예제 #1
0
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))
예제 #2
0
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))
예제 #3
0
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
예제 #4
0
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