from sympy.testing.pytest import raises from sympy.tensor.toperators import PartialDerivative from sympy.tensor.tensor import (TensorIndexType, tensor_indices, TensorHead, tensor_heads) from sympy.core.numbers import Rational from sympy.core.symbol import symbols from sympy.matrices.dense import diag from sympy.tensor.array import Array from sympy.core.random import randint L = TensorIndexType("L") i, j, k, m, m1, m2, m3, m4 = tensor_indices("i j k m m1 m2 m3 m4", L) i0 = tensor_indices("i0", L) L_0, L_1 = tensor_indices("L_0 L_1", L) A, B, C, D = tensor_heads("A B C D", [L]) H = TensorHead("H", [L, L]) def test_invalid_partial_derivative_valence(): raises(ValueError, lambda: PartialDerivative(C(j), D(-j))) raises(ValueError, lambda: PartialDerivative(C(-j), D(j))) def test_tensor_partial_deriv(): # Test flatten: expr = PartialDerivative(PartialDerivative(A(i), A(j)), A(i)) assert expr.expr == A(L_0) assert expr.variables == (A(j), A(L_0)) expr1 = PartialDerivative(A(i), A(j)) assert expr1.expr == A(i) assert expr1.variables == (A(j),) expr2 = A(i)*PartialDerivative(H(k, -i), A(j)) assert expr2.get_indices() == [L_0, k, -L_0, -j] expr2b = A(i)*PartialDerivative(H(k, -i), A(-j)) assert expr2b.get_indices() == [L_0, k, -L_0, j] expr3 = A(i)*PartialDerivative(B(k)*C(-i) + 3*H(k, -i), A(j)) assert expr3.get_indices() == [L_0, k, -L_0, -j] expr4 = (A(i) + B(i))*PartialDerivative(C(j), D(j)) assert expr4.get_indices() == [i, L_0, -L_0] expr4b = (A(i) + B(i))*PartialDerivative(C(-j), D(-j)) assert expr4b.get_indices() == [i, -L_0, L_0] expr5 = (A(i) + B(i))*PartialDerivative(C(-i), D(j)) assert expr5.get_indices() == [L_0, -L_0, -j] def test_replace_arrays_partial_derivative(): x, y, z, t = symbols("x y z t") # d(A^i)/d(A_j) = d(g^ik A_k)/d(A_j) = g^ik delta_jk expr = PartialDerivative(A(i), A(-j)) assert expr.get_free_indices() == [i, j] assert expr.get_indices() == [i, j] assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, 1)}, [i, j]) == Array([[1, 0], [0, 1]]) assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, -1)}, [i, j]) == Array([[1, 0], [0, -1]]) assert expr.replace_with_arrays({A(-i): [x, y], L: diag(1, 1)}, [i, j]) == Array([[1, 0], [0, 1]]) assert expr.replace_with_arrays({A(-i): [x, y], L: diag(1, -1)}, [i, j]) == Array([[1, 0], [0, -1]]) expr = PartialDerivative(A(i), A(j)) assert expr.get_free_indices() == [i, -j] assert expr.get_indices() == [i, -j] assert expr.replace_with_arrays({A(i): [x, y]}, [i, -j]) == Array([[1, 0], [0, 1]]) assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, 1)}, [i, -j]) == Array([[1, 0], [0, 1]]) assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, -1)}, [i, -j]) == Array([[1, 0], [0, 1]]) assert expr.replace_with_arrays({A(-i): [x, y], L: diag(1, 1)}, [i, -j]) == Array([[1, 0], [0, 1]]) assert expr.replace_with_arrays({A(-i): [x, y], L: diag(1, -1)}, [i, -j]) == Array([[1, 0], [0, 1]]) expr = PartialDerivative(A(-i), A(-j)) assert expr.get_free_indices() == [-i, j] assert expr.get_indices() == [-i, j] assert expr.replace_with_arrays({A(-i): [x, y]}, [-i, j]) == Array([[1, 0], [0, 1]]) assert expr.replace_with_arrays({A(-i): [x, y], L: diag(1, 1)}, [-i, j]) == Array([[1, 0], [0, 1]]) assert expr.replace_with_arrays({A(-i): [x, y], L: diag(1, -1)}, [-i, j]) == Array([[1, 0], [0, 1]]) assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, 1)}, [-i, j]) == Array([[1, 0], [0, 1]]) assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, -1)}, [-i, j]) == Array([[1, 0], [0, 1]]) expr = PartialDerivative(A(i), A(i)) assert expr.get_free_indices() == [] assert expr.get_indices() == [L_0, -L_0] assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, 1)}, []) == 2 assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, -1)}, []) == 2 expr = PartialDerivative(A(-i), A(-i)) assert expr.get_free_indices() == [] assert expr.get_indices() == [-L_0, L_0] assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, 1)}, []) == 2 assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, -1)}, []) == 2 expr = PartialDerivative(H(i, j) + H(j, i), A(i)) assert expr.get_indices() == [L_0, j, -L_0] assert expr.get_free_indices() == [j] expr = PartialDerivative(H(i, j) + H(j, i), A(k))*B(-i) assert expr.get_indices() == [L_0, j, -k, -L_0] assert expr.get_free_indices() == [j, -k] expr = PartialDerivative(A(i)*(H(-i, j) + H(j, -i)), A(j)) assert expr.get_indices() == [L_0, -L_0, L_1, -L_1] assert expr.get_free_indices() == [] expr = A(j)*A(-j) + expr assert expr.get_indices() == [L_0, -L_0, L_1, -L_1] assert expr.get_free_indices() == [] expr = A(i)*(B(j)*PartialDerivative(C(-j), D(i)) + C(j)*PartialDerivative(D(-j), B(i))) assert expr.get_indices() == [L_0, L_1, -L_1, -L_0] assert expr.get_free_indices() == [] expr = A(i)*PartialDerivative(C(-j), D(i)) assert expr.get_indices() == [L_0, -j, -L_0] assert expr.get_free_indices() == [-j] def test_expand_partial_derivative_sum_rule(): tau = symbols("tau") # check sum rule for D(tensor, symbol) expr1aa = PartialDerivative(A(i), tau) assert expr1aa._expand_partial_derivative() == PartialDerivative(A(i), tau) expr1ab = PartialDerivative(A(i) + B(i), tau) assert (expr1ab._expand_partial_derivative() == PartialDerivative(A(i), tau) + PartialDerivative(B(i), tau)) expr1ac = PartialDerivative(A(i) + B(i) + C(i), tau) assert (expr1ac._expand_partial_derivative() == PartialDerivative(A(i), tau) + PartialDerivative(B(i), tau) + PartialDerivative(C(i), tau)) # check sum rule for D(tensor, D(j)) expr1ba = PartialDerivative(A(i), D(j)) assert expr1ba._expand_partial_derivative() ==\ PartialDerivative(A(i), D(j)) expr1bb = PartialDerivative(A(i) + B(i), D(j)) assert (expr1bb._expand_partial_derivative() == PartialDerivative(A(i), D(j)) + PartialDerivative(B(i), D(j))) expr1bc = PartialDerivative(A(i) + B(i) + C(i), D(j)) assert expr1bc._expand_partial_derivative() ==\ PartialDerivative(A(i), D(j))\ + PartialDerivative(B(i), D(j))\ + PartialDerivative(C(i), D(j)) # check sum rule for D(tensor, H(j, k)) expr1ca = PartialDerivative(A(i), H(j, k)) assert expr1ca._expand_partial_derivative() ==\ PartialDerivative(A(i), H(j, k)) expr1cb = PartialDerivative(A(i) + B(i), H(j, k)) assert (expr1cb._expand_partial_derivative() == PartialDerivative(A(i), H(j, k)) + PartialDerivative(B(i), H(j, k))) expr1cc = PartialDerivative(A(i) + B(i) + C(i), H(j, k)) assert (expr1cc._expand_partial_derivative() == PartialDerivative(A(i), H(j, k)) + PartialDerivative(B(i), H(j, k)) + PartialDerivative(C(i), H(j, k))) # check sum rule for D(D(tensor, D(j)), H(k, m)) expr1da = PartialDerivative(A(i), (D(j), H(k, m))) assert expr1da._expand_partial_derivative() ==\ PartialDerivative(A(i), (D(j), H(k, m))) expr1db = PartialDerivative(A(i) + B(i), (D(j), H(k, m))) assert expr1db._expand_partial_derivative() ==\ PartialDerivative(A(i), (D(j), H(k, m)))\ + PartialDerivative(B(i), (D(j), H(k, m))) expr1dc = PartialDerivative(A(i) + B(i) + C(i), (D(j), H(k, m))) assert expr1dc._expand_partial_derivative() ==\ PartialDerivative(A(i), (D(j), H(k, m)))\ + PartialDerivative(B(i), (D(j), H(k, m)))\ + PartialDerivative(C(i), (D(j), H(k, m))) def test_expand_partial_derivative_constant_factor_rule(): nneg = randint(0, 1000) pos = randint(1, 1000) neg = -randint(1, 1000) c1 = Rational(nneg, pos) c2 = Rational(neg, pos) c3 = Rational(nneg, neg) expr2a = PartialDerivative(nneg*A(i), D(j)) assert expr2a._expand_partial_derivative() ==\ nneg*PartialDerivative(A(i), D(j)) expr2b = PartialDerivative(neg*A(i), D(j)) assert expr2b._expand_partial_derivative() ==\ neg*PartialDerivative(A(i), D(j)) expr2ca = PartialDerivative(c1*A(i), D(j)) assert expr2ca._expand_partial_derivative() ==\ c1*PartialDerivative(A(i), D(j)) expr2cb = PartialDerivative(c2*A(i), D(j)) assert expr2cb._expand_partial_derivative() ==\ c2*PartialDerivative(A(i), D(j)) expr2cc = PartialDerivative(c3*A(i), D(j)) assert expr2cc._expand_partial_derivative() ==\ c3*PartialDerivative(A(i), D(j)) def test_expand_partial_derivative_full_linearity(): nneg = randint(0, 1000) pos = randint(1, 1000) neg = -randint(1, 1000) c1 = Rational(nneg, pos) c2 = Rational(neg, pos) c3 = Rational(nneg, neg) # check full linearity p = PartialDerivative(42, D(j)) assert p and not p._expand_partial_derivative() expr3a = PartialDerivative(nneg*A(i) + pos*B(i), D(j)) assert expr3a._expand_partial_derivative() ==\ nneg*PartialDerivative(A(i), D(j))\ + pos*PartialDerivative(B(i), D(j)) expr3b = PartialDerivative(nneg*A(i) + neg*B(i), D(j)) assert expr3b._expand_partial_derivative() ==\ nneg*PartialDerivative(A(i), D(j))\ + neg*PartialDerivative(B(i), D(j)) expr3c = PartialDerivative(neg*A(i) + pos*B(i), D(j)) assert expr3c._expand_partial_derivative() ==\ neg*PartialDerivative(A(i), D(j))\ + pos*PartialDerivative(B(i), D(j)) expr3d = PartialDerivative(c1*A(i) + c2*B(i), D(j)) assert expr3d._expand_partial_derivative() ==\ c1*PartialDerivative(A(i), D(j))\ + c2*PartialDerivative(B(i), D(j)) expr3e = PartialDerivative(c2*A(i) + c1*B(i), D(j)) assert expr3e._expand_partial_derivative() ==\ c2*PartialDerivative(A(i), D(j))\ + c1*PartialDerivative(B(i), D(j)) expr3f = PartialDerivative(c2*A(i) + c3*B(i), D(j)) assert expr3f._expand_partial_derivative() ==\ c2*PartialDerivative(A(i), D(j))\ + c3*PartialDerivative(B(i), D(j)) expr3g = PartialDerivative(c3*A(i) + c2*B(i), D(j)) assert expr3g._expand_partial_derivative() ==\ c3*PartialDerivative(A(i), D(j))\ + c2*PartialDerivative(B(i), D(j)) expr3h = PartialDerivative(c3*A(i) + c1*B(i), D(j)) assert expr3h._expand_partial_derivative() ==\ c3*PartialDerivative(A(i), D(j))\ + c1*PartialDerivative(B(i), D(j)) expr3i = PartialDerivative(c1*A(i) + c3*B(i), D(j)) assert expr3i._expand_partial_derivative() ==\ c1*PartialDerivative(A(i), D(j))\ + c3*PartialDerivative(B(i), D(j)) def test_expand_partial_derivative_product_rule(): # check product rule expr4a = PartialDerivative(A(i)*B(j), D(k)) assert expr4a._expand_partial_derivative() == \ PartialDerivative(A(i), D(k))*B(j)\ + A(i)*PartialDerivative(B(j), D(k)) expr4b = PartialDerivative(A(i)*B(j)*C(k), D(m)) assert expr4b._expand_partial_derivative() ==\ PartialDerivative(A(i), D(m))*B(j)*C(k)\ + A(i)*PartialDerivative(B(j), D(m))*C(k)\ + A(i)*B(j)*PartialDerivative(C(k), D(m)) expr4c = PartialDerivative(A(i)*B(j), C(k), D(m)) assert expr4c._expand_partial_derivative() ==\ PartialDerivative(A(i), C(k), D(m))*B(j) \ + PartialDerivative(A(i), C(k))*PartialDerivative(B(j), D(m))\ + PartialDerivative(A(i), D(m))*PartialDerivative(B(j), C(k))\ + A(i)*PartialDerivative(B(j), C(k), D(m)) 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_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 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_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_expr1(): tau, alpha = symbols("tau alpha") # this is only some special expression # tested: vector derivative # tested: scalar derivative # tested: tensor derivative base_expr1 = A(i)*H(-i, j) + A(i)*A(-i)*A(j) + tau**alpha*A(j) tensor_derivative = PartialDerivative(base_expr1, H(k, m))._perform_derivative() vector_derivative = PartialDerivative(base_expr1, A(k))._perform_derivative() scalar_derivative = PartialDerivative(base_expr1, tau)._perform_derivative() assert (tensor_derivative - A(L_0)*L.metric(-L_0, -L_1)*L.delta(L_1, -k)*L.delta(j, -m)) == 0 assert (vector_derivative - (tau**alpha*L.delta(j, -k) + L.delta(L_0, -k)*A(-L_0)*A(j) + A(L_0)*L.metric(-L_0, -L_1)*L.delta(L_1, -k)*A(j) + A(L_0)*A(-L_0)*L.delta(j, -k) + L.delta(L_0, -k)*H(-L_0, j))).expand() == 0 assert (vector_derivative.contract_metric(L.metric).contract_delta(L.delta) - (tau**alpha*L.delta(j, -k) + A(L_0)*A(-L_0)*L.delta(j, -k) + H(-k, j) + 2*A(j)*A(-k))).expand() == 0 assert scalar_derivative - alpha*1/tau*tau**alpha*A(j) == 0 def test_eval_partial_derivative_mixed_scalar_tensor_expr2(): tau, alpha = symbols("tau alpha") base_expr2 = A(i)*A(-i) + tau**2 vector_expression = PartialDerivative(base_expr2, A(k))._perform_derivative() assert (vector_expression - (L.delta(L_0, -k)*A(-L_0) + A(L_0)*L.metric(-L_0, -L_1)*L.delta(L_1, -k))).expand() == 0 scalar_expression = PartialDerivative(base_expr2, tau)._perform_derivative() assert scalar_expression == 2*tau