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 test_differential_operators_curvilinear_system():
A = CoordSys3D(
"A", transformation="spherical", variable_names=["r", "theta", "phi"]
)
B = CoordSys3D(
"B", transformation="cylindrical", variable_names=["r", "theta", "z"]
)
# Test for spherical coordinate system and gradient
assert gradient(3 * A.r + 4 * A.theta) == 3 * A.i + 4 / A.r * A.j
assert (
gradient(3 * A.r * A.phi + 4 * A.theta)
== 3 * A.phi * A.i + 4 / A.r * A.j + (3 / sin(A.theta)) * A.k
)
assert gradient(0 * A.r + 0 * A.theta + 0 * A.phi) == Vector.zero
assert (
gradient(A.r * A.theta * A.phi)
== A.theta * A.phi * A.i + A.phi * A.j + (A.theta / sin(A.theta)) * A.k
)
# Test for spherical coordinate system and divergence
assert divergence(A.r * A.i + A.theta * A.j + A.phi * A.k) == (
sin(A.theta) * A.r + cos(A.theta) * A.r * A.theta
) / (sin(A.theta) * A.r ** 2) + 3 + 1 / (sin(A.theta) * A.r)
assert divergence(
3 * A.r * A.phi * A.i + A.theta * A.j + A.r * A.theta * A.phi * A.k
) == (sin(A.theta) * A.r + cos(A.theta) * A.r * A.theta) / (
sin(A.theta) * A.r ** 2
) + 9 * A.phi + A.theta / sin(
A.theta
)
assert divergence(Vector.zero) == 0
assert divergence(0 * A.i + 0 * A.j + 0 * A.k) == 0
# Test for spherical coordinate system and curl
assert (
curl(A.r * A.i + A.theta * A.j + A.phi * A.k)
== (cos(A.theta) * A.phi / (sin(A.theta) * A.r)) * A.i
+ (-A.phi / A.r) * A.j
+ A.theta / A.r * A.k
)
assert (
curl(A.r * A.j + A.phi * A.k)
== (cos(A.theta) * A.phi / (sin(A.theta) * A.r)) * A.i
+ (-A.phi / A.r) * A.j
+ 2 * A.k
)
# Test for cylindrical coordinate system and gradient
assert gradient(0 * B.r + 0 * B.theta + 0 * B.z) == Vector.zero
assert (
gradient(B.r * B.theta * B.z)
== B.theta * B.z * B.i + B.z * B.j + B.r * B.theta * B.k
)
assert gradient(3 * B.r) == 3 * B.i
assert gradient(2 * B.theta) == 2 / B.r * B.j
assert gradient(4 * B.z) == 4 * B.k
# Test for cylindrical coordinate system and divergence
assert divergence(B.r * B.i + B.theta * B.j + B.z * B.k) == 3 + 1 / B.r
assert divergence(B.r * B.j + B.z * B.k) == 1
# Test for cylindrical coordinate system and curl
assert curl(B.r * B.j + B.z * B.k) == 2 * B.k
assert curl(3 * B.i + 2 / B.r * B.j + 4 * B.k) == Vector.zero
def test_differential_operators_curvilinear_system():
A = CoordSys3D('A', transformation="spherical", variable_names=["r", "theta", "phi"])
B = CoordSys3D('B', transformation='cylindrical', variable_names=["r", "theta", "z"])
# Test for spherical coordinate system and gradient
assert gradient(3*A.r + 4*A.theta) == 3*A.i + 4/A.r*A.j
assert gradient(3*A.r*A.phi + 4*A.theta) == 3*A.phi*A.i + 4/A.r*A.j + (3/sin(A.theta))*A.k
assert gradient(0*A.r + 0*A.theta+0*A.phi) == Vector.zero
assert gradient(A.r*A.theta*A.phi) == A.theta*A.phi*A.i + A.phi*A.j + (A.theta/sin(A.theta))*A.k
# Test for spherical coordinate system and divergence
assert divergence(A.r * A.i + A.theta * A.j + A.phi * A.k) == \
(sin(A.theta)*A.r + cos(A.theta)*A.r*A.theta)/(sin(A.theta)*A.r**2) + 3 + 1/(sin(A.theta)*A.r)
assert divergence(3*A.r*A.phi*A.i + A.theta*A.j + A.r*A.theta*A.phi*A.k) == \
(sin(A.theta)*A.r + cos(A.theta)*A.r*A.theta)/(sin(A.theta)*A.r**2) + 9*A.phi + A.theta/sin(A.theta)
assert divergence(Vector.zero) == 0
assert divergence(0*A.i + 0*A.j + 0*A.k) == 0
# Test for spherical coordinate system and curl
assert curl(A.r*A.i + A.theta*A.j + A.phi*A.k) == \
(cos(A.theta)*A.phi/(sin(A.theta)*A.r))*A.i + (-A.phi/A.r)*A.j + A.theta/A.r*A.k
assert curl(A.r*A.j + A.phi*A.k) == (cos(A.theta)*A.phi/(sin(A.theta)*A.r))*A.i + (-A.phi/A.r)*A.j + 2*A.k
# Test for cylindrical coordinate system and gradient
assert gradient(0*B.r + 0*B.theta+0*B.z) == Vector.zero
assert gradient(B.r*B.theta*B.z) == B.theta*B.z*B.i + B.z*B.j + B.r*B.theta*B.k
assert gradient(3*B.r) == 3*B.i
assert gradient(2*B.theta) == 2/B.r * B.j
assert gradient(4*B.z) == 4*B.k
# Test for cylindrical coordinate system and divergence
assert divergence(B.r*B.i + B.theta*B.j + B.z*B.k) == 3 + 1/B.r
assert divergence(B.r*B.j + B.z*B.k) == 1
# Test for cylindrical coordinate system and curl
assert curl(B.r*B.j + B.z*B.k) == 2*B.k
assert curl(3*B.i + 2/B.r*B.j + 4*B.k) == Vector.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 test_differential_operators_curvilinear_system():
A = CoordSys3D('A')
A._set_lame_coefficient_mapping('spherical')
B = CoordSys3D('B')
B._set_lame_coefficient_mapping('cylindrical')
# Test for spherical coordinate system and gradient
assert gradient(3 * A.x + 4 * A.y) == 3 * A.i + 4 / A.x * A.j
assert gradient(
3 * A.x * A.z +
4 * A.y) == 3 * A.z * A.i + 4 / A.x * A.j + (3 / sin(A.y)) * A.k
assert gradient(0 * A.x + 0 * A.y + 0 * A.z) == Vector.zero
assert gradient(
A.x * A.y *
A.z) == A.y * A.z * A.i + A.z * A.j + (A.y / sin(A.y)) * A.k
# Test for spherical coordinate system and divergence
assert divergence(A.x * A.i + A.y * A.j + A.z * A.k) == \
(sin(A.y)*A.x + cos(A.y)*A.x*A.y)/(sin(A.y)*A.x**2) + 3 + 1/(sin(A.y)*A.x)
assert divergence(3*A.x*A.z*A.i + A.y*A.j + A.x*A.y*A.z*A.k) == \
(sin(A.y)*A.x + cos(A.y)*A.x*A.y)/(sin(A.y)*A.x**2) + 9*A.z + A.y/sin(A.y)
assert divergence(Vector.zero) == 0
assert divergence(0 * A.i + 0 * A.j + 0 * A.k) == 0
# Test for cylindrical coordinate system and divergence
assert divergence(B.x * B.i + B.y * B.j + B.z * B.k) == 2 + 1 / B.y
assert divergence(B.x * B.j + B.z * B.k) == 1
# Test for spherical coordinate system and divergence
assert curl(A.x*A.i + A.y*A.j + A.z*A.k) == \
(cos(A.y)*A.z/(sin(A.y)*A.x))*A.i + (-A.z/A.x)*A.j + A.y/A.x*A.k
assert curl(A.x * A.j + A.z *
A.k) == (cos(A.y) * A.z /
(sin(A.y) * A.x)) * A.i + (-A.z / A.x) * A.j + 2 * A.k
def is_conservative(field):
"""
Checks if a field is conservative.
Paramaters
==========
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_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 cross(self, vect, doit=False):
"""
Represents the cross product between this operator and a given
vector - equal to the curl of the vector field.
Parameters
==========
vect : Vector
The vector whose curl is to be calculated.
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, Del
>>> C = CoordSys3D('C')
>>> delop = Del()
>>> v = C.x*C.y*C.z * (C.i + C.j + C.k)
>>> delop.cross(v, doit = True)
(-C.x*C.y + C.x*C.z)*C.i + (C.x*C.y - C.y*C.z)*C.j +
(-C.x*C.z + C.y*C.z)*C.k
>>> (delop ^ C.i).doit()
0
"""
return curl(vect, doit=doit)
def cross(self, vect, doit=False):
"""
Represents the cross product between this operator and a given
vector - equal to the curl of the vector field.
Parameters
==========
vect : Vector
The vector whose curl is to be calculated.
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, Del
>>> C = CoordSys3D('C')
>>> delop = Del()
>>> v = C.x*C.y*C.z * (C.i + C.j + C.k)
>>> delop.cross(v, doit = True)
(-C.x*C.y + C.x*C.z)*C.i + (C.x*C.y - C.y*C.z)*C.j +
(-C.x*C.z + C.y*C.z)*C.k
>>> (delop ^ C.i).doit()
0
"""
return curl(vect, doit=doit)
def test_differential_operators_curvilinear_system():
A = CoordSys3D('A')
A._set_lame_coefficient_mapping('spherical')
B = CoordSys3D('B')
B._set_lame_coefficient_mapping('cylindrical')
# Test for spherical coordinate system and gradient
assert gradient(3*A.x + 4*A.y) == 3*A.i + 4/A.x*A.j
assert gradient(3*A.x*A.z + 4*A.y) == 3*A.z*A.i + 4/A.x*A.j + (3/sin(A.y))*A.k
assert gradient(0*A.x + 0*A.y+0*A.z) == Vector.zero
assert gradient(A.x*A.y*A.z) == A.y*A.z*A.i + A.z*A.j + (A.y/sin(A.y))*A.k
# Test for spherical coordinate system and divergence
assert divergence(A.x * A.i + A.y * A.j + A.z * A.k) == \
(sin(A.y)*A.x + cos(A.y)*A.x*A.y)/(sin(A.y)*A.x**2) + 3 + 1/(sin(A.y)*A.x)
assert divergence(3*A.x*A.z*A.i + A.y*A.j + A.x*A.y*A.z*A.k) == \
(sin(A.y)*A.x + cos(A.y)*A.x*A.y)/(sin(A.y)*A.x**2) + 9*A.z + A.y/sin(A.y)
assert divergence(Vector.zero) == 0
assert divergence(0*A.i + 0*A.j + 0*A.k) == 0
# Test for cylindrical coordinate system and divergence
assert divergence(B.x*B.i + B.y*B.j + B.z*B.k) == 2 + 1/B.y
assert divergence(B.x*B.j + B.z*B.k) == 1
# Test for spherical coordinate system and divergence
assert curl(A.x*A.i + A.y*A.j + A.z*A.k) == \
(cos(A.y)*A.z/(sin(A.y)*A.x))*A.i + (-A.z/A.x)*A.j + A.y/A.x*A.k
assert curl(A.x*A.j + A.z*A.k) == (cos(A.y)*A.z/(sin(A.y)*A.x))*A.i + (-A.z/A.x)*A.j + 2*A.k
def test_del_operator():
# Tests for curl
assert delop ^ Vector.zero == Vector.zero
assert (delop ^ Vector.zero).doit() == Vector.zero == curl(Vector.zero)
assert delop.cross(Vector.zero) == delop ^ Vector.zero
assert (delop ^ i).doit() == Vector.zero
assert delop.cross(2 * y ** 2 * j, doit=True) == Vector.zero
assert delop.cross(2 * y ** 2 * j) == delop ^ 2 * y ** 2 * j
v = x * y * z * (i + j + k)
assert (
(delop ^ v).doit()
== (-x * y + x * z) * i + (x * y - y * z) * j + (-x * z + y * z) * k
== curl(v)
)
assert delop ^ v == delop.cross(v)
assert (
delop.cross(2 * x ** 2 * j)
== (Derivative(0, C.y) - Derivative(2 * C.x ** 2, C.z)) * C.i
+ (-Derivative(0, C.x) + Derivative(0, C.z)) * C.j
+ (-Derivative(0, C.y) + Derivative(2 * C.x ** 2, C.x)) * C.k
)
assert delop.cross(2 * x ** 2 * j, doit=True) == 4 * x * k == curl(2 * x ** 2 * j)
# Tests for divergence
assert delop & Vector.zero is S.Zero == divergence(Vector.zero)
assert (delop & Vector.zero).doit() is S.Zero
assert delop.dot(Vector.zero) == delop & Vector.zero
assert (delop & i).doit() is S.Zero
assert (delop & x ** 2 * i).doit() == 2 * x == divergence(x ** 2 * i)
assert delop.dot(v, doit=True) == x * y + y * z + z * x == divergence(v)
assert delop & v == delop.dot(v)
assert delop.dot(1 / (x * y * z) * (i + j + k), doit=True) == -1 / (
x * y * z ** 2
) - 1 / (x * y ** 2 * z) - 1 / (x ** 2 * y * z)
v = x * i + y * j + z * k
assert delop & v == Derivative(C.x, C.x) + Derivative(C.y, C.y) + Derivative(
C.z, C.z
)
assert delop.dot(v, doit=True) == 3 == divergence(v)
assert delop & v == delop.dot(v)
assert simplify((delop & v).doit()) == 3
# Tests for gradient
assert delop.gradient(0, doit=True) == Vector.zero == gradient(0)
assert delop.gradient(0) == delop(0)
assert (delop(S.Zero)).doit() == Vector.zero
assert (
delop(x)
== (Derivative(C.x, C.x)) * C.i
+ (Derivative(C.x, C.y)) * C.j
+ (Derivative(C.x, C.z)) * C.k
)
assert (delop(x)).doit() == i == gradient(x)
assert (
delop(x * y * z)
== (Derivative(C.x * C.y * C.z, C.x)) * C.i
+ (Derivative(C.x * C.y * C.z, C.y)) * C.j
+ (Derivative(C.x * C.y * C.z, C.z)) * C.k
)
assert (
delop.gradient(x * y * z, doit=True)
== y * z * i + z * x * j + x * y * k
== gradient(x * y * z)
)
assert delop(x * y * z) == delop.gradient(x * y * z)
assert (delop(2 * x ** 2)).doit() == 4 * x * i
assert (delop(a * sin(y) / x)).doit() == -a * sin(y) / x ** 2 * i + a * cos(
y
) / x * j
# Tests for directional derivative
assert (Vector.zero & delop)(a) is S.Zero
assert ((Vector.zero & delop)(a)).doit() is S.Zero
assert ((v & delop)(Vector.zero)).doit() == Vector.zero
assert ((v & delop)(S.Zero)).doit() is S.Zero
assert ((i & delop)(x)).doit() == 1
assert ((j & delop)(y)).doit() == 1
assert ((k & delop)(z)).doit() == 1
assert ((i & delop)(x * y * z)).doit() == y * z
assert ((v & delop)(x)).doit() == x
assert ((v & delop)(x * y * z)).doit() == 3 * x * y * z
assert (v & delop)(x + y + z) == C.x + C.y + C.z
assert ((v & delop)(x + y + z)).doit() == x + y + z
assert ((v & delop)(v)).doit() == v
assert ((i & delop)(v)).doit() == i
assert ((j & delop)(v)).doit() == j
assert ((k & delop)(v)).doit() == k
assert ((v & delop)(Vector.zero)).doit() == Vector.zero
# Tests for laplacian on scalar fields
assert laplacian(x * y * z) is S.Zero
assert laplacian(x ** 2) == S(2)
assert (
laplacian(x ** 2 * y ** 2 * z ** 2)
== 2 * y ** 2 * z ** 2 + 2 * x ** 2 * z ** 2 + 2 * x ** 2 * y ** 2
)
A = CoordSys3D(
"A", transformation="spherical", variable_names=["r", "theta", "phi"]
)
B = CoordSys3D(
"B", transformation="cylindrical", variable_names=["r", "theta", "z"]
)
assert laplacian(A.r + A.theta + A.phi) == 2 / A.r + cos(A.theta) / (
A.r ** 2 * sin(A.theta)
)
assert laplacian(B.r + B.theta + B.z) == 1 / B.r
# Tests for laplacian on vector fields
assert laplacian(x * y * z * (i + j + k)) == Vector.zero
assert (
laplacian(x * y ** 2 * z * (i + j + k))
== 2 * x * z * i + 2 * x * z * j + 2 * x * z * k
)
# Fifth product rule
lhs = (delop ^ (f * v)).doit()
rhs = (((delop(f)) ^ v) + (f * (delop ^ v))).doit()
assert simplify(lhs) == simplify(rhs)
# Sixth product rule
lhs = (delop ^ (u ^ v)).doit()
rhs = ((u * (delop & v) - v * (delop & u) + (v & delop)(u) - (u & delop)(v))).doit()
assert simplify(lhs) == simplify(rhs)
P = C.orient_new_axis("P", q, C.k) # type: ignore
scalar_field = 2 * x ** 2 * y * z
grad_field = gradient(scalar_field)
vector_field = y ** 2 * i + 3 * x * j + 5 * y * z * k
curl_field = curl(vector_field)
def test_conservative():
assert is_conservative(Vector.zero) is True
assert is_conservative(i) is True
assert is_conservative(2 * i + 3 * j + 4 * k) is True
assert is_conservative(y * z * i + x * z * j + x * y * k) is True
assert is_conservative(x * j) is False
assert is_conservative(grad_field) is True
assert is_conservative(curl_field) is False
assert is_conservative(4 * x * y * z * i + 2 * x ** 2 * z * j) is False
assert is_conservative(z * P.i + P.x * k) is True
def test_solenoidal():
def test_del_operator():
# Tests for curl
assert delop ^ Vector.zero == Vector.zero
assert ((delop ^ Vector.zero).doit() == Vector.zero ==
curl(Vector.zero))
assert delop.cross(Vector.zero) == delop ^ Vector.zero
assert (delop ^ i).doit() == Vector.zero
assert delop.cross(2*y**2*j, doit=True) == Vector.zero
assert delop.cross(2*y**2*j) == delop ^ 2*y**2*j
v = x*y*z * (i + j + k)
assert ((delop ^ v).doit() ==
(-x*y + x*z)*i + (x*y - y*z)*j + (-x*z + y*z)*k ==
curl(v))
assert delop ^ v == delop.cross(v)
assert (delop.cross(2*x**2*j) ==
(Derivative(0, C.y) - Derivative(2*C.x**2, C.z))*C.i +
(-Derivative(0, C.x) + Derivative(0, C.z))*C.j +
(-Derivative(0, C.y) + Derivative(2*C.x**2, C.x))*C.k)
assert (delop.cross(2*x**2*j, doit=True) == 4*x*k ==
curl(2*x**2*j))
#Tests for divergence
assert delop & Vector.zero == S(0) == divergence(Vector.zero)
assert (delop & Vector.zero).doit() == S(0)
assert delop.dot(Vector.zero) == delop & Vector.zero
assert (delop & i).doit() == S(0)
assert (delop & x**2*i).doit() == 2*x == divergence(x**2*i)
assert (delop.dot(v, doit=True) == x*y + y*z + z*x ==
divergence(v))
assert delop & v == delop.dot(v)
assert delop.dot(1/(x*y*z) * (i + j + k), doit=True) == \
- 1 / (x*y*z**2) - 1 / (x*y**2*z) - 1 / (x**2*y*z)
v = x*i + y*j + z*k
assert (delop & v == Derivative(C.x, C.x) +
Derivative(C.y, C.y) + Derivative(C.z, C.z))
assert delop.dot(v, doit=True) == 3 == divergence(v)
assert delop & v == delop.dot(v)
assert simplify((delop & v).doit()) == 3
#Tests for gradient
assert (delop.gradient(0, doit=True) == Vector.zero ==
gradient(0))
assert delop.gradient(0) == delop(0)
assert (delop(S(0))).doit() == Vector.zero
assert (delop(x) == (Derivative(C.x, C.x))*C.i +
(Derivative(C.x, C.y))*C.j + (Derivative(C.x, C.z))*C.k)
assert (delop(x)).doit() == i == gradient(x)
assert (delop(x*y*z) ==
(Derivative(C.x*C.y*C.z, C.x))*C.i +
(Derivative(C.x*C.y*C.z, C.y))*C.j +
(Derivative(C.x*C.y*C.z, C.z))*C.k)
assert (delop.gradient(x*y*z, doit=True) ==
y*z*i + z*x*j + x*y*k ==
gradient(x*y*z))
assert delop(x*y*z) == delop.gradient(x*y*z)
assert (delop(2*x**2)).doit() == 4*x*i
assert ((delop(a*sin(y) / x)).doit() ==
-a*sin(y)/x**2 * i + a*cos(y)/x * j)
#Tests for directional derivative
assert (Vector.zero & delop)(a) == S(0)
assert ((Vector.zero & delop)(a)).doit() == S(0)
assert ((v & delop)(Vector.zero)).doit() == Vector.zero
assert ((v & delop)(S(0))).doit() == S(0)
assert ((i & delop)(x)).doit() == 1
assert ((j & delop)(y)).doit() == 1
assert ((k & delop)(z)).doit() == 1
assert ((i & delop)(x*y*z)).doit() == y*z
assert ((v & delop)(x)).doit() == x
assert ((v & delop)(x*y*z)).doit() == 3*x*y*z
assert (v & delop)(x + y + z) == C.x + C.y + C.z
assert ((v & delop)(x + y + z)).doit() == x + y + z
assert ((v & delop)(v)).doit() == v
assert ((i & delop)(v)).doit() == i
assert ((j & delop)(v)).doit() == j
assert ((k & delop)(v)).doit() == k
assert ((v & delop)(Vector.zero)).doit() == Vector.zero
# Tests for laplacian on scalar fields
assert laplacian(x*y*z) == S.Zero
assert laplacian(x**2) == S(2)
assert laplacian(x**2*y**2*z**2) == \
2*y**2*z**2 + 2*x**2*z**2 + 2*x**2*y**2
# Tests for laplacian on vector fields
assert laplacian(x*y*z*(i + j + k)) == Vector.zero
assert laplacian(x*y**2*z*(i + j + k)) == \
2*x*z*i + 2*x*z*j + 2*x*z*k
lhs = (delop ^ (f * v)).doit()
rhs = (((delop(f)) ^ v) + (f * (delop ^ v))).doit()
assert simplify(lhs) == simplify(rhs)
# Sixth product rule
lhs = (delop ^ (u ^ v)).doit()
rhs = ((u * (delop & v) - v * (delop & u) +
(v & delop)(u) - (u & delop)(v))).doit()
assert simplify(lhs) == simplify(rhs)
P = C.orient_new_axis('P', q, C.k)
scalar_field = 2*x**2*y*z
grad_field = gradient(scalar_field)
vector_field = y**2*i + 3*x*j + 5*y*z*k
curl_field = curl(vector_field)
def test_conservative():
assert is_conservative(Vector.zero) is True
assert is_conservative(i) is True
assert is_conservative(2 * i + 3 * j + 4 * k) is True
assert (is_conservative(y*z*i + x*z*j + x*y*k) is
True)
assert is_conservative(x * j) is False
assert is_conservative(grad_field) is True
assert is_conservative(curl_field) is False
assert (is_conservative(4*x*y*z*i + 2*x**2*z*j) is
False)
assert is_conservative(z*P.i + P.x*k) is True
def test_del_operator():
# Tests for curl
assert delop ^ Vector.zero == Vector.zero
assert ((delop ^ Vector.zero).doit() == Vector.zero ==
curl(Vector.zero))
assert delop.cross(Vector.zero) == delop ^ Vector.zero
assert (delop ^ i).doit() == Vector.zero
assert delop.cross(2*y**2*j, doit=True) == Vector.zero
assert delop.cross(2*y**2*j) == delop ^ 2*y**2*j
v = x*y*z * (i + j + k)
assert ((delop ^ v).doit() ==
(-x*y + x*z)*i + (x*y - y*z)*j + (-x*z + y*z)*k ==
curl(v))
assert delop ^ v == delop.cross(v)
assert (delop.cross(2*x**2*j) ==
(Derivative(0, C.y) - Derivative(2*C.x**2, C.z))*C.i +
(-Derivative(0, C.x) + Derivative(0, C.z))*C.j +
(-Derivative(0, C.y) + Derivative(2*C.x**2, C.x))*C.k)
assert (delop.cross(2*x**2*j, doit=True) == 4*x*k ==
curl(2*x**2*j))
#Tests for divergence
assert delop & Vector.zero == S(0) == divergence(Vector.zero)
assert (delop & Vector.zero).doit() == S(0)
assert delop.dot(Vector.zero) == delop & Vector.zero
assert (delop & i).doit() == S(0)
assert (delop & x**2*i).doit() == 2*x == divergence(x**2*i)
assert (delop.dot(v, doit=True) == x*y + y*z + z*x ==
divergence(v))
assert delop & v == delop.dot(v)
assert delop.dot(1/(x*y*z) * (i + j + k), doit=True) == \
- 1 / (x*y*z**2) - 1 / (x*y**2*z) - 1 / (x**2*y*z)
v = x*i + y*j + z*k
assert (delop & v == Derivative(C.x, C.x) +
Derivative(C.y, C.y) + Derivative(C.z, C.z))
assert delop.dot(v, doit=True) == 3 == divergence(v)
assert delop & v == delop.dot(v)
assert simplify((delop & v).doit()) == 3
#Tests for gradient
assert (delop.gradient(0, doit=True) == Vector.zero ==
gradient(0))
assert delop.gradient(0) == delop(0)
assert (delop(S(0))).doit() == Vector.zero
assert (delop(x) == (Derivative(C.x, C.x))*C.i +
(Derivative(C.x, C.y))*C.j + (Derivative(C.x, C.z))*C.k)
assert (delop(x)).doit() == i == gradient(x)
assert (delop(x*y*z) ==
(Derivative(C.x*C.y*C.z, C.x))*C.i +
(Derivative(C.x*C.y*C.z, C.y))*C.j +
(Derivative(C.x*C.y*C.z, C.z))*C.k)
assert (delop.gradient(x*y*z, doit=True) ==
y*z*i + z*x*j + x*y*k ==
gradient(x*y*z))
assert delop(x*y*z) == delop.gradient(x*y*z)
assert (delop(2*x**2)).doit() == 4*x*i
assert ((delop(a*sin(y) / x)).doit() ==
-a*sin(y)/x**2 * i + a*cos(y)/x * j)
#Tests for directional derivative
assert (Vector.zero & delop)(a) == S(0)
assert ((Vector.zero & delop)(a)).doit() == S(0)
assert ((v & delop)(Vector.zero)).doit() == Vector.zero
assert ((v & delop)(S(0))).doit() == S(0)
assert ((i & delop)(x)).doit() == 1
assert ((j & delop)(y)).doit() == 1
assert ((k & delop)(z)).doit() == 1
assert ((i & delop)(x*y*z)).doit() == y*z
assert ((v & delop)(x)).doit() == x
assert ((v & delop)(x*y*z)).doit() == 3*x*y*z
assert (v & delop)(x + y + z) == C.x + C.y + C.z
assert ((v & delop)(x + y + z)).doit() == x + y + z
assert ((v & delop)(v)).doit() == v
assert ((i & delop)(v)).doit() == i
assert ((j & delop)(v)).doit() == j
assert ((k & delop)(v)).doit() == k
assert ((v & delop)(Vector.zero)).doit() == Vector.zero
