def test_correct_arguments():
raises(ValueError, lambda: R2.e_x(R2.e_x))
raises(ValueError, lambda: R2.e_x(R2.dx))
raises(ValueError, lambda: Commutator(R2.e_x, R2.x))
raises(ValueError, lambda: Commutator(R2.dx, R2.e_x))
raises(ValueError, lambda: Differential(Differential(R2.e_x)))
raises(ValueError, lambda: R2.dx(R2.x))
raises(ValueError, lambda: LieDerivative(R2.dx, R2.dx))
raises(ValueError, lambda: LieDerivative(R2.x, R2.dx))
raises(ValueError, lambda: CovarDerivativeOp(R2.dx, []))
raises(ValueError, lambda: CovarDerivativeOp(R2.x, []))
a = Symbol('a')
raises(ValueError, lambda: intcurve_series(R2.dx, a, R2_r.point([1, 2])))
raises(ValueError, lambda: intcurve_series(R2.x, a, R2_r.point([1, 2])))
raises(ValueError, lambda: intcurve_diffequ(R2.dx, a, R2_r.point([1, 2])))
raises(ValueError, lambda: intcurve_diffequ(R2.x, a, R2_r.point([1, 2])))
raises(ValueError, lambda: contravariant_order(R2.e_x + R2.dx))
raises(ValueError, lambda: covariant_order(R2.e_x + R2.dx))
raises(ValueError, lambda: contravariant_order(R2.e_x * R2.e_y))
raises(ValueError, lambda: covariant_order(R2.dx * R2.dy))
def test_correct_arguments():
raises(ValueError, lambda : R2.e_x(R2.e_x))
raises(ValueError, lambda : R2.e_x(R2.dx))
raises(ValueError, lambda : Commutator(R2.e_x, R2.x))
raises(ValueError, lambda : Commutator(R2.dx, R2.e_x))
raises(ValueError, lambda : Differential(Differential(R2.e_x)))
raises(ValueError, lambda : R2.dx(R2.x))
raises(ValueError, lambda : TensorProduct(R2.e_x, R2.dx))
raises(ValueError, lambda : LieDerivative(R2.dx, R2.dx))
raises(ValueError, lambda : LieDerivative(R2.x, R2.dx))
raises(ValueError, lambda : CovarDerivativeOp(R2.dx, []))
raises(ValueError, lambda : CovarDerivativeOp(R2.x, []))
a = Symbol('a')
raises(ValueError, lambda : intcurve_series(R2.dx, a, R2_r.point([1,2])))
raises(ValueError, lambda : intcurve_series(R2.x, a, R2_r.point([1,2])))
raises(ValueError, lambda : intcurve_diffequ(R2.dx, a, R2_r.point([1,2])))
raises(ValueError, lambda : intcurve_diffequ(R2.x, a, R2_r.point([1,2])))
raises(ValueError, lambda : contravariant_order(R2.e_x + R2.dx))
raises(ValueError, lambda : covariant_order(R2.e_x + R2.dx))
raises(ValueError, lambda : contravariant_order(R2.e_x*R2.e_y))
raises(ValueError, lambda : covariant_order(R2.dx*R2.dy))
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)
one_vector = R2.e_x + R2.e_y
two_vector = TensorProduct(R2.e_x, R2.e_y)
three_vector = TensorProduct(R2.e_x, R2.e_y, R2.e_x)
two_wp = WedgeProduct(R2.e_x, R2.e_y)
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
assert contravariant_order(one_vector) == 1
assert contravariant_order(two_vector) == 2
assert contravariant_order(three_vector) == 3
assert contravariant_order(two_vector + two_wp) == 2
raises(ValueError, lambda: covariant_order(misform_a))
raises(ValueError, lambda: covariant_order(misform_b))
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]])
raises(ValueError, lambda: twoform_to_matrix(one_form))
raises(ValueError, lambda: twoform_to_matrix(three_form))
raises(ValueError, lambda: twoform_to_matrix(metric_ambig))
raises(ValueError, lambda: metric_to_Christoffel_1st(twoform_not_sym))
raises(ValueError, lambda: metric_to_Christoffel_2nd(twoform_not_sym))
raises(ValueError, lambda: metric_to_Riemann_components(twoform_not_sym))
raises(ValueError, lambda: metric_to_Ricci_components(twoform_not_sym))
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)
one_vector = R2.e_x + R2.e_y
two_vector = TensorProduct(R2.e_x, R2.e_y)
three_vector = TensorProduct(R2.e_x, R2.e_y, R2.e_x)
two_wp = WedgeProduct(R2.e_x,R2.e_y)
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
assert contravariant_order(one_vector) == 1
assert contravariant_order(two_vector) == 2
assert contravariant_order(three_vector) == 3
assert contravariant_order(two_vector + two_wp) == 2
raises(ValueError, lambda: covariant_order(misform_a))
raises(ValueError, lambda: covariant_order(misform_b))
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]])
raises(ValueError, lambda: twoform_to_matrix(one_form))
raises(ValueError, lambda: twoform_to_matrix(three_form))
raises(ValueError, lambda: twoform_to_matrix(metric_ambig))
raises(ValueError, lambda: metric_to_Christoffel_1st(twoform_not_sym))
raises(ValueError, lambda: metric_to_Christoffel_2nd(twoform_not_sym))
raises(ValueError, lambda: metric_to_Riemann_components(twoform_not_sym))
raises(ValueError, lambda: metric_to_Ricci_components(twoform_not_sym))
def exteriorderivative(self, f):
order = covariant_order(f)
# Automatically return 0 if f's grade is equal to manifold dimension
if order == self.dimension:
return 0
if order == 0:
return sum([
Differential(f)(D_x) * dx
for (D_x, dx) in zip(self.base_vectors, self.base_oneforms)
])
# If's f's grade is 1 or higher, we still need to implement this
if order > 0:
raise NotImplementedError(
'Exterior derivative of forms higher order than 1 is not implemented.'
)