def test_eval_partial_derivative_single_1st_rank_tensors_by_tensor():
expr1 = PartialDerivative(A(i), A(j))
assert expr1._perform_derivative() - L.delta(i, -j) == 0
expr2 = PartialDerivative(A(i), A(-j))
assert expr2._perform_derivative() - L.metric(i, L_0) * L.delta(-L_0,
j) == 0
expr3 = PartialDerivative(A(-i), A(-j))
assert expr3._perform_derivative() - L.delta(-i, j) == 0
expr4 = PartialDerivative(A(-i), A(j))
assert expr4._perform_derivative() - L.metric(-i, -L_0) * L.delta(L_0,
-j) == 0
expr5 = PartialDerivative(A(i), B(j))
expr6 = PartialDerivative(A(i), C(j))
expr7 = PartialDerivative(A(i), D(j))
expr8 = PartialDerivative(A(i), H(j, k))
assert expr5._perform_derivative() == 0
assert expr6._perform_derivative() == 0
assert expr7._perform_derivative() == 0
assert expr8._perform_derivative() == 0
expr9 = PartialDerivative(A(i), A(i))
assert expr9._perform_derivative() - L.delta(L_0, -L_0) == 0
expr10 = PartialDerivative(A(-i), A(-i))
assert expr10._perform_derivative() - L.delta(-L_0, L_0) == 0
def test_eval_partial_derivative_expr_by_symbol():
tau, alpha = symbols("tau alpha")
expr1 = PartialDerivative(tau**alpha, tau)
assert expr1._perform_derivative() == alpha * 1 / tau * tau**alpha
expr2 = PartialDerivative(2 * tau + 3 * tau**4, tau)
assert expr2._perform_derivative() == 2 + 12 * tau**3
expr3 = PartialDerivative(2 * tau + 3 * tau**4, alpha)
assert expr3._perform_derivative() == 0
def test_eval_partial_derivative_single_2nd_rank_tensors_by_tensor():
expr1 = PartialDerivative(H(i, j), H(m, m1))
assert expr1._perform_derivative() - L.delta(i, -m) * L.delta(j, -m1) == 0
expr2 = PartialDerivative(H(i, j), H(-m, m1))
assert expr2._perform_derivative() - L.metric(i, L_0) * L.delta(-L_0, m) * L.delta(j, -m1) == 0
expr3 = PartialDerivative(H(i, j), H(m, -m1))
assert expr3._perform_derivative() - L.delta(i, -m) * L.metric(j, L_0) * L.delta(-L_0, m1) == 0
expr4 = PartialDerivative(H(i, j), H(-m, -m1))
assert expr4._perform_derivative() - L.metric(i, L_0) * L.delta(-L_0, m) * L.metric(j, L_1) * L.delta(-L_1, m1) == 0
def test_eval_partial_derivative_divergence_type():
expr1a = PartialDerivative(A(i), A(i))
expr1b = PartialDerivative(A(i), A(k))
expr1c = PartialDerivative(L.delta(-i, k) * A(i), A(k))
assert (expr1a._perform_derivative()
- (L.delta(-i, k) * expr1b._perform_derivative())).contract_delta(L.delta) == 0
assert (expr1a._perform_derivative()
- expr1c._perform_derivative()).contract_delta(L.delta) == 0
expr2a = PartialDerivative(H(i, j), H(i, j))
expr2b = PartialDerivative(H(i, j), H(k, m))
expr2c = PartialDerivative(L.delta(-i, k) * L.delta(-j, m) * H(i, j), H(k, m))
assert (expr2a._perform_derivative()
- (L.delta(-i, k) * L.delta(-j, m) * expr2b._perform_derivative())).contract_delta(L.delta) == 0
assert (expr2a._perform_derivative()
- expr2c._perform_derivative()).contract_delta(L.delta) == 0
def test_eval_partial_derivative_single_tensors_by_scalar():
tau, mu = symbols("tau mu")
expr = PartialDerivative(tau**mu, tau)
assert expr._perform_derivative() == mu * tau**mu / tau
expr1a = PartialDerivative(A(i), tau)
assert expr1a._perform_derivative() == 0
expr1b = PartialDerivative(A(-i), tau)
assert expr1b._perform_derivative() == 0
expr2a = PartialDerivative(H(i, j), tau)
assert expr2a._perform_derivative() == 0
expr2b = PartialDerivative(H(i, -j), tau)
assert expr2b._perform_derivative() == 0
expr2c = PartialDerivative(H(-i, j), tau)
assert expr2c._perform_derivative() == 0
expr2d = PartialDerivative(H(-i, -j), tau)
assert expr2d._perform_derivative() == 0