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_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 curl(vect, coord_sys=None, 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
coord_sys : CoordSys3D
The coordinate system to calculate the gradient in.
Deprecated since version 1.1
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_sys_from_expr(vect, coord_sys)
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 curl(vect, coord_sys=None, 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
coord_sys : CoordSys3D
The coordinate system to calculate the gradient in.
Deprecated since version 1.1
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_sys_from_expr(vect, coord_sys)
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)