from sympy.core.assumptions import StdFactKB
from sympy.core import S, Pow, Symbol
from sympy.core.expr import AtomicExpr
from sympy.core.compatibility import range
from sympy import diff as df, sqrt, ImmutableMatrix as Matrix
from sympy.vector.coordsysrect import CoordSysCartesian
from sympy.vector.basisdependent import (BasisDependent, BasisDependentAdd,
BasisDependentMul, BasisDependentZero)
from sympy.vector.dyadic import BaseDyadic, Dyadic, DyadicAdd
from sympy.core.compatibility import u
[docs]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_Vector = True
_op_priority = 12.0
@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 CoordSysCartesian
>>> C = CoordSysCartesian('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
[docs] def magnitude(self):
"""
Returns the magnitude of this vector.
"""
return sqrt(self & self)
[docs] def normalize(self):
"""
Returns the normalized version of this vector.
"""
return self / self.magnitude()
[docs] 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
derivate 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 CoordSysCartesian
>>> C = CoordSysCartesian('C')
>>> 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 & C.delop)(C.x*C.y*C.z)
C.y*C.z
>>> d = C.i.outer(C.i)
>>> C.i.dot(d)
C.i
"""
from sympy.vector.functions import express
# 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, Vector) and not isinstance(other, Del):
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):
field = express(field, other.system, variables=True)
out = self.dot(other._i) * df(field, other._x)
out += self.dot(other._j) * df(field, other._y)
out += self.dot(other._k) * df(field, other._z)
if out == 0 and isinstance(field, Vector):
out = Vector.zero
return out
return directional_derivative
if isinstance(self, VectorZero) or isinstance(other, VectorZero):
return S(0)
v1 = express(self, other._sys)
v2 = express(other, other._sys)
dotproduct = S(0)
for x in other._sys.base_vectors():
dotproduct += (v1.components.get(x, 0) *
v2.components.get(x, 0))
return dotproduct
def __and__(self, other):
return self.dot(other)
__and__.__doc__ = dot.__doc__
[docs] 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 CoordSysCartesian
>>> C = CoordSysCartesian('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
elif not isinstance(other, Vector):
raise TypeError(str(other) + " is not a vector")
elif (isinstance(self, VectorZero) or
isinstance(other, VectorZero)):
return Vector.zero
# Compute cross product
def _det(mat):
"""This is needed as a little method for to find the determinant
of a list in python.
SymPy's Matrix won't take in Vector, so need a custom function.
The user shouldn't be calling this.
"""
return (mat[0][0] * (mat[1][1] * mat[2][2] - mat[1][2] *
mat[2][1]) +
mat[0][1] * (mat[1][2] * mat[2][0] - mat[1][0] *
mat[2][2]) +
mat[0][2] * (mat[1][0] * mat[2][1] - mat[1][1] *
mat[2][0]))
outvec = Vector.zero
for system, vect in other.separate().items():
tempi = system.i
tempj = system.j
tempk = system.k
tempm = [[tempi, tempj, tempk],
[self & tempi, self & tempj, self & tempk],
[vect & tempi, vect & tempj, vect & tempk]]
outvec += _det(tempm)
return outvec
def __xor__(self, other):
return self.cross(other)
__xor__.__doc__ = cross.__doc__
[docs] 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 CoordSysCartesian
>>> N = CoordSysCartesian('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 = []
for k1, v1 in self.components.items():
for k2, v2 in other.components.items():
args.append((v1 * v2) * BaseDyadic(k1, k2))
return DyadicAdd(*args)
[docs] def projection(self, other, scalar=False):
"""
Returns the vector or scalar projection of the 'other' on 'self'.
Examples
========
>>> from sympy.vector.coordsysrect import CoordSysCartesian
>>> from sympy.vector.vector import Vector, BaseVector
>>> C = CoordSysCartesian('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
def __or__(self, other):
return self.outer(other)
__or__.__doc__ = outer.__doc__
[docs] def to_matrix(self, system):
"""
Returns the matrix form of this vector with respect to the
specified coordinate system.
Parameters
==========
system : CoordSysCartesian
The system wrt which the matrix form is to be computed
Examples
========
>>> from sympy.vector import CoordSysCartesian
>>> C = CoordSysCartesian('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()])
[docs] def separate(self):
"""
The constituents of this vector in different coordinate systems,
as per its definition.
Returns a dict mapping each CoordSysCartesian to the corresponding
constituent Vector.
Examples
========
>>> from sympy.vector import CoordSysCartesian
>>> R1 = CoordSysCartesian('R1')
>>> R2 = CoordSysCartesian('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
class BaseVector(Vector, AtomicExpr):
"""
Class to denote a base vector.
"""
def __new__(cls, name, index, system, pretty_str, latex_str):
name = str(name)
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, CoordSysCartesian):
raise TypeError("system should be a CoordSysCartesian")
# Initialize an object
obj = super(BaseVector, cls).__new__(cls, Symbol(name), S(index),
system, Symbol(pretty_str),
Symbol(latex_str))
# Assign important attributes
obj._base_instance = obj
obj._components = {obj: S(1)}
obj._measure_number = S(1)
obj._name = name
obj._pretty_form = u(pretty_str)
obj._latex_form = latex_str
obj._system = 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 __str__(self, printer=None):
return self._name
@property
def free_symbols(self):
return {self}
__repr__ = __str__
_sympystr = __str__
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 __str__(self, printer=None):
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 += temp_vect.__str__(printer) + " + "
return ret_str[:-3]
__repr__ = __str__
_sympystr = __str__
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 defition of
this VectorMul.
"""
return self._measure_number
class VectorZero(BasisDependentZero, Vector):
"""
Class to denote a zero vector
"""
_op_priority = 12.1
_pretty_form = u'0'
_latex_form = '\mathbf{\hat{0}}'
def __new__(cls):
obj = BasisDependentZero.__new__(cls)
return obj
def _vect_div(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")
Vector._expr_type = Vector
Vector._mul_func = VectorMul
Vector._add_func = VectorAdd
Vector._zero_func = VectorZero
Vector._base_func = BaseVector
Vector._div_helper = _vect_div
Vector.zero = VectorZero()
