def test_del_operator():
#Tests for curl
assert (delop ^ Vector.zero ==
(Derivative(0, C.y) - Derivative(0, C.z))*C.i +
(-Derivative(0, C.x) + Derivative(0, C.z))*C.j +
(Derivative(0, C.x) - Derivative(0, C.y))*C.k)
assert ((delop ^ Vector.zero).doit() == Vector.zero ==
curl(Vector.zero, C))
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, C))
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, C))
#Tests for divergence
assert delop & Vector.zero == S(0) == divergence(Vector.zero, C)
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, C)
assert (delop.dot(v, doit = True) == x*y + y*z + z*x ==
divergence(v, C))
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, C)
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, C))
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, C)
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, C))
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
def test_del_operator():
#Tests for curl
assert (delop
^ Vector.zero == (Derivative(0, C.y) - Derivative(0, C.z)) * C.i +
(-Derivative(0, C.x) + Derivative(0, C.z)) * C.j +
(Derivative(0, C.x) - Derivative(0, C.y)) * C.k)
assert ((delop ^ Vector.zero).doit() == Vector.zero == curl(
Vector.zero, C))
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, C))
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, C))
#Tests for divergence
assert delop & Vector.zero == S(0) == divergence(Vector.zero, C)
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, C)
assert (delop.dot(v, doit=True) == x * y + y * z + z * x == divergence(
v, C))
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, C)
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, C))
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, C)
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, C))
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
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, C)
vector_field = y**2*i + 3*x*j + 5*y*z*k
curl_field = curl(vector_field, C)
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
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, C)
vector_field = y**2 * i + 3 * x * j + 5 * y * z * k
curl_field = curl(vector_field, C)
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():