add read me

venv/lib/python3.12/site-packages/sympy/tensor/tests/test_functions.py

@@ -0,0 +1,57 @@
+from sympy.tensor.functions import TensorProduct
+from sympy.matrices.dense import Matrix
+from sympy.matrices.expressions.matexpr import MatrixSymbol
+from sympy.tensor.array import Array
+from sympy.abc import x, y, z
+from sympy.abc import i, j, k, l
+
+
+A = MatrixSymbol("A", 3, 3)
+B = MatrixSymbol("B", 3, 3)
+C = MatrixSymbol("C", 3, 3)
+
+
+def test_TensorProduct_construction():
+    assert TensorProduct(3, 4) == 12
+    assert isinstance(TensorProduct(A, A), TensorProduct)
+
+    expr = TensorProduct(TensorProduct(x, y), z)
+    assert expr == x*y*z
+
+    expr = TensorProduct(TensorProduct(A, B), C)
+    assert expr == TensorProduct(A, B, C)
+
+    expr = TensorProduct(Matrix.eye(2), Array([[0, -1], [1, 0]]))
+    assert expr == Array([
+        [
+            [[0, -1], [1, 0]],
+            [[0, 0], [0, 0]]
+        ],
+        [
+            [[0, 0], [0, 0]],
+            [[0, -1], [1, 0]]
+        ]
+    ])
+
+
+def test_TensorProduct_shape():
+
+    expr = TensorProduct(3, 4, evaluate=False)
+    assert expr.shape == ()
+    assert expr.rank() == 0
+
+    expr = TensorProduct(Array([1, 2]), Array([x, y]), evaluate=False)
+    assert expr.shape == (2, 2)
+    assert expr.rank() == 2
+    expr = TensorProduct(expr, expr, evaluate=False)
+    assert expr.shape == (2, 2, 2, 2)
+    assert expr.rank() == 4
+
+    expr = TensorProduct(Matrix.eye(2), Array([[0, -1], [1, 0]]), evaluate=False)
+    assert expr.shape == (2, 2, 2, 2)
+    assert expr.rank() == 4
+
+
+def test_TensorProduct_getitem():
+    expr = TensorProduct(A, B)
+    assert expr[i, j, k, l] == A[i, j]*B[k, l]

venv/lib/python3.12/site-packages/sympy/tensor/tests/test_index_methods.py

@@ -0,0 +1,227 @@
+from sympy.core import symbols, S, Pow, Function
+from sympy.functions import exp
+from sympy.testing.pytest import raises
+from sympy.tensor.indexed import Idx, IndexedBase
+from sympy.tensor.index_methods import IndexConformanceException
+
+from sympy.tensor.index_methods import (get_contraction_structure, get_indices)
+
+
+def test_trivial_indices():
+    x, y = symbols('x y')
+    assert get_indices(x) == (set(), {})
+    assert get_indices(x*y) == (set(), {})
+    assert get_indices(x + y) == (set(), {})
+    assert get_indices(x**y) == (set(), {})
+
+
+def test_get_indices_Indexed():
+    x = IndexedBase('x')
+    i, j = Idx('i'), Idx('j')
+    assert get_indices(x[i, j]) == ({i, j}, {})
+    assert get_indices(x[j, i]) == ({j, i}, {})
+
+
+def test_get_indices_Idx():
+    f = Function('f')
+    i, j = Idx('i'), Idx('j')
+    assert get_indices(f(i)*j) == ({i, j}, {})
+    assert get_indices(f(j, i)) == ({j, i}, {})
+    assert get_indices(f(i)*i) == (set(), {})
+
+
+def test_get_indices_mul():
+    x = IndexedBase('x')
+    y = IndexedBase('y')
+    i, j = Idx('i'), Idx('j')
+    assert get_indices(x[j]*y[i]) == ({i, j}, {})
+    assert get_indices(x[i]*y[j]) == ({i, j}, {})
+
+
+def test_get_indices_exceptions():
+    x = IndexedBase('x')
+    y = IndexedBase('y')
+    i, j = Idx('i'), Idx('j')
+    raises(IndexConformanceException, lambda: get_indices(x[i] + y[j]))
+
+
+def test_scalar_broadcast():
+    x = IndexedBase('x')
+    y = IndexedBase('y')
+    i, j = Idx('i'), Idx('j')
+    assert get_indices(x[i] + y[i, i]) == ({i}, {})
+    assert get_indices(x[i] + y[j, j]) == ({i}, {})
+
+
+def test_get_indices_add():
+    x = IndexedBase('x')
+    y = IndexedBase('y')
+    A = IndexedBase('A')
+    i, j, k = Idx('i'), Idx('j'), Idx('k')
+    assert get_indices(x[i] + 2*y[i]) == ({i}, {})
+    assert get_indices(y[i] + 2*A[i, j]*x[j]) == ({i}, {})
+    assert get_indices(y[i] + 2*(x[i] + A[i, j]*x[j])) == ({i}, {})
+    assert get_indices(y[i] + x[i]*(A[j, j] + 1)) == ({i}, {})
+    assert get_indices(
+        y[i] + x[i]*x[j]*(y[j] + A[j, k]*x[k])) == ({i}, {})
+
+
+def test_get_indices_Pow():
+    x = IndexedBase('x')
+    y = IndexedBase('y')
+    A = IndexedBase('A')
+    i, j, k = Idx('i'), Idx('j'), Idx('k')
+    assert get_indices(Pow(x[i], y[j])) == ({i, j}, {})
+    assert get_indices(Pow(x[i, k], y[j, k])) == ({i, j, k}, {})
+    assert get_indices(Pow(A[i, k], y[k] + A[k, j]*x[j])) == ({i, k}, {})
+    assert get_indices(Pow(2, x[i])) == get_indices(exp(x[i]))
+
+    # test of a design decision, this may change:
+    assert get_indices(Pow(x[i], 2)) == ({i}, {})
+
+
+def test_get_contraction_structure_basic():
+    x = IndexedBase('x')
+    y = IndexedBase('y')
+    i, j = Idx('i'), Idx('j')
+    assert get_contraction_structure(x[i]*y[j]) == {None: {x[i]*y[j]}}
+    assert get_contraction_structure(x[i] + y[j]) == {None: {x[i], y[j]}}
+    assert get_contraction_structure(x[i]*y[i]) == {(i,): {x[i]*y[i]}}
+    assert get_contraction_structure(
+        1 + x[i]*y[i]) == {None: {S.One}, (i,): {x[i]*y[i]}}
+    assert get_contraction_structure(x[i]**y[i]) == {None: {x[i]**y[i]}}
+
+
+def test_get_contraction_structure_complex():
+    x = IndexedBase('x')
+    y = IndexedBase('y')
+    A = IndexedBase('A')
+    i, j, k = Idx('i'), Idx('j'), Idx('k')
+    expr1 = y[i] + A[i, j]*x[j]
+    d1 = {None: {y[i]}, (j,): {A[i, j]*x[j]}}
+    assert get_contraction_structure(expr1) == d1
+    expr2 = expr1*A[k, i] + x[k]
+    d2 = {None: {x[k]}, (i,): {expr1*A[k, i]}, expr1*A[k, i]: [d1]}
+    assert get_contraction_structure(expr2) == d2
+
+
+def test_contraction_structure_simple_Pow():
+    x = IndexedBase('x')
+    y = IndexedBase('y')
+    i, j, k = Idx('i'), Idx('j'), Idx('k')
+    ii_jj = x[i, i]**y[j, j]
+    assert get_contraction_structure(ii_jj) == {
+        None: {ii_jj},
+        ii_jj: [
+            {(i,): {x[i, i]}},
+            {(j,): {y[j, j]}}
+        ]
+    }
+
+    ii_jk = x[i, i]**y[j, k]
+    assert get_contraction_structure(ii_jk) == {
+        None: {x[i, i]**y[j, k]},
+        x[i, i]**y[j, k]: [
+            {(i,): {x[i, i]}}
+        ]
+    }
+
+
+def test_contraction_structure_Mul_and_Pow():
+    x = IndexedBase('x')
+    y = IndexedBase('y')
+    i, j, k = Idx('i'), Idx('j'), Idx('k')
+
+    i_ji = x[i]**(y[j]*x[i])
+    assert get_contraction_structure(i_ji) == {None: {i_ji}}
+    ij_i = (x[i]*y[j])**(y[i])
+    assert get_contraction_structure(ij_i) == {None: {ij_i}}
+    j_ij_i = x[j]*(x[i]*y[j])**(y[i])
+    assert get_contraction_structure(j_ij_i) == {(j,): {j_ij_i}}
+    j_i_ji = x[j]*x[i]**(y[j]*x[i])
+    assert get_contraction_structure(j_i_ji) == {(j,): {j_i_ji}}
+    ij_exp_kki = x[i]*y[j]*exp(y[i]*y[k, k])
+    result = get_contraction_structure(ij_exp_kki)
+    expected = {
+        (i,): {ij_exp_kki},
+        ij_exp_kki: [{
+                     None: {exp(y[i]*y[k, k])},
+                exp(y[i]*y[k, k]): [{
+                    None: {y[i]*y[k, k]},
+                    y[i]*y[k, k]: [{(k,): {y[k, k]}}]
+                }]}
+        ]
+    }
+    assert result == expected
+
+
+def test_contraction_structure_Add_in_Pow():
+    x = IndexedBase('x')
+    y = IndexedBase('y')
+    i, j, k = Idx('i'), Idx('j'), Idx('k')
+    s_ii_jj_s = (1 + x[i, i])**(1 + y[j, j])
+    expected = {
+        None: {s_ii_jj_s},
+        s_ii_jj_s: [
+            {None: {S.One}, (i,): {x[i, i]}},
+            {None: {S.One}, (j,): {y[j, j]}}
+        ]
+    }
+    result = get_contraction_structure(s_ii_jj_s)
+    assert result == expected
+
+    s_ii_jk_s = (1 + x[i, i]) ** (1 + y[j, k])
+    expected_2 = {
+        None: {(x[i, i] + 1)**(y[j, k] + 1)},
+        s_ii_jk_s: [
+            {None: {S.One}, (i,): {x[i, i]}}
+        ]
+    }
+    result_2 = get_contraction_structure(s_ii_jk_s)
+    assert result_2 == expected_2
+
+
+def test_contraction_structure_Pow_in_Pow():
+    x = IndexedBase('x')
+    y = IndexedBase('y')
+    z = IndexedBase('z')
+    i, j, k = Idx('i'), Idx('j'), Idx('k')
+    ii_jj_kk = x[i, i]**y[j, j]**z[k, k]
+    expected = {
+        None: {ii_jj_kk},
+        ii_jj_kk: [
+            {(i,): {x[i, i]}},
+            {
+                None: {y[j, j]**z[k, k]},
+                y[j, j]**z[k, k]: [
+                    {(j,): {y[j, j]}},
+                    {(k,): {z[k, k]}}
+                ]
+            }
+        ]
+    }
+    assert get_contraction_structure(ii_jj_kk) == expected
+
+
+def test_ufunc_support():
+    f = Function('f')
+    g = Function('g')
+    x = IndexedBase('x')
+    y = IndexedBase('y')
+    i, j = Idx('i'), Idx('j')
+    a = symbols('a')
+
+    assert get_indices(f(x[i])) == ({i}, {})
+    assert get_indices(f(x[i], y[j])) == ({i, j}, {})
+    assert get_indices(f(y[i])*g(x[i])) == (set(), {})
+    assert get_indices(f(a, x[i])) == ({i}, {})
+    assert get_indices(f(a, y[i], x[j])*g(x[i])) == ({j}, {})
+    assert get_indices(g(f(x[i]))) == ({i}, {})
+
+    assert get_contraction_structure(f(x[i])) == {None: {f(x[i])}}
+    assert get_contraction_structure(
+        f(y[i])*g(x[i])) == {(i,): {f(y[i])*g(x[i])}}
+    assert get_contraction_structure(
+        f(y[i])*g(f(x[i]))) == {(i,): {f(y[i])*g(f(x[i]))}}
+    assert get_contraction_structure(
+        f(x[j], y[i])*g(x[i])) == {(i,): {f(x[j], y[i])*g(x[i])}}

venv/lib/python3.12/site-packages/sympy/tensor/tests/test_indexed.py

@@ -0,0 +1,511 @@
+from sympy.core import symbols, Symbol, Tuple, oo, Dummy
+from sympy.tensor.indexed import IndexException
+from sympy.testing.pytest import raises
+from sympy.utilities.iterables import iterable
+
+# import test:
+from sympy.concrete.summations import Sum
+from sympy.core.function import Function, Subs, Derivative
+from sympy.core.relational import (StrictLessThan, GreaterThan,
+    StrictGreaterThan, LessThan)
+from sympy.core.singleton import S
+from sympy.functions.elementary.exponential import exp, log
+from sympy.functions.elementary.trigonometric import cos, sin
+from sympy.functions.special.tensor_functions import KroneckerDelta
+from sympy.series.order import Order
+from sympy.sets.fancysets import Range
+from sympy.tensor.indexed import IndexedBase, Idx, Indexed
+
+
+def test_Idx_construction():
+    i, a, b = symbols('i a b', integer=True)
+    assert Idx(i) != Idx(i, 1)
+    assert Idx(i, a) == Idx(i, (0, a - 1))
+    assert Idx(i, oo) == Idx(i, (0, oo))
+
+    x = symbols('x', integer=False)
+    raises(TypeError, lambda: Idx(x))
+    raises(TypeError, lambda: Idx(0.5))
+    raises(TypeError, lambda: Idx(i, x))
+    raises(TypeError, lambda: Idx(i, 0.5))
+    raises(TypeError, lambda: Idx(i, (x, 5)))
+    raises(TypeError, lambda: Idx(i, (2, x)))
+    raises(TypeError, lambda: Idx(i, (2, 3.5)))
+
+
+def test_Idx_properties():
+    i, a, b = symbols('i a b', integer=True)
+    assert Idx(i).is_integer
+    assert Idx(i).name == 'i'
+    assert Idx(i + 2).name == 'i + 2'
+    assert Idx('foo').name == 'foo'
+
+
+def test_Idx_bounds():
+    i, a, b = symbols('i a b', integer=True)
+    assert Idx(i).lower is None
+    assert Idx(i).upper is None
+    assert Idx(i, a).lower == 0
+    assert Idx(i, a).upper == a - 1
+    assert Idx(i, 5).lower == 0
+    assert Idx(i, 5).upper == 4
+    assert Idx(i, oo).lower == 0
+    assert Idx(i, oo).upper is oo
+    assert Idx(i, (a, b)).lower == a
+    assert Idx(i, (a, b)).upper == b
+    assert Idx(i, (1, 5)).lower == 1
+    assert Idx(i, (1, 5)).upper == 5
+    assert Idx(i, (-oo, oo)).lower is -oo
+    assert Idx(i, (-oo, oo)).upper is oo
+
+
+def test_Idx_fixed_bounds():
+    i, a, b, x = symbols('i a b x', integer=True)
+    assert Idx(x).lower is None
+    assert Idx(x).upper is None
+    assert Idx(x, a).lower == 0
+    assert Idx(x, a).upper == a - 1
+    assert Idx(x, 5).lower == 0
+    assert Idx(x, 5).upper == 4
+    assert Idx(x, oo).lower == 0
+    assert Idx(x, oo).upper is oo
+    assert Idx(x, (a, b)).lower == a
+    assert Idx(x, (a, b)).upper == b
+    assert Idx(x, (1, 5)).lower == 1
+    assert Idx(x, (1, 5)).upper == 5
+    assert Idx(x, (-oo, oo)).lower is -oo
+    assert Idx(x, (-oo, oo)).upper is oo
+
+
+def test_Idx_inequalities():
+    i14 = Idx("i14", (1, 4))
+    i79 = Idx("i79", (7, 9))
+    i46 = Idx("i46", (4, 6))
+    i35 = Idx("i35", (3, 5))
+
+    assert i14 <= 5
+    assert i14 < 5
+    assert not (i14 >= 5)
+    assert not (i14 > 5)
+
+    assert 5 >= i14
+    assert 5 > i14
+    assert not (5 <= i14)
+    assert not (5 < i14)
+
+    assert LessThan(i14, 5)
+    assert StrictLessThan(i14, 5)
+    assert not GreaterThan(i14, 5)
+    assert not StrictGreaterThan(i14, 5)
+
+    assert i14 <= 4
+    assert isinstance(i14 < 4, StrictLessThan)
+    assert isinstance(i14 >= 4, GreaterThan)
+    assert not (i14 > 4)
+
+    assert isinstance(i14 <= 1, LessThan)
+    assert not (i14 < 1)
+    assert i14 >= 1
+    assert isinstance(i14 > 1, StrictGreaterThan)
+
+    assert not (i14 <= 0)
+    assert not (i14 < 0)
+    assert i14 >= 0
+    assert i14 > 0
+
+    from sympy.abc import x
+
+    assert isinstance(i14 < x, StrictLessThan)
+    assert isinstance(i14 > x, StrictGreaterThan)
+    assert isinstance(i14 <= x, LessThan)
+    assert isinstance(i14 >= x, GreaterThan)
+
+    assert i14 < i79
+    assert i14 <= i79
+    assert not (i14 > i79)
+    assert not (i14 >= i79)
+
+    assert i14 <= i46
+    assert isinstance(i14 < i46, StrictLessThan)
+    assert isinstance(i14 >= i46, GreaterThan)
+    assert not (i14 > i46)
+
+    assert isinstance(i14 < i35, StrictLessThan)
+    assert isinstance(i14 > i35, StrictGreaterThan)
+    assert isinstance(i14 <= i35, LessThan)
+    assert isinstance(i14 >= i35, GreaterThan)
+
+    iNone1 = Idx("iNone1")
+    iNone2 = Idx("iNone2")
+
+    assert isinstance(iNone1 < iNone2, StrictLessThan)
+    assert isinstance(iNone1 > iNone2, StrictGreaterThan)
+    assert isinstance(iNone1 <= iNone2, LessThan)
+    assert isinstance(iNone1 >= iNone2, GreaterThan)
+
+
+def test_Idx_inequalities_current_fails():
+    i14 = Idx("i14", (1, 4))
+
+    assert S(5) >= i14
+    assert S(5) > i14
+    assert not (S(5) <= i14)
+    assert not (S(5) < i14)
+
+
+def test_Idx_func_args():
+    i, a, b = symbols('i a b', integer=True)
+    ii = Idx(i)
+    assert ii.func(*ii.args) == ii
+    ii = Idx(i, a)
+    assert ii.func(*ii.args) == ii
+    ii = Idx(i, (a, b))
+    assert ii.func(*ii.args) == ii
+
+
+def test_Idx_subs():
+    i, a, b = symbols('i a b', integer=True)
+    assert Idx(i, a).subs(a, b) == Idx(i, b)
+    assert Idx(i, a).subs(i, b) == Idx(b, a)
+
+    assert Idx(i).subs(i, 2) == Idx(2)
+    assert Idx(i, a).subs(a, 2) == Idx(i, 2)
+    assert Idx(i, (a, b)).subs(i, 2) == Idx(2, (a, b))
+
+
+def test_IndexedBase_sugar():
+    i, j = symbols('i j', integer=True)
+    a = symbols('a')
+    A1 = Indexed(a, i, j)
+    A2 = IndexedBase(a)
+    assert A1 == A2[i, j]
+    assert A1 == A2[(i, j)]
+    assert A1 == A2[[i, j]]
+    assert A1 == A2[Tuple(i, j)]
+    assert all(a.is_Integer for a in A2[1, 0].args[1:])
+
+
+def test_IndexedBase_subs():
+    i = symbols('i', integer=True)
+    a, b = symbols('a b')
+    A = IndexedBase(a)
+    B = IndexedBase(b)
+    assert A[i] == B[i].subs(b, a)
+    C = {1: 2}
+    assert C[1] == A[1].subs(A, C)
+
+
+def test_IndexedBase_shape():
+    i, j, m, n = symbols('i j m n', integer=True)
+    a = IndexedBase('a', shape=(m, m))
+    b = IndexedBase('a', shape=(m, n))
+    assert b.shape == Tuple(m, n)
+    assert a[i, j] != b[i, j]
+    assert a[i, j] == b[i, j].subs(n, m)
+    assert b.func(*b.args) == b
+    assert b[i, j].func(*b[i, j].args) == b[i, j]
+    raises(IndexException, lambda: b[i])
+    raises(IndexException, lambda: b[i, i, j])
+    F = IndexedBase("F", shape=m)
+    assert F.shape == Tuple(m)
+    assert F[i].subs(i, j) == F[j]
+    raises(IndexException, lambda: F[i, j])
+
+
+def test_IndexedBase_assumptions():
+    i = Symbol('i', integer=True)
+    a = Symbol('a')
+    A = IndexedBase(a, positive=True)
+    for c in (A, A[i]):
+        assert c.is_real
+        assert c.is_complex
+        assert not c.is_imaginary
+        assert c.is_nonnegative
+        assert c.is_nonzero
+        assert c.is_commutative
+        assert log(exp(c)) == c
+
+    assert A != IndexedBase(a)
+    assert A == IndexedBase(a, positive=True, real=True)
+    assert A[i] != Indexed(a, i)
+
+
+def test_IndexedBase_assumptions_inheritance():
+    I = Symbol('I', integer=True)
+    I_inherit = IndexedBase(I)
+    I_explicit = IndexedBase('I', integer=True)
+
+    assert I_inherit.is_integer
+    assert I_explicit.is_integer
+    assert I_inherit.label.is_integer
+    assert I_explicit.label.is_integer
+    assert I_inherit == I_explicit
+
+
+def test_issue_17652():
+    """Regression test issue #17652.
+
+    IndexedBase.label should not upcast subclasses of Symbol
+    """
+    class SubClass(Symbol):
+        pass
+
+    x = SubClass('X')
+    assert type(x) == SubClass
+    base = IndexedBase(x)
+    assert type(x) == SubClass
+    assert type(base.label) == SubClass
+
+
+def test_Indexed_constructor():
+    i, j = symbols('i j', integer=True)
+    A = Indexed('A', i, j)
+    assert A == Indexed(Symbol('A'), i, j)
+    assert A == Indexed(IndexedBase('A'), i, j)
+    raises(TypeError, lambda: Indexed(A, i, j))
+    raises(IndexException, lambda: Indexed("A"))
+    assert A.free_symbols == {A, A.base.label, i, j}
+
+
+def test_Indexed_func_args():
+    i, j = symbols('i j', integer=True)
+    a = symbols('a')
+    A = Indexed(a, i, j)
+    assert A == A.func(*A.args)
+
+
+def test_Indexed_subs():
+    i, j, k = symbols('i j k', integer=True)
+    a, b = symbols('a b')
+    A = IndexedBase(a)
+    B = IndexedBase(b)
+    assert A[i, j] == B[i, j].subs(b, a)
+    assert A[i, j] == A[i, k].subs(k, j)
+
+
+def test_Indexed_properties():
+    i, j = symbols('i j', integer=True)
+    A = Indexed('A', i, j)
+    assert A.name == 'A[i, j]'
+    assert A.rank == 2
+    assert A.indices == (i, j)
+    assert A.base == IndexedBase('A')
+    assert A.ranges == [None, None]
+    raises(IndexException, lambda: A.shape)
+
+    n, m = symbols('n m', integer=True)
+    assert Indexed('A', Idx(
+        i, m), Idx(j, n)).ranges == [Tuple(0, m - 1), Tuple(0, n - 1)]
+    assert Indexed('A', Idx(i, m), Idx(j, n)).shape == Tuple(m, n)
+    raises(IndexException, lambda: Indexed("A", Idx(i, m), Idx(j)).shape)
+
+
+def test_Indexed_shape_precedence():
+    i, j = symbols('i j', integer=True)
+    o, p = symbols('o p', integer=True)
+    n, m = symbols('n m', integer=True)
+    a = IndexedBase('a', shape=(o, p))
+    assert a.shape == Tuple(o, p)
+    assert Indexed(
+        a, Idx(i, m), Idx(j, n)).ranges == [Tuple(0, m - 1), Tuple(0, n - 1)]
+    assert Indexed(a, Idx(i, m), Idx(j, n)).shape == Tuple(o, p)
+    assert Indexed(
+        a, Idx(i, m), Idx(j)).ranges == [Tuple(0, m - 1), (None, None)]
+    assert Indexed(a, Idx(i, m), Idx(j)).shape == Tuple(o, p)
+
+
+def test_complex_indices():
+    i, j = symbols('i j', integer=True)
+    A = Indexed('A', i, i + j)
+    assert A.rank == 2
+    assert A.indices == (i, i + j)
+
+
+def test_not_interable():
+    i, j = symbols('i j', integer=True)
+    A = Indexed('A', i, i + j)
+    assert not iterable(A)
+
+
+def test_Indexed_coeff():
+    N = Symbol('N', integer=True)
+    len_y = N
+    i = Idx('i', len_y-1)
+    y = IndexedBase('y', shape=(len_y,))
+    a = (1/y[i+1]*y[i]).coeff(y[i])
+    b = (y[i]/y[i+1]).coeff(y[i])
+    assert a == b
+
+
+def test_differentiation():
+    from sympy.functions.special.tensor_functions import KroneckerDelta
+    i, j, k, l = symbols('i j k l', cls=Idx)
+    a = symbols('a')
+    m, n = symbols("m, n", integer=True, finite=True)
+    assert m.is_real
+    h, L = symbols('h L', cls=IndexedBase)
+    hi, hj = h[i], h[j]
+
+    expr = hi
+    assert expr.diff(hj) == KroneckerDelta(i, j)
+    assert expr.diff(hi) == KroneckerDelta(i, i)
+
+    expr = S(2) * hi
+    assert expr.diff(hj) == S(2) * KroneckerDelta(i, j)
+    assert expr.diff(hi) == S(2) * KroneckerDelta(i, i)
+    assert expr.diff(a) is S.Zero
+
+    assert Sum(expr, (i, -oo, oo)).diff(hj) == Sum(2*KroneckerDelta(i, j), (i, -oo, oo))
+    assert Sum(expr.diff(hj), (i, -oo, oo)) == Sum(2*KroneckerDelta(i, j), (i, -oo, oo))
+    assert Sum(expr, (i, -oo, oo)).diff(hj).doit() == 2
+
+    assert Sum(expr.diff(hi), (i, -oo, oo)).doit() == Sum(2, (i, -oo, oo)).doit()
+    assert Sum(expr, (i, -oo, oo)).diff(hi).doit() is oo
+
+    expr = a * hj * hj / S(2)
+    assert expr.diff(hi) == a * h[j] * KroneckerDelta(i, j)
+    assert expr.diff(a) == hj * hj / S(2)
+    assert expr.diff(a, 2) is S.Zero
+
+    assert Sum(expr, (i, -oo, oo)).diff(hi) == Sum(a*KroneckerDelta(i, j)*h[j], (i, -oo, oo))
+    assert Sum(expr.diff(hi), (i, -oo, oo)) == Sum(a*KroneckerDelta(i, j)*h[j], (i, -oo, oo))
+    assert Sum(expr, (i, -oo, oo)).diff(hi).doit() == a*h[j]
+
+    assert Sum(expr, (j, -oo, oo)).diff(hi) == Sum(a*KroneckerDelta(i, j)*h[j], (j, -oo, oo))
+    assert Sum(expr.diff(hi), (j, -oo, oo)) == Sum(a*KroneckerDelta(i, j)*h[j], (j, -oo, oo))
+    assert Sum(expr, (j, -oo, oo)).diff(hi).doit() == a*h[i]
+
+    expr = a * sin(hj * hj)
+    assert expr.diff(hi) == 2*a*cos(hj * hj) * hj * KroneckerDelta(i, j)
+    assert expr.diff(hj) == 2*a*cos(hj * hj) * hj
+
+    expr = a * L[i, j] * h[j]
+    assert expr.diff(hi) == a*L[i, j]*KroneckerDelta(i, j)
+    assert expr.diff(hj) == a*L[i, j]
+    assert expr.diff(L[i, j]) == a*h[j]
+    assert expr.diff(L[k, l]) == a*KroneckerDelta(i, k)*KroneckerDelta(j, l)*h[j]
+    assert expr.diff(L[i, l]) == a*KroneckerDelta(j, l)*h[j]
+
+    assert Sum(expr, (j, -oo, oo)).diff(L[k, l]) == Sum(a * KroneckerDelta(i, k) * KroneckerDelta(j, l) * h[j], (j, -oo, oo))
+    assert Sum(expr, (j, -oo, oo)).diff(L[k, l]).doit() == a * KroneckerDelta(i, k) * h[l]
+
+    assert h[m].diff(h[m]) == 1
+    assert h[m].diff(h[n]) == KroneckerDelta(m, n)
+    assert Sum(a*h[m], (m, -oo, oo)).diff(h[n]) == Sum(a*KroneckerDelta(m, n), (m, -oo, oo))
+    assert Sum(a*h[m], (m, -oo, oo)).diff(h[n]).doit() == a
+    assert Sum(a*h[m], (n, -oo, oo)).diff(h[n]) == Sum(a*KroneckerDelta(m, n), (n, -oo, oo))
+    assert Sum(a*h[m], (m, -oo, oo)).diff(h[m]).doit() == oo*a
+
+
+def test_indexed_series():
+    A = IndexedBase("A")
+    i = symbols("i", integer=True)
+    assert sin(A[i]).series(A[i]) == A[i] - A[i]**3/6 + A[i]**5/120 + Order(A[i]**6, A[i])
+
+
+def test_indexed_is_constant():
+    A = IndexedBase("A")
+    i, j, k = symbols("i,j,k")
+    assert not A[i].is_constant()
+    assert A[i].is_constant(j)
+    assert not A[1+2*i, k].is_constant()
+    assert not A[1+2*i, k].is_constant(i)
+    assert A[1+2*i, k].is_constant(j)
+    assert not A[1+2*i, k].is_constant(k)
+
+
+def test_issue_12533():
+    d = IndexedBase('d')
+    assert IndexedBase(range(5)) == Range(0, 5, 1)
+    assert d[0].subs(Symbol("d"), range(5)) == 0
+    assert d[0].subs(d, range(5)) == 0
+    assert d[1].subs(d, range(5)) == 1
+    assert Indexed(Range(5), 2) == 2
+
+
+def test_issue_12780():
+    n = symbols("n")
+    i = Idx("i", (0, n))
+    raises(TypeError, lambda: i.subs(n, 1.5))
+
+
+def test_issue_18604():
+    m = symbols("m")
+    assert Idx("i", m).name == 'i'
+    assert Idx("i", m).lower == 0
+    assert Idx("i", m).upper == m - 1
+    m = symbols("m", real=False)
+    raises(TypeError, lambda: Idx("i", m))
+
+def test_Subs_with_Indexed():
+    A = IndexedBase("A")
+    i, j, k = symbols("i,j,k")
+    x, y, z = symbols("x,y,z")
+    f = Function("f")
+
+    assert Subs(A[i], A[i], A[j]).diff(A[j]) == 1
+    assert Subs(A[i], A[i], x).diff(A[i]) == 0
+    assert Subs(A[i], A[i], x).diff(A[j]) == 0
+    assert Subs(A[i], A[i], x).diff(x) == 1
+    assert Subs(A[i], A[i], x).diff(y) == 0
+    assert Subs(A[i], A[i], A[j]).diff(A[k]) == KroneckerDelta(j, k)
+    assert Subs(x, x, A[i]).diff(A[j]) == KroneckerDelta(i, j)
+    assert Subs(f(A[i]), A[i], x).diff(A[j]) == 0
+    assert Subs(f(A[i]), A[i], A[k]).diff(A[j]) == Derivative(f(A[k]), A[k])*KroneckerDelta(j, k)
+    assert Subs(x, x, A[i]**2).diff(A[j]) == 2*KroneckerDelta(i, j)*A[i]
+    assert Subs(A[i], A[i], A[j]**2).diff(A[k]) == 2*KroneckerDelta(j, k)*A[j]
+
+    assert Subs(A[i]*x, x, A[i]).diff(A[i]) == 2*A[i]
+    assert Subs(A[i]*x, x, A[i]).diff(A[j]) == 2*A[i]*KroneckerDelta(i, j)
+    assert Subs(A[i]*x, x, A[j]).diff(A[i]) == A[j] + A[i]*KroneckerDelta(i, j)
+    assert Subs(A[i]*x, x, A[j]).diff(A[j]) == A[i] + A[j]*KroneckerDelta(i, j)
+    assert Subs(A[i]*x, x, A[i]).diff(A[k]) == 2*A[i]*KroneckerDelta(i, k)
+    assert Subs(A[i]*x, x, A[j]).diff(A[k]) == KroneckerDelta(i, k)*A[j] + KroneckerDelta(j, k)*A[i]
+
+    assert Subs(A[i]*x, A[i], x).diff(A[i]) == 0
+    assert Subs(A[i]*x, A[i], x).diff(A[j]) == 0
+    assert Subs(A[i]*x, A[j], x).diff(A[i]) == x
+    assert Subs(A[i]*x, A[j], x).diff(A[j]) == x*KroneckerDelta(i, j)
+    assert Subs(A[i]*x, A[i], x).diff(A[k]) == 0
+    assert Subs(A[i]*x, A[j], x).diff(A[k]) == x*KroneckerDelta(i, k)
+
+
+def test_complicated_derivative_with_Indexed():
+    x, y = symbols("x,y", cls=IndexedBase)
+    sigma = symbols("sigma")
+    i, j, k = symbols("i,j,k")
+    m0,m1,m2,m3,m4,m5 = symbols("m0:6")
+    f = Function("f")
+
+    expr = f((x[i] - y[i])**2/sigma)
+    _xi_1 = symbols("xi_1", cls=Dummy)
+    assert expr.diff(x[m0]).dummy_eq(
+        (x[i] - y[i])*KroneckerDelta(i, m0)*\
+        2*Subs(
+            Derivative(f(_xi_1), _xi_1),
+            (_xi_1,),
+            ((x[i] - y[i])**2/sigma,)
+        )/sigma
+    )
+    assert expr.diff(x[m0]).diff(x[m1]).dummy_eq(
+        2*KroneckerDelta(i, m0)*\
+        KroneckerDelta(i, m1)*Subs(
+            Derivative(f(_xi_1), _xi_1),
+            (_xi_1,),
+            ((x[i] - y[i])**2/sigma,)
+         )/sigma + \
+        4*(x[i] - y[i])**2*KroneckerDelta(i, m0)*KroneckerDelta(i, m1)*\
+        Subs(
+            Derivative(f(_xi_1), _xi_1, _xi_1),
+            (_xi_1,),
+            ((x[i] - y[i])**2/sigma,)
+        )/sigma**2
+    )
+
+
+def test_IndexedBase_commutative():
+    t = IndexedBase('t', commutative=False)
+    u = IndexedBase('u', commutative=False)
+    v = IndexedBase('v')
+    assert t[0]*v[0] == v[0]*t[0]
+    assert t[0]*u[0] != u[0]*t[0]

venv/lib/python3.12/site-packages/sympy/tensor/tests/test_printing.py

@@ -0,0 +1,13 @@
+from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorHead
+from sympy import I
+
+def test_printing_TensMul():
+    R3 = TensorIndexType('R3', dim=3)
+    p, q = tensor_indices("p q", R3)
+    K = TensorHead("K", [R3])
+
+    assert repr(2*K(p)) == "2*K(p)"
+    assert repr(-K(p)) == "-K(p)"
+    assert repr(-2*K(p)*K(q)) == "-2*K(p)*K(q)"
+    assert repr(-I*K(p)) == "-I*K(p)"
+    assert repr(I*K(p)) == "I*K(p)"

venv/lib/python3.12/site-packages/sympy/tensor/tests/test_tensor_element.py

@@ -0,0 +1,31 @@
+from sympy.tensor.tensor import (Tensor, TensorIndexType, TensorSymmetry,
+        tensor_indices, TensorHead, TensorElement)
+from sympy.tensor import Array
+from sympy.core.symbol import Symbol
+
+
+def test_tensor_element():
+    L = TensorIndexType("L")
+    i, j, k, l, m, n = tensor_indices("i j k l m n", L)
+
+    A = TensorHead("A", [L, L], TensorSymmetry.no_symmetry(2))
+
+    a = A(i, j)
+
+    assert isinstance(TensorElement(a, {}), Tensor)
+    assert isinstance(TensorElement(a, {k: 1}), Tensor)
+
+    te1 = TensorElement(a, {Symbol("i"): 1})
+    assert te1.free == [(j, 0)]
+    assert te1.get_free_indices() == [j]
+    assert te1.dum == []
+
+    te2 = TensorElement(a, {i: 1})
+    assert te2.free == [(j, 0)]
+    assert te2.get_free_indices() == [j]
+    assert te2.dum == []
+
+    assert te1 == te2
+
+    array = Array([[1, 2], [3, 4]])
+    assert te1.replace_with_arrays({A(i, j): array}, [j]) == array[1, :]

venv/lib/python3.12/site-packages/sympy/tensor/tests/test_tensor_operators.py

@@ -0,0 +1,464 @@
+from sympy import sin, cos
+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(k))
+    assert expr == PartialDerivative(A(i), A(j), A(k))
+    assert expr.expr == A(i)
+    assert expr.variables == (A(j), A(k))
+    assert expr.get_indices() == [i, -j, -k]
+    assert expr.get_free_indices() == [i, -j, -k]
+
+    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")
+
+    expr = PartialDerivative(A(i), B(j))
+    repl = expr.replace_with_arrays({A(i): [sin(x)*cos(y), x**3*y**2], B(i): [x, y]})
+    assert repl == Array([[cos(x)*cos(y), -sin(x)*sin(y)], [3*x**2*y**2, 2*x**3*y]])
+    repl = expr.replace_with_arrays({A(i): [sin(x)*cos(y), x**3*y**2], B(i): [x, y]}, [-j, i])
+    assert repl == Array([[cos(x)*cos(y), 3*x**2*y**2], [-sin(x)*sin(y), 2*x**3*y]])
+
+    # 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().doit() == 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().doit() == 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().doit() == 0
+
+    scalar_expression = PartialDerivative(base_expr2, tau)._perform_derivative()
+    assert scalar_expression == 2*tau