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_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_H2():
TP = diofant.diffgeom.TensorProduct
R2 = diofant.diffgeom.rn.R2
y = R2.y
dy = R2.dy
dx = R2.dx
g = (TP(dx, dx) + TP(dy, dy)) * y**(-2)
automat = twoform_to_matrix(g)
mat = diag(y**(-2), y**(-2))
assert mat == automat
gamma1 = metric_to_Christoffel_1st(g)
assert gamma1[0, 0, 0] == 0
assert gamma1[0, 0, 1] == -y**(-3)
assert gamma1[0, 1, 0] == -y**(-3)
assert gamma1[0, 1, 1] == 0
assert gamma1[1, 1, 1] == -y**(-3)
assert gamma1[1, 1, 0] == 0
assert gamma1[1, 0, 1] == 0
assert gamma1[1, 0, 0] == y**(-3)
gamma2 = metric_to_Christoffel_2nd(g)
assert gamma2[0, 0, 0] == 0
assert gamma2[0, 0, 1] == -y**(-1)
assert gamma2[0, 1, 0] == -y**(-1)
assert gamma2[0, 1, 1] == 0
assert gamma2[1, 1, 1] == -y**(-1)
assert gamma2[1, 1, 0] == 0
assert gamma2[1, 0, 1] == 0
assert gamma2[1, 0, 0] == y**(-1)
Rm = metric_to_Riemann_components(g)
assert Rm[0, 0, 0, 0] == 0
assert Rm[0, 0, 0, 1] == 0
assert Rm[0, 0, 1, 0] == 0
assert Rm[0, 0, 1, 1] == 0
assert Rm[0, 1, 0, 0] == 0
assert Rm[0, 1, 0, 1] == -y**(-2)
assert Rm[0, 1, 1, 0] == y**(-2)
assert Rm[0, 1, 1, 1] == 0
assert Rm[1, 0, 0, 0] == 0
assert Rm[1, 0, 0, 1] == y**(-2)
assert Rm[1, 0, 1, 0] == -y**(-2)
assert Rm[1, 0, 1, 1] == 0
assert Rm[1, 1, 0, 0] == 0
assert Rm[1, 1, 0, 1] == 0
assert Rm[1, 1, 1, 0] == 0
assert Rm[1, 1, 1, 1] == 0
Ric = metric_to_Ricci_components(g)
assert Ric[0, 0] == -y**(-2)
assert Ric[0, 1] == 0
assert Ric[1, 0] == 0
assert Ric[0, 0] == -y**(-2)
assert Ric == ImmutableDenseNDimArray([-y**(-2), 0, 0, -y**(-2)], (2, 2))
# scalar curvature is -2
# TODO - it would be nice to have index contraction built-in
R = (Ric[0, 0] + Ric[1, 1]) * y**2
assert R == -2
# Gauss curvature is -1
assert R / 2 == -1
def test_H2():
TP = diofant.diffgeom.TensorProduct
R2 = diofant.diffgeom.rn.R2
y = R2.y
dy = R2.dy
dx = R2.dx
g = (TP(dx, dx) + TP(dy, dy))*y**(-2)
automat = twoform_to_matrix(g)
mat = diag(y**(-2), y**(-2))
assert mat == automat
gamma1 = metric_to_Christoffel_1st(g)
assert gamma1[0, 0, 0] == 0
assert gamma1[0, 0, 1] == -y**(-3)
assert gamma1[0, 1, 0] == -y**(-3)
assert gamma1[0, 1, 1] == 0
assert gamma1[1, 1, 1] == -y**(-3)
assert gamma1[1, 1, 0] == 0
assert gamma1[1, 0, 1] == 0
assert gamma1[1, 0, 0] == y**(-3)
gamma2 = metric_to_Christoffel_2nd(g)
assert gamma2[0, 0, 0] == 0
assert gamma2[0, 0, 1] == -y**(-1)
assert gamma2[0, 1, 0] == -y**(-1)
assert gamma2[0, 1, 1] == 0
assert gamma2[1, 1, 1] == -y**(-1)
assert gamma2[1, 1, 0] == 0
assert gamma2[1, 0, 1] == 0
assert gamma2[1, 0, 0] == y**(-1)
Rm = metric_to_Riemann_components(g)
assert Rm[0, 0, 0, 0] == 0
assert Rm[0, 0, 0, 1] == 0
assert Rm[0, 0, 1, 0] == 0
assert Rm[0, 0, 1, 1] == 0
assert Rm[0, 1, 0, 0] == 0
assert Rm[0, 1, 0, 1] == -y**(-2)
assert Rm[0, 1, 1, 0] == y**(-2)
assert Rm[0, 1, 1, 1] == 0
assert Rm[1, 0, 0, 0] == 0
assert Rm[1, 0, 0, 1] == y**(-2)
assert Rm[1, 0, 1, 0] == -y**(-2)
assert Rm[1, 0, 1, 1] == 0
assert Rm[1, 1, 0, 0] == 0
assert Rm[1, 1, 0, 1] == 0
assert Rm[1, 1, 1, 0] == 0
assert Rm[1, 1, 1, 1] == 0
Ric = metric_to_Ricci_components(g)
assert Ric[0, 0] == -y**(-2)
assert Ric[0, 1] == 0
assert Ric[1, 0] == 0
assert Ric[0, 0] == -y**(-2)
assert Ric == ImmutableDenseNDimArray([-y**(-2), 0, 0, -y**(-2)], (2, 2))
# scalar curvature is -2
# TODO - it would be nice to have index contraction built-in
R = (Ric[0, 0] + Ric[1, 1])*y**2
assert R == -2
# Gauss curvature is -1
assert R/2 == -1