    raises(ValueError, lambda: ParametricRegion((a*t**2, 2*a*t), (a, -2)))
+    raises(ValueError, lambda: ParametricRegion((a, b), (a**2, sin(b)), (a, 2, 4, 6)))
+
+
+def test_parametric_region_list():
+
+    point = Point(-5, 12)
+    assert parametric_region_list(point) == [ParametricRegion((-5, 12))]
+
+    e = Ellipse(Point(2, 8), 2, 6)
+    assert parametric_region_list(e, t) == [ParametricRegion((2*cos(t) + 2, 6*sin(t) + 8), (t, 0, 2*pi))]
+
+    c = Curve((t, t**3), (t, 5, 3))
+    assert parametric_region_list(c) == [ParametricRegion((t, t**3), (t, 5, 3))]
+
+    s = Segment(Point(2, 11, -6), Point(0, 2, 5))
+    assert parametric_region_list(s, t) == [ParametricRegion((2 - 2*t, 11 - 9*t, 11*t - 6), (t, 0, 1))]
+    s1 = Segment(Point(0, 0), (1, 0))
+    assert parametric_region_list(s1, t) == [ParametricRegion((t, 0), (t, 0, 1))]
+    s2 = Segment(Point(1, 2, 3), Point(1, 2, 5))
+    assert parametric_region_list(s2, t) == [ParametricRegion((1, 2, 2*t + 3), (t, 0, 1))]
+    s3 = Segment(Point(12, 56), Point(12, 56))
+    assert parametric_region_list(s3) == [ParametricRegion((12, 56))]
+
+    poly = Polygon((1,3), (-3, 8), (2, 4))
+    assert parametric_region_list(poly, t) == [ParametricRegion((1 - 4*t, 5*t + 3), (t, 0, 1)), ParametricRegion((5*t - 3, 8 - 4*t), (t, 0, 1)), ParametricRegion((2 - t, 4 - t), (t, 0, 1))]
+
+    p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7,8)))
+    raises(ValueError, lambda: parametric_region_list(p1))

rl/Lib/site-packages/sympy/vector/tests/test_printing.py

@@ -0,0 +1,221 @@
+# -*- coding: utf-8 -*-
+from sympy.core.function import Function
+from sympy.integrals.integrals import Integral
+from sympy.printing.latex import latex
+from sympy.printing.pretty import pretty as xpretty
+from sympy.vector import CoordSys3D, Del, Vector, express
+from sympy.abc import a, b, c
+from sympy.testing.pytest import XFAIL
+
+
+def pretty(expr):
+    """ASCII pretty-printing"""
+    return xpretty(expr, use_unicode=False, wrap_line=False)
+
+
+def upretty(expr):
+    """Unicode pretty-printing"""
+    return xpretty(expr, use_unicode=True, wrap_line=False)
+
+
+# Initialize the basic and tedious vector/dyadic expressions
+# needed for testing.
+# Some of the pretty forms shown denote how the expressions just
+# above them should look with pretty printing.
+N = CoordSys3D('N')
+C = N.orient_new_axis('C', a, N.k)  # type: ignore
+v = []
+d = []
+v.append(Vector.zero)
+v.append(N.i)  # type: ignore
+v.append(-N.i)  # type: ignore
+v.append(N.i + N.j)  # type: ignore
+v.append(a*N.i)  # type: ignore
+v.append(a*N.i - b*N.j)  # type: ignore
+v.append((a**2 + N.x)*N.i + N.k)  # type: ignore
+v.append((a**2 + b)*N.i + 3*(C.y - c)*N.k)  # type: ignore
+f = Function('f')
+v.append(N.j - (Integral(f(b)) - C.x**2)*N.k)  # type: ignore
+upretty_v_8 = """\
+      ⎛   2   ⌠        ⎞    \n\
+j_N + ⎜x_C  - ⎮ f(b) db⎟ k_N\n\
+      ⎝       ⌡        ⎠    \
+"""
+pretty_v_8 = """\
+j_N + /         /       \\\n\
+      |   2    |        |\n\
+      |x_C  -  | f(b) db|\n\
+      |        |        |\n\
+      \\       /         / \
+"""
+
+v.append(N.i + C.k)  # type: ignore
+v.append(express(N.i, C))  # type: ignore
+v.append((a**2 + b)*N.i + (Integral(f(b)))*N.k)  # type: ignore
+upretty_v_11 = """\
+⎛ 2    ⎞        ⎛⌠        ⎞    \n\
+⎝a  + b⎠ i_N  + ⎜⎮ f(b) db⎟ k_N\n\
+                ⎝⌡        ⎠    \
+"""
+pretty_v_11 = """\
+/ 2    \\ + /  /       \\\n\
+\\a  + b/ i_N| |        |\n\
+           | | f(b) db|\n\
+           | |        |\n\
+           \\/         / \
+"""
+
+for x in v:
+    d.append(x | N.k)  # type: ignore
+s = 3*N.x**2*C.y  # type: ignore
+upretty_s = """\
+         2\n\
+3⋅y_C⋅x_N \
+"""
+pretty_s = """\
+         2\n\
+3*y_C*x_N \
+"""
+
+# This is the pretty form for ((a**2 + b)*N.i + 3*(C.y - c)*N.k) | N.k
+upretty_d_7 = """\
+⎛ 2    ⎞                                     \n\
+⎝a  + b⎠ (i_N|k_N)  + (3⋅y_C - 3⋅c) (k_N|k_N)\
+"""
+pretty_d_7 = """\
+/ 2    \\ (i_N|k_N) + (3*y_C - 3*c) (k_N|k_N)\n\
+\\a  + b/                                    \
+"""
+
+
+def test_str_printing():
+    assert str(v[0]) == '0'
+    assert str(v[1]) == 'N.i'
+    assert str(v[2]) == '(-1)*N.i'
+    assert str(v[3]) == 'N.i + N.j'
+    assert str(v[8]) == 'N.j + (C.x**2 - Integral(f(b), b))*N.k'
+    assert str(v[9]) == 'C.k + N.i'
+    assert str(s) == '3*C.y*N.x**2'
+    assert str(d[0]) == '0'
+    assert str(d[1]) == '(N.i|N.k)'
+    assert str(d[4]) == 'a*(N.i|N.k)'
+    assert str(d[5]) == 'a*(N.i|N.k) + (-b)*(N.j|N.k)'
+    assert str(d[8]) == ('(N.j|N.k) + (C.x**2 - ' +
+                         'Integral(f(b), b))*(N.k|N.k)')
+
+
+@XFAIL
+def test_pretty_printing_ascii():
+    assert pretty(v[0]) == '0'
+    assert pretty(v[1]) == 'i_N'
+    assert pretty(v[5]) == '(a) i_N + (-b) j_N'
+    assert pretty(v[8]) == pretty_v_8
+    assert pretty(v[2]) == '(-1) i_N'
+    assert pretty(v[11]) == pretty_v_11
+    assert pretty(s) == pretty_s
+    assert pretty(d[0]) == '(0|0)'
+    assert pretty(d[5]) == '(a) (i_N|k_N) + (-b) (j_N|k_N)'
+    assert pretty(d[7]) == pretty_d_7
+    assert pretty(d[10]) == '(cos(a)) (i_C|k_N) + (-sin(a)) (j_C|k_N)'
+
+
+def test_pretty_print_unicode_v():
+    assert upretty(v[0]) == '0'
+    assert upretty(v[1]) == 'i_N'
+    assert upretty(v[5]) == '(a) i_N + (-b) j_N'
+    # Make sure the printing works in other objects
+    assert upretty(v[5].args) == '((a) i_N, (-b) j_N)'
+    assert upretty(v[8]) == upretty_v_8
+    assert upretty(v[2]) == '(-1) i_N'
+    assert upretty(v[11]) == upretty_v_11
+    assert upretty(s) == upretty_s
+    assert upretty(d[0]) == '(0|0)'
+    assert upretty(d[5]) == '(a) (i_N|k_N) + (-b) (j_N|k_N)'
+    assert upretty(d[7]) == upretty_d_7
+    assert upretty(d[10]) == '(cos(a)) (i_C|k_N) + (-sin(a)) (j_C|k_N)'
+
+
+def test_latex_printing():
+    assert latex(v[0]) == '\\mathbf{\\hat{0}}'
+    assert latex(v[1]) == '\\mathbf{\\hat{i}_{N}}'
+    assert latex(v[2]) == '- \\mathbf{\\hat{i}_{N}}'
+    assert latex(v[5]) == ('\\left(a\\right)\\mathbf{\\hat{i}_{N}} + ' +
+                           '\\left(- b\\right)\\mathbf{\\hat{j}_{N}}')
+    assert latex(v[6]) == ('\\left(\\mathbf{{x}_{N}} + a^{2}\\right)\\mathbf{\\hat{i}_' +
+                          '{N}} + \\mathbf{\\hat{k}_{N}}')
+    assert latex(v[8]) == ('\\mathbf{\\hat{j}_{N}} + \\left(\\mathbf{{x}_' +
+                           '{C}}^{2} - \\int f{\\left(b \\right)}\\,' +
+                           ' db\\right)\\mathbf{\\hat{k}_{N}}')
+    assert latex(s) == '3 \\mathbf{{y}_{C}} \\mathbf{{x}_{N}}^{2}'
+    assert latex(d[0]) == '(\\mathbf{\\hat{0}}|\\mathbf{\\hat{0}})'
+    assert latex(d[4]) == ('\\left(a\\right)\\left(\\mathbf{\\hat{i}_{N}}{\\middle|}' +
+                           '\\mathbf{\\hat{k}_{N}}\\right)')
+    assert latex(d[9]) == ('\\left(\\mathbf{\\hat{k}_{C}}{\\middle|}' +
+                           '\\mathbf{\\hat{k}_{N}}\\right) + \\left(' +
+                           '\\mathbf{\\hat{i}_{N}}{\\middle|}\\mathbf{' +
+                           '\\hat{k}_{N}}\\right)')
+    assert latex(d[11]) == ('\\left(a^{2} + b\\right)\\left(\\mathbf{\\hat{i}_{N}}' +
+                            '{\\middle|}\\mathbf{\\hat{k}_{N}}\\right) + ' +
+                            '\\left(\\int f{\\left(b \\right)}\\, db\\right)\\left(' +
+                            '\\mathbf{\\hat{k}_{N}}{\\middle|}\\mathbf{' +
+                            '\\hat{k}_{N}}\\right)')
+
+def test_issue_23058():
+    from sympy import symbols, sin, cos, pi, UnevaluatedExpr
+
+    delop = Del()
+    CC_   = CoordSys3D("C")
+    y     = CC_.y
+    xhat  = CC_.i
+
+    t = symbols("t")
+    ten = symbols("10", positive=True)
+    eps, mu = 4*pi*ten**(-11), ten**(-5)
+
+    Bx = 2 * ten**(-4) * cos(ten**5 * t) * sin(ten**(-3) * y)
+    vecB = Bx * xhat
+    vecE = (1/eps) * Integral(delop.cross(vecB/mu).doit(), t)
+    vecE = vecE.doit()
+
+    vecB_str = """\
+⎛     ⎛y_C⎞    ⎛  5  ⎞⎞    \n\
+⎜2⋅sin⎜───⎟⋅cos⎝10 ⋅t⎠⎟ i_C\n\
+⎜     ⎜  3⎟           ⎟    \n\
+⎜     ⎝10 ⎠           ⎟    \n\
+⎜─────────────────────⎟    \n\
+⎜           4         ⎟    \n\
+⎝         10          ⎠    \
+"""
+    vecE_str = """\
+⎛   4    ⎛  5  ⎞    ⎛y_C⎞ ⎞    \n\
+⎜-10 ⋅sin⎝10 ⋅t⎠⋅cos⎜───⎟ ⎟ k_C\n\
+⎜                   ⎜  3⎟ ⎟    \n\
+⎜                   ⎝10 ⎠ ⎟    \n\
+⎜─────────────────────────⎟    \n\
+⎝           2⋅π           ⎠    \
+"""
+
+    assert upretty(vecB) == vecB_str
+    assert upretty(vecE) == vecE_str
+
+    ten = UnevaluatedExpr(10)
+    eps, mu = 4*pi*ten**(-11), ten**(-5)
+
+    Bx = 2 * ten**(-4) * cos(ten**5 * t) * sin(ten**(-3) * y)
+    vecB = Bx * xhat
+
+    vecB_str = """\
+⎛    -4    ⎛    5⎞    ⎛      -3⎞⎞     \n\
+⎝2⋅10  ⋅cos⎝t⋅10 ⎠⋅sin⎝y_C⋅10  ⎠⎠ i_C \
+"""
+    assert upretty(vecB) == vecB_str
+
+def test_custom_names():
+    A = CoordSys3D('A', vector_names=['x', 'y', 'z'],
+                   variable_names=['i', 'j', 'k'])
+    assert A.i.__str__() == 'A.i'
+    assert A.x.__str__() == 'A.x'
+    assert A.i._pretty_form == 'i_A'
+    assert A.x._pretty_form == 'x_A'
+    assert A.i._latex_form == r'\mathbf{{i}_{A}}'
+    assert A.x._latex_form == r"\mathbf{\hat{x}_{A}}"

rl/Lib/site-packages/sympy/vector/tests/test_vector.py

@@ -0,0 +1,266 @@
+from sympy.core import Rational, S
+from sympy.simplify import simplify, trigsimp
+from sympy.core.function import (Derivative, Function, diff)
+from sympy.core.numbers import pi
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.integrals.integrals import Integral
+from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix
+from sympy.vector.vector import Vector, BaseVector, VectorAdd, \
+     VectorMul, VectorZero
+from sympy.vector.coordsysrect import CoordSys3D
+from sympy.vector.vector import Cross, Dot, cross
+from sympy.testing.pytest import raises
+
+C = CoordSys3D('C')
+
+i, j, k = C.base_vectors()
+a, b, c = symbols('a b c')
+
+
+def test_cross():
+    v1 = C.x * i + C.z * C.z * j
+    v2 = C.x * i + C.y * j + C.z * k
+    assert Cross(v1, v2) == Cross(C.x*C.i + C.z**2*C.j, C.x*C.i + C.y*C.j + C.z*C.k)
+    assert Cross(v1, v2).doit() == C.z**3*C.i + (-C.x*C.z)*C.j + (C.x*C.y - C.x*C.z**2)*C.k
+    assert cross(v1, v2) == C.z**3*C.i + (-C.x*C.z)*C.j + (C.x*C.y - C.x*C.z**2)*C.k
+    assert Cross(v1, v2) == -Cross(v2, v1)
+    assert Cross(v1, v2) + Cross(v2, v1) == Vector.zero
+
+
+def test_dot():
+    v1 = C.x * i + C.z * C.z * j
+    v2 = C.x * i + C.y * j + C.z * k
+    assert Dot(v1, v2) == Dot(C.x*C.i + C.z**2*C.j, C.x*C.i + C.y*C.j + C.z*C.k)
+    assert Dot(v1, v2).doit() == C.x**2 + C.y*C.z**2
+    assert Dot(v1, v2).doit() == C.x**2 + C.y*C.z**2
+    assert Dot(v1, v2) == Dot(v2, v1)
+
+
+def test_vector_sympy():
+    """
+    Test whether the Vector framework confirms to the hashing
+    and equality testing properties of SymPy.
+    """
+    v1 = 3*j
+    assert v1 == j*3
+    assert v1.components == {j: 3}
+    v2 = 3*i + 4*j + 5*k
+    v3 = 2*i + 4*j + i + 4*k + k
+    assert v3 == v2
+    assert v3.__hash__() == v2.__hash__()
+
+
+def test_vector():
+    assert isinstance(i, BaseVector)
+    assert i != j
+    assert j != k
+    assert k != i
+    assert i - i == Vector.zero
+    assert i + Vector.zero == i
+    assert i - Vector.zero == i
+    assert Vector.zero != 0
+    assert -Vector.zero == Vector.zero
+
+    v1 = a*i + b*j + c*k
+    v2 = a**2*i + b**2*j + c**2*k
+    v3 = v1 + v2
+    v4 = 2 * v1
+    v5 = a * i
+
+    assert isinstance(v1, VectorAdd)
+    assert v1 - v1 == Vector.zero
+    assert v1 + Vector.zero == v1
+    assert v1.dot(i) == a
+    assert v1.dot(j) == b
+    assert v1.dot(k) == c
+    assert i.dot(v2) == a**2
+    assert j.dot(v2) == b**2
+    assert k.dot(v2) == c**2
+    assert v3.dot(i) == a**2 + a
+    assert v3.dot(j) == b**2 + b
+    assert v3.dot(k) == c**2 + c
+
+    assert v1 + v2 == v2 + v1
+    assert v1 - v2 == -1 * (v2 - v1)
+    assert a * v1 == v1 * a
+
+    assert isinstance(v5, VectorMul)
+    assert v5.base_vector == i
+    assert v5.measure_number == a
+    assert isinstance(v4, Vector)
+    assert isinstance(v4, VectorAdd)
+    assert isinstance(v4, Vector)
+    assert isinstance(Vector.zero, VectorZero)
+    assert isinstance(Vector.zero, Vector)
+    assert isinstance(v1 * 0, VectorZero)
+
+    assert v1.to_matrix(C) == Matrix([[a], [b], [c]])
+
+    assert i.components == {i: 1}
+    assert v5.components == {i: a}
+    assert v1.components == {i: a, j: b, k: c}
+
+    assert VectorAdd(v1, Vector.zero) == v1
+    assert VectorMul(a, v1) == v1*a
+    assert VectorMul(1, i) == i
+    assert VectorAdd(v1, Vector.zero) == v1
+    assert VectorMul(0, Vector.zero) == Vector.zero
+    raises(TypeError, lambda: v1.outer(1))
+    raises(TypeError, lambda: v1.dot(1))
+
+
+def test_vector_magnitude_normalize():
+    assert Vector.zero.magnitude() == 0
+    assert Vector.zero.normalize() == Vector.zero
+
+    assert i.magnitude() == 1
+    assert j.magnitude() == 1
+    assert k.magnitude() == 1
+    assert i.normalize() == i
+    assert j.normalize() == j
+    assert k.normalize() == k
+
+    v1 = a * i
+    assert v1.normalize() == (a/sqrt(a**2))*i
+    assert v1.magnitude() == sqrt(a**2)
+
+    v2 = a*i + b*j + c*k
+    assert v2.magnitude() == sqrt(a**2 + b**2 + c**2)
+    assert v2.normalize() == v2 / v2.magnitude()
+
+    v3 = i + j
+    assert v3.normalize() == (sqrt(2)/2)*C.i + (sqrt(2)/2)*C.j
+
+
+def test_vector_simplify():
+    A, s, k, m = symbols('A, s, k, m')
+
+    test1 = (1 / a + 1 / b) * i
+    assert (test1 & i) != (a + b) / (a * b)
+    test1 = simplify(test1)
+    assert (test1 & i) == (a + b) / (a * b)
+    assert test1.simplify() == simplify(test1)
+
+    test2 = (A**2 * s**4 / (4 * pi * k * m**3)) * i
+    test2 = simplify(test2)
+    assert (test2 & i) == (A**2 * s**4 / (4 * pi * k * m**3))
+
+    test3 = ((4 + 4 * a - 2 * (2 + 2 * a)) / (2 + 2 * a)) * i
+    test3 = simplify(test3)
+    assert (test3 & i) == 0
+
+    test4 = ((-4 * a * b**2 - 2 * b**3 - 2 * a**2 * b) / (a + b)**2) * i
+    test4 = simplify(test4)
+    assert (test4 & i) == -2 * b
+
+    v = (sin(a)+cos(a))**2*i - j
+    assert trigsimp(v) == (2*sin(a + pi/4)**2)*i + (-1)*j
+    assert trigsimp(v) == v.trigsimp()
+
+    assert simplify(Vector.zero) == Vector.zero
+
+
+def test_vector_dot():
+    assert i.dot(Vector.zero) == 0
+    assert Vector.zero.dot(i) == 0
+    assert i & Vector.zero == 0
+
+    assert i.dot(i) == 1
+    assert i.dot(j) == 0
+    assert i.dot(k) == 0
+    assert i & i == 1
+    assert i & j == 0
+    assert i & k == 0
+
+    assert j.dot(i) == 0
+    assert j.dot(j) == 1
+    assert j.dot(k) == 0
+    assert j & i == 0
+    assert j & j == 1
+    assert j & k == 0
+
+    assert k.dot(i) == 0
+    assert k.dot(j) == 0
+    assert k.dot(k) == 1
+    assert k & i == 0
+    assert k & j == 0
+    assert k & k == 1
+
+    raises(TypeError, lambda: k.dot(1))
+
+
+def test_vector_cross():
+    assert i.cross(Vector.zero) == Vector.zero
+    assert Vector.zero.cross(i) == Vector.zero
+
+    assert i.cross(i) == Vector.zero
+    assert i.cross(j) == k
+    assert i.cross(k) == -j
+    assert i ^ i == Vector.zero
+    assert i ^ j == k
+    assert i ^ k == -j
+
+    assert j.cross(i) == -k
+    assert j.cross(j) == Vector.zero
+    assert j.cross(k) == i
+    assert j ^ i == -k
+    assert j ^ j == Vector.zero
+    assert j ^ k == i
+
+    assert k.cross(i) == j
+    assert k.cross(j) == -i
+    assert k.cross(k) == Vector.zero
+    assert k ^ i == j
+    assert k ^ j == -i
+    assert k ^ k == Vector.zero
+
+    assert k.cross(1) == Cross(k, 1)
+
+
+def test_projection():
+    v1 = i + j + k
+    v2 = 3*i + 4*j
+    v3 = 0*i + 0*j
+    assert v1.projection(v1) == i + j + k
+    assert v1.projection(v2) == Rational(7, 3)*C.i + Rational(7, 3)*C.j + Rational(7, 3)*C.k
+    assert v1.projection(v1, scalar=True) == S.One
+    assert v1.projection(v2, scalar=True) == Rational(7, 3)
+    assert v3.projection(v1) == Vector.zero
+    assert v3.projection(v1, scalar=True) == S.Zero
+
+
+def test_vector_diff_integrate():
+    f = Function('f')
+    v = f(a)*C.i + a**2*C.j - C.k
+    assert Derivative(v, a) == Derivative((f(a))*C.i +
+                                          a**2*C.j + (-1)*C.k, a)
+    assert (diff(v, a) == v.diff(a) == Derivative(v, a).doit() ==
+            (Derivative(f(a), a))*C.i + 2*a*C.j)
+    assert (Integral(v, a) == (Integral(f(a), a))*C.i +
+            (Integral(a**2, a))*C.j + (Integral(-1, a))*C.k)
+
+
+def test_vector_args():
+    raises(ValueError, lambda: BaseVector(3, C))
+    raises(TypeError, lambda: BaseVector(0, Vector.zero))
+
+
+def test_srepr():
+    from sympy.printing.repr import srepr
+    res = "CoordSys3D(Str('C'), Tuple(ImmutableDenseMatrix([[Integer(1), "\
+            "Integer(0), Integer(0)], [Integer(0), Integer(1), Integer(0)], "\
+            "[Integer(0), Integer(0), Integer(1)]]), VectorZero())).i"
+    assert srepr(C.i) == res
+
+
+def test_scalar():
+    from sympy.vector import CoordSys3D
+    C = CoordSys3D('C')
+    v1 = 3*C.i + 4*C.j + 5*C.k
+    v2 = 3*C.i - 4*C.j + 5*C.k
+    assert v1.is_Vector is True
+    assert v1.is_scalar is False
+    assert (v1.dot(v2)).is_scalar is True
+    assert (v1.cross(v2)).is_scalar is False

rl/Lib/site-packages/sympy/vector/vector.py

@@ -0,0 +1,623 @@
+from __future__ import annotations
+from itertools import product
+
+from sympy.core.add import Add
+from sympy.core.assumptions import StdFactKB
+from sympy.core.expr import AtomicExpr, Expr
+from sympy.core.power import Pow
+from sympy.core.singleton import S
+from sympy.core.sorting import default_sort_key
+from sympy.core.sympify import sympify
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix
+from sympy.vector.basisdependent import (BasisDependentZero,
+    BasisDependent, BasisDependentMul, BasisDependentAdd)
+from sympy.vector.coordsysrect import CoordSys3D
+from sympy.vector.dyadic import Dyadic, BaseDyadic, DyadicAdd
+
+
+class Vector(BasisDependent):
+    """
+    Super class for all Vector classes.
+    Ideally, neither this class nor any of its subclasses should be
+    instantiated by the user.
+    """
+
+    is_scalar = False
+    is_Vector = True
+    _op_priority = 12.0
+
+    _expr_type: type[Vector]
+    _mul_func: type[Vector]
+    _add_func: type[Vector]
+    _zero_func: type[Vector]
+    _base_func: type[Vector]
+    zero: VectorZero
+
+    @property
+    def components(self):
+        """
+        Returns the components of this vector in the form of a
+        Python dictionary mapping BaseVector instances to the
+        corresponding measure numbers.
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D
+        >>> C = CoordSys3D('C')
+        >>> v = 3*C.i + 4*C.j + 5*C.k
+        >>> v.components
+        {C.i: 3, C.j: 4, C.k: 5}
+
+        """
+        # The '_components' attribute is defined according to the
+        # subclass of Vector the instance belongs to.
+        return self._components
+
+    def magnitude(self):
+        """
+        Returns the magnitude of this vector.
+        """
+        return sqrt(self & self)
+
+    def normalize(self):
+        """
+        Returns the normalized version of this vector.
+        """
+        return self / self.magnitude()
+
+    def dot(self, other):
+        """
+        Returns the dot product of this Vector, either with another
+        Vector, or a Dyadic, or a Del operator.
+        If 'other' is a Vector, returns the dot product scalar (SymPy
+        expression).
+        If 'other' is a Dyadic, the dot product is returned as a Vector.
+        If 'other' is an instance of Del, returns the directional
+        derivative operator as a Python function. If this function is
+        applied to a scalar expression, it returns the directional
+        derivative of the scalar field wrt this Vector.
+
+        Parameters
+        ==========
+
+        other: Vector/Dyadic/Del
+            The Vector or Dyadic we are dotting with, or a Del operator .
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D, Del
+        >>> C = CoordSys3D('C')
+        >>> delop = Del()
+        >>> C.i.dot(C.j)
+        0
+        >>> C.i & C.i
+        1
+        >>> v = 3*C.i + 4*C.j + 5*C.k
+        >>> v.dot(C.k)
+        5
+        >>> (C.i & delop)(C.x*C.y*C.z)
+        C.y*C.z
+        >>> d = C.i.outer(C.i)
+        >>> C.i.dot(d)
+        C.i
+
+        """
+
+        # Check special cases
+        if isinstance(other, Dyadic):
+            if isinstance(self, VectorZero):
+                return Vector.zero
+            outvec = Vector.zero
+            for k, v in other.components.items():
+                vect_dot = k.args[0].dot(self)
+                outvec += vect_dot * v * k.args[1]
+            return outvec
+        from sympy.vector.deloperator import Del
+        if not isinstance(other, (Del, Vector)):
+            raise TypeError(str(other) + " is not a vector, dyadic or " +
+                            "del operator")
+
+        # Check if the other is a del operator
+        if isinstance(other, Del):
+            def directional_derivative(field):
+                from sympy.vector.functions import directional_derivative
+                return directional_derivative(field, self)
+            return directional_derivative
+
+        return dot(self, other)
+
+    def __and__(self, other):
+        return self.dot(other)
+
+    __and__.__doc__ = dot.__doc__
+
+    def cross(self, other):
+        """
+        Returns the cross product of this Vector with another Vector or
+        Dyadic instance.
+        The cross product is a Vector, if 'other' is a Vector. If 'other'
+        is a Dyadic, this returns a Dyadic instance.
+
+        Parameters
+        ==========
+
+        other: Vector/Dyadic
+            The Vector or Dyadic we are crossing with.
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D
+        >>> C = CoordSys3D('C')
+        >>> C.i.cross(C.j)
+        C.k
+        >>> C.i ^ C.i
+        0
+        >>> v = 3*C.i + 4*C.j + 5*C.k
+        >>> v ^ C.i
+        5*C.j + (-4)*C.k
+        >>> d = C.i.outer(C.i)
+        >>> C.j.cross(d)
+        (-1)*(C.k|C.i)
+
+        """
+
+        # Check special cases
+        if isinstance(other, Dyadic):
+            if isinstance(self, VectorZero):
+                return Dyadic.zero
+            outdyad = Dyadic.zero
+            for k, v in other.components.items():
+                cross_product = self.cross(k.args[0])
+                outer = cross_product.outer(k.args[1])
+                outdyad += v * outer
+            return outdyad
+
+        return cross(self, other)
+
+    def __xor__(self, other):
+        return self.cross(other)
+
+    __xor__.__doc__ = cross.__doc__
+
+    def outer(self, other):
+        """
+        Returns the outer product of this vector with another, in the
+        form of a Dyadic instance.
+
+        Parameters
+        ==========
+
+        other : Vector
+            The Vector with respect to which the outer product is to
+            be computed.
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D
+        >>> N = CoordSys3D('N')
+        >>> N.i.outer(N.j)
+        (N.i|N.j)
+
+        """
+
+        # Handle the special cases
+        if not isinstance(other, Vector):
+            raise TypeError("Invalid operand for outer product")
+        elif (isinstance(self, VectorZero) or
+                isinstance(other, VectorZero)):
+            return Dyadic.zero
+
+        # Iterate over components of both the vectors to generate
+        # the required Dyadic instance
+        args = [(v1 * v2) * BaseDyadic(k1, k2) for (k1, v1), (k2, v2)
+                in product(self.components.items(), other.components.items())]
+
+        return DyadicAdd(*args)
+
+    def projection(self, other, scalar=False):
+        """
+        Returns the vector or scalar projection of the 'other' on 'self'.
+
+        Examples
+        ========
+
+        >>> from sympy.vector.coordsysrect import CoordSys3D
+        >>> C = CoordSys3D('C')
+        >>> i, j, k = C.base_vectors()
+        >>> v1 = i + j + k
+        >>> v2 = 3*i + 4*j
+        >>> v1.projection(v2)
+        7/3*C.i + 7/3*C.j + 7/3*C.k
+        >>> v1.projection(v2, scalar=True)
+        7/3
+
+        """
+        if self.equals(Vector.zero):
+            return S.Zero if scalar else Vector.zero
+
+        if scalar:
+            return self.dot(other) / self.dot(self)
+        else:
+            return self.dot(other) / self.dot(self) * self
+
+    @property
+    def _projections(self):
+        """
+        Returns the components of this vector but the output includes
+        also zero values components.
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D, Vector
+        >>> C = CoordSys3D('C')
+        >>> v1 = 3*C.i + 4*C.j + 5*C.k
+        >>> v1._projections
+        (3, 4, 5)
+        >>> v2 = C.x*C.y*C.z*C.i
+        >>> v2._projections
+        (C.x*C.y*C.z, 0, 0)
+        >>> v3 = Vector.zero
+        >>> v3._projections
+        (0, 0, 0)
+        """
+
+        from sympy.vector.operators import _get_coord_systems
+        if isinstance(self, VectorZero):
+            return (S.Zero, S.Zero, S.Zero)
+        base_vec = next(iter(_get_coord_systems(self))).base_vectors()
+        return tuple([self.dot(i) for i in base_vec])
+
+    def __or__(self, other):
+        return self.outer(other)
+
+    __or__.__doc__ = outer.__doc__
+
+    def to_matrix(self, system):
+        """
+        Returns the matrix form of this vector with respect to the
+        specified coordinate system.
+
+        Parameters
+        ==========
+
+        system : CoordSys3D
+            The system wrt which the matrix form is to be computed
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D
+        >>> C = CoordSys3D('C')
+        >>> from sympy.abc import a, b, c
+        >>> v = a*C.i + b*C.j + c*C.k
+        >>> v.to_matrix(C)
+        Matrix([
+        [a],
+        [b],
+        [c]])
+
+        """
+
+        return Matrix([self.dot(unit_vec) for unit_vec in
+                       system.base_vectors()])
+
+    def separate(self):
+        """
+        The constituents of this vector in different coordinate systems,
+        as per its definition.
+
+        Returns a dict mapping each CoordSys3D to the corresponding
+        constituent Vector.
+
+        Examples
+        ========
+
+        >>> from sympy.vector import CoordSys3D
+        >>> R1 = CoordSys3D('R1')
+        >>> R2 = CoordSys3D('R2')
+        >>> v = R1.i + R2.i
+        >>> v.separate() == {R1: R1.i, R2: R2.i}
+        True
+
+        """
+
+        parts = {}
+        for vect, measure in self.components.items():
+            parts[vect.system] = (parts.get(vect.system, Vector.zero) +
+                                  vect * measure)
+        return parts
+
+    def _div_helper(one, other):
+        """ Helper for division involving vectors. """
+        if isinstance(one, Vector) and isinstance(other, Vector):
+            raise TypeError("Cannot divide two vectors")
+        elif isinstance(one, Vector):
+            if other == S.Zero:
+                raise ValueError("Cannot divide a vector by zero")
+            return VectorMul(one, Pow(other, S.NegativeOne))
+        else:
+            raise TypeError("Invalid division involving a vector")
+
+
+class BaseVector(Vector, AtomicExpr):
+    """
+    Class to denote a base vector.
+
+    """
+
+    def __new__(cls, index, system, pretty_str=None, latex_str=None):
+        if pretty_str is None:
+            pretty_str = "x{}".format(index)
+        if latex_str is None:
+            latex_str = "x_{}".format(index)
+        pretty_str = str(pretty_str)
+        latex_str = str(latex_str)
+        # Verify arguments
+        if index not in range(0, 3):
+            raise ValueError("index must be 0, 1 or 2")
+        if not isinstance(system, CoordSys3D):
+            raise TypeError("system should be a CoordSys3D")
+        name = system._vector_names[index]
+        # Initialize an object
+        obj = super().__new__(cls, S(index), system)
+        # Assign important attributes
+        obj._base_instance = obj
+        obj._components = {obj: S.One}
+        obj._measure_number = S.One
+        obj._name = system._name + '.' + name
+        obj._pretty_form = '' + pretty_str
+        obj._latex_form = latex_str
+        obj._system = system
+        # The _id is used for printing purposes
+        obj._id = (index, system)
+        assumptions = {'commutative': True}
+        obj._assumptions = StdFactKB(assumptions)
+
+        # This attr is used for re-expression to one of the systems
+        # involved in the definition of the Vector. Applies to
+        # VectorMul and VectorAdd too.
+        obj._sys = system
+
+        return obj
+
+    @property
+    def system(self):
+        return self._system
+
+    def _sympystr(self, printer):
+        return self._name
+
+    def _sympyrepr(self, printer):
+        index, system = self._id
+        return printer._print(system) + '.' + system._vector_names[index]
+
+    @property
+    def free_symbols(self):
+        return {self}
+
+
+class VectorAdd(BasisDependentAdd, Vector):
+    """
+    Class to denote sum of Vector instances.
+    """
+
+    def __new__(cls, *args, **options):
+        obj = BasisDependentAdd.__new__(cls, *args, **options)
+        return obj
+
+    def _sympystr(self, printer):
+        ret_str = ''
+        items = list(self.separate().items())
+        items.sort(key=lambda x: x[0].__str__())
+        for system, vect in items:
+            base_vects = system.base_vectors()
+            for x in base_vects:
+                if x in vect.components:
+                    temp_vect = self.components[x] * x
+                    ret_str += printer._print(temp_vect) + " + "
+        return ret_str[:-3]
+
+
+class VectorMul(BasisDependentMul, Vector):
+    """
+    Class to denote products of scalars and BaseVectors.
+    """
+
+    def __new__(cls, *args, **options):
+        obj = BasisDependentMul.__new__(cls, *args, **options)
+        return obj
+
+    @property
+    def base_vector(self):
+        """ The BaseVector involved in the product. """
+        return self._base_instance
+
+    @property
+    def measure_number(self):
+        """ The scalar expression involved in the definition of
+        this VectorMul.
+        """
+        return self._measure_number
+
+
+class VectorZero(BasisDependentZero, Vector):
+    """
+    Class to denote a zero vector
+    """
+
+    _op_priority = 12.1
+    _pretty_form = '0'
+    _latex_form = r'\mathbf{\hat{0}}'
+
+    def __new__(cls):
+        obj = BasisDependentZero.__new__(cls)
+        return obj
+
+
+class Cross(Vector):
+    """
+    Represents unevaluated Cross product.
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, Cross
+    >>> R = CoordSys3D('R')
+    >>> v1 = R.i + R.j + R.k
+    >>> v2 = R.x * R.i + R.y * R.j + R.z * R.k
+    >>> Cross(v1, v2)
+    Cross(R.i + R.j + R.k, R.x*R.i + R.y*R.j + R.z*R.k)
+    >>> Cross(v1, v2).doit()
+    (-R.y + R.z)*R.i + (R.x - R.z)*R.j + (-R.x + R.y)*R.k
+
+    """
+
+    def __new__(cls, expr1, expr2):
+        expr1 = sympify(expr1)
+        expr2 = sympify(expr2)
+        if default_sort_key(expr1) > default_sort_key(expr2):
+            return -Cross(expr2, expr1)
+        obj = Expr.__new__(cls, expr1, expr2)
+        obj._expr1 = expr1
+        obj._expr2 = expr2
+        return obj
+
+    def doit(self, **hints):
+        return cross(self._expr1, self._expr2)
+
+
+class Dot(Expr):
+    """
+    Represents unevaluated Dot product.
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D, Dot
+    >>> from sympy import symbols
+    >>> R = CoordSys3D('R')
+    >>> a, b, c = symbols('a b c')
+    >>> v1 = R.i + R.j + R.k
+    >>> v2 = a * R.i + b * R.j + c * R.k
+    >>> Dot(v1, v2)
+    Dot(R.i + R.j + R.k, a*R.i + b*R.j + c*R.k)
+    >>> Dot(v1, v2).doit()
+    a + b + c
+
+    """
+
+    def __new__(cls, expr1, expr2):
+        expr1 = sympify(expr1)
+        expr2 = sympify(expr2)
+        expr1, expr2 = sorted([expr1, expr2], key=default_sort_key)
+        obj = Expr.__new__(cls, expr1, expr2)
+        obj._expr1 = expr1
+        obj._expr2 = expr2
+        return obj
+
+    def doit(self, **hints):
+        return dot(self._expr1, self._expr2)
+
+
+def cross(vect1, vect2):
+    """
+    Returns cross product of two vectors.
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D
+    >>> from sympy.vector.vector import cross
+    >>> R = CoordSys3D('R')
+    >>> v1 = R.i + R.j + R.k
+    >>> v2 = R.x * R.i + R.y * R.j + R.z * R.k
+    >>> cross(v1, v2)
+    (-R.y + R.z)*R.i + (R.x - R.z)*R.j + (-R.x + R.y)*R.k
+
+    """
+    if isinstance(vect1, Add):
+        return VectorAdd.fromiter(cross(i, vect2) for i in vect1.args)
+    if isinstance(vect2, Add):
+        return VectorAdd.fromiter(cross(vect1, i) for i in vect2.args)
+    if isinstance(vect1, BaseVector) and isinstance(vect2, BaseVector):
+        if vect1._sys == vect2._sys:
+            n1 = vect1.args[0]
+            n2 = vect2.args[0]
+            if n1 == n2:
+                return Vector.zero
+            n3 = ({0,1,2}.difference({n1, n2})).pop()
+            sign = 1 if ((n1 + 1) % 3 == n2) else -1
+            return sign*vect1._sys.base_vectors()[n3]
+        from .functions import express
+        try:
+            v = express(vect1, vect2._sys)
+        except ValueError:
+            return Cross(vect1, vect2)
+        else:
+            return cross(v, vect2)
+    if isinstance(vect1, VectorZero) or isinstance(vect2, VectorZero):
+        return Vector.zero
+    if isinstance(vect1, VectorMul):
+        v1, m1 = next(iter(vect1.components.items()))
+        return m1*cross(v1, vect2)
+    if isinstance(vect2, VectorMul):
+        v2, m2 = next(iter(vect2.components.items()))
+        return m2*cross(vect1, v2)
+
+    return Cross(vect1, vect2)
+
+
+def dot(vect1, vect2):
+    """
+    Returns dot product of two vectors.
+
+    Examples
+    ========
+
+    >>> from sympy.vector import CoordSys3D
+    >>> from sympy.vector.vector import dot
+    >>> R = CoordSys3D('R')
+    >>> v1 = R.i + R.j + R.k
+    >>> v2 = R.x * R.i + R.y * R.j + R.z * R.k
+    >>> dot(v1, v2)
+    R.x + R.y + R.z
+
+    """
+    if isinstance(vect1, Add):
+        return Add.fromiter(dot(i, vect2) for i in vect1.args)
+    if isinstance(vect2, Add):
+        return Add.fromiter(dot(vect1, i) for i in vect2.args)
+    if isinstance(vect1, BaseVector) and isinstance(vect2, BaseVector):
+        if vect1._sys == vect2._sys:
+            return S.One if vect1 == vect2 else S.Zero
+        from .functions import express
+        try:
+            v = express(vect2, vect1._sys)
+        except ValueError:
+            return Dot(vect1, vect2)
+        else:
+            return dot(vect1, v)
+    if isinstance(vect1, VectorZero) or isinstance(vect2, VectorZero):
+        return S.Zero
+    if isinstance(vect1, VectorMul):
+        v1, m1 = next(iter(vect1.components.items()))
+        return m1*dot(v1, vect2)
+    if isinstance(vect2, VectorMul):
+        v2, m2 = next(iter(vect2.components.items()))
+        return m2*dot(vect1, v2)
+
+    return Dot(vect1, vect2)
+
+
+Vector._expr_type = Vector
+Vector._mul_func = VectorMul
+Vector._add_func = VectorAdd
+Vector._zero_func = VectorZero
+Vector._base_func = BaseVector
+Vector.zero = VectorZero()