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 + Dot(b.j, c.z*b.i + b.x*c.k)
assert divergence(3*cos(q)*a.x*b.x*b.i*c.x) == \
Dot(b.i, (3*b.x*c.x*cos(q))*a.i + (3*a.x*c.x*cos(q))*b.i + (3*a.x*b.x*cos(q))*c.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)) + \
Dot(Cross(a.x*a.i, a.y*b.j), b.x*c.x*a.i + a.x*c.x*b.i + a.x*b.x*c.i)
assert divergence(a.x*b.x*c.x*(a.x*a.i + b.x*b.i)) ==\
Dot(a.i, 2*a.x*b.x*c.x*a.i + a.x**2*c.x*b.i + a.x**2*b.x*c.i) + \
Dot(b.i, b.x**2*c.x*a.i + 2*a.x*b.x*c.x*b.i + a.x*b.x**2*c.i)
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 divergence(vect, coord_sys=None, 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
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, 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_sys_from_expr(vect, coord_sys)
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 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
)
def divergence(vect, coord_sys=None, 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
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, 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_sys_from_expr(vect, coord_sys)
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)