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