def test_functional_diffgeom_ch6():
u0, u1, u2, v0, v1, v2, w0, w1, w2 = symbols('u0:3, v0:3, w0:3',
extended_real=True)
u = u0 * R2.e_x + u1 * R2.e_y
v = v0 * R2.e_x + v1 * R2.e_y
wp = WedgeProduct(R2.dx, R2.dy)
assert wp(u, v) == u0 * v1 - u1 * v0
u = u0 * R3_r.e_x + u1 * R3_r.e_y + u2 * R3_r.e_z
v = v0 * R3_r.e_x + v1 * R3_r.e_y + v2 * R3_r.e_z
w = w0 * R3_r.e_x + w1 * R3_r.e_y + w2 * R3_r.e_z
wp = WedgeProduct(R3_r.dx, R3_r.dy, R3_r.dz)
assert wp(u, v, w) == Matrix(3, 3,
[u0, u1, u2, v0, v1, v2, w0, w1, w2]).det()
a, b, c = symbols('a, b, c', cls=Function)
a_f = a(R3_r.x, R3_r.y, R3_r.z)
b_f = b(R3_r.x, R3_r.y, R3_r.z)
c_f = c(R3_r.x, R3_r.y, R3_r.z)
theta = a_f * R3_r.dx + b_f * R3_r.dy + c_f * R3_r.dz
dtheta = Differential(theta)
da = Differential(a_f)
db = Differential(b_f)
dc = Differential(c_f)
expr = dtheta - WedgeProduct(da, R3_r.dx) - WedgeProduct(
db, R3_r.dy) - WedgeProduct(dc, R3_r.dz)
assert expr.rcall(R3_r.e_x, R3_r.e_y) == 0
def test_differential():
xdy = R2.x * R2.dy
dxdy = Differential(xdy)
assert xdy.rcall(None) == xdy
assert dxdy(R2.e_x, R2.e_y) == 1
assert dxdy(R2.e_x, R2.x * R2.e_y) == R2.x
assert Differential(dxdy) == 0
def test_correct_arguments():
pytest.raises(ValueError, lambda: R2.e_x(R2.e_x))
pytest.raises(ValueError, lambda: R2.e_x(R2.dx))
pytest.raises(ValueError, lambda: Commutator(R2.e_x, R2.x))
pytest.raises(ValueError, lambda: Commutator(R2.dx, R2.e_x))
pytest.raises(ValueError, lambda: Differential(Differential(R2.e_x)))
pytest.raises(ValueError, lambda: R2.dx(R2.x))
pytest.raises(ValueError, lambda: TensorProduct(R2.e_x, R2.dx))
pytest.raises(ValueError, lambda: LieDerivative(R2.dx, R2.dx))
pytest.raises(ValueError, lambda: LieDerivative(R2.x, R2.dx))
pytest.raises(ValueError, lambda: CovarDerivativeOp(R2.dx, []))
pytest.raises(ValueError, lambda: CovarDerivativeOp(R2.x, []))
a = Symbol('a')
pytest.raises(ValueError, lambda: intcurve_series(R2.dx, a, R2_r.point([1, 2])))
pytest.raises(ValueError, lambda: intcurve_series(R2.x, a, R2_r.point([1, 2])))
pytest.raises(ValueError, lambda: intcurve_diffequ(R2.dx, a, R2_r.point([1, 2])))
pytest.raises(ValueError, lambda: intcurve_diffequ(R2.x, a, R2_r.point([1, 2])))
pytest.raises(ValueError, lambda: contravariant_order(R2.e_x + R2.dx))
pytest.raises(ValueError, lambda: contravariant_order(R2.dx**2))
pytest.raises(ValueError, lambda: covariant_order(R2.e_x + R2.dx))
pytest.raises(ValueError, lambda: contravariant_order(R2.e_x*R2.e_y))
pytest.raises(ValueError, lambda: covariant_order(R2.dx*R2.dy))
assert covariant_order(Integer(0), True) == -1
assert contravariant_order(Integer(0), True) == -1
def test_Differential():
tp = TensorProduct(R2.dx, R2.dy)
assert sstr(LieDerivative(R2.e_x, tp)) == 'LieDerivative(e_x, TensorProduct(dx, dy))'
g = Function('g')
s_field = g(R2.x, R2.y)
assert sstr(Differential(s_field)) == 'd(g(x, y))'
def test_helpers_and_coordinate_dependent():
one_form = R2.dr + R2.dx
two_form = Differential(R2.x * R2.dr + R2.r * R2.dx)
three_form = Differential(R2.y * two_form) + Differential(
R2.x * Differential(R2.r * R2.dr))
metric = TensorProduct(R2.dx, R2.dx) + TensorProduct(R2.dy, R2.dy)
metric_ambig = TensorProduct(R2.dx, R2.dx) + TensorProduct(R2.dr, R2.dr)
misform_a = TensorProduct(R2.dr, R2.dr) + R2.dr
misform_b = R2.dr**4
misform_c = R2.dx * R2.dy
twoform_not_sym = TensorProduct(R2.dx, R2.dx) + TensorProduct(R2.dx, R2.dy)
twoform_not_TP = WedgeProduct(R2.dx, R2.dy)
assert covariant_order(one_form) == 1
assert covariant_order(two_form) == 2
assert covariant_order(three_form) == 3
assert covariant_order(two_form + metric) == 2
assert covariant_order(two_form + metric_ambig) == 2
assert covariant_order(two_form + twoform_not_sym) == 2
assert covariant_order(two_form + twoform_not_TP) == 2
pytest.raises(ValueError, lambda: covariant_order(misform_a))
pytest.raises(ValueError, lambda: covariant_order(misform_b))
pytest.raises(ValueError, lambda: covariant_order(misform_c))
assert twoform_to_matrix(metric) == Matrix([[1, 0], [0, 1]])
assert twoform_to_matrix(twoform_not_sym) == Matrix([[1, 0], [1, 0]])
assert twoform_to_matrix(twoform_not_TP) == Matrix([[0, -1], [1, 0]])
pytest.raises(ValueError, lambda: twoform_to_matrix(one_form))
pytest.raises(ValueError, lambda: twoform_to_matrix(three_form))
pytest.raises(ValueError, lambda: twoform_to_matrix(metric_ambig))
pytest.raises(ValueError,
lambda: metric_to_Christoffel_1st(twoform_not_sym))
pytest.raises(ValueError,
lambda: metric_to_Christoffel_2nd(twoform_not_sym))
pytest.raises(ValueError,
lambda: metric_to_Riemann_components(twoform_not_sym))
pytest.raises(ValueError,
lambda: metric_to_Ricci_components(twoform_not_sym))
def test_functional_diffgeom_ch4():
x0, y0, theta0 = symbols('x0, y0, theta0', extended_real=True)
x, y, r, theta = symbols('x, y, r, theta', extended_real=True)
r0 = symbols('r0', positive=True)
f = Function('f')
b1 = Function('b1')
b2 = Function('b2')
p_r = R2_r.point([x0, y0])
p_p = R2_p.point([r0, theta0])
f_field = b1(R2.x, R2.y) * R2.dx + b2(R2.x, R2.y) * R2.dy
assert f_field.rcall(R2.e_x).rcall(p_r) == b1(x0, y0)
assert f_field.rcall(R2.e_y).rcall(p_r) == b2(x0, y0)
s_field_r = f(R2.x, R2.y)
df = Differential(s_field_r)
assert df(R2.e_x).rcall(p_r).doit() == Derivative(f(x0, y0), x0)
assert df(R2.e_y).rcall(p_r).doit() == Derivative(f(x0, y0), y0)
s_field_p = f(R2.r, R2.theta)
df = Differential(s_field_p)
assert trigsimp(df(R2.e_x).rcall(p_p).doit()) == (
cos(theta0) * Derivative(f(r0, theta0), r0) -
sin(theta0) * Derivative(f(r0, theta0), theta0) / r0)
assert trigsimp(df(R2.e_y).rcall(p_p).doit()) == (
sin(theta0) * Derivative(f(r0, theta0), r0) +
cos(theta0) * Derivative(f(r0, theta0), theta0) / r0)
assert R2.dx(R2.e_x).rcall(p_r) == 1
assert R2.dx(R2.e_x) == 1
assert R2.dx(R2.e_y).rcall(p_r) == 0
assert R2.dx(R2.e_y) == 0
circ = -R2.y * R2.e_x + R2.x * R2.e_y
assert R2.dx(circ).rcall(p_r).doit() == -y0
assert R2.dy(circ).rcall(p_r) == x0
assert R2.dr(circ).rcall(p_r) == 0
assert simplify(R2.dtheta(circ).rcall(p_r)) == 1
assert (circ - R2.e_theta).rcall(s_field_r).rcall(p_r) == 0
