def gradient(scalar_field, coord_sys=None, 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
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, 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_sys_from_expr(scalar_field, coord_sys)
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)
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 gradient(scalar_field, coord_sys=None, 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
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, 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_sys_from_expr(scalar_field, coord_sys)
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)
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)
