gh-35193: symbolics: add derivative operator
    
This allows for also using the syntax which sagemath already uses in the
output of symbolic derivatives of functions like `D[0,1](f)(x+y,x-y)` in
user input.

Resolves #32554
Resolves #17445
    
URL: #35193
Reported by: Marius Gerbershagen
Reviewer(s): Eric Gourgoulhon, Matthias Köppe

src/sage/symbolic/all.py

@@ -18,3 +18,5 @@
 from .units import units
 
 π = pi
+
+from .operators import D

src/sage/symbolic/operators.py

@@ -196,3 +196,65 @@ def parameter_set(self):
             [0, 1]
         """
         return self._parameter_set
+
+class DerivativeOperator():
+    """
+    Derivative operator.
+
+    Acting with this operator onto a function gives a new operator (of
+    type :class:`FDerivativeOperator`) representing the function
+    differentiated with respect to one or multiple of its arguments.
+
+    This operator takes a list of indices specifying the position of
+    the arguments to differentiate. For example, D[0, 0, 1] is an
+    operator that differentiates a function twice with respect to its
+    first argument and once with respect to its second argument.
+
+    EXAMPLES::
+
+        sage: x, y = var('x,y'); f = function('f')
+        sage: D[0](f)(x)
+        diff(f(x), x)
+        sage: D[0](f)(x, y)
+        diff(f(x, y), x)
+        sage: D[0, 1](f)(x, y)
+        diff(f(x, y), x, y)
+        sage: D[0, 1](f)(x, x^2)
+        D[0, 1](f)(x, x^2)
+
+    """
+    class DerivativeOperatorWithParameters():
+        def __init__(self, parameter_set):
+            self._parameter_set = parameter_set
+        def __call__(self, function):
+            return FDerivativeOperator(function, self._parameter_set)
+        def __repr__(self):
+            """
+            Return the string representation of this derivative operator.
+
+            EXAMPLES::
+
+                sage: D[0]
+                D[0]
+                sage: D[0, 1]
+                D[0, 1]
+            """
+            return "D[%s]" % (", ".join(map(repr, self._parameter_set)))
+
+    def __getitem__(self, args):
+        """
+        TESTS:
+
+        The order in which the indices are given should not matter::
+
+            sage: x, y = var('x,y'); f = function('f')
+            sage: bool(D[0, 1, 0](f)(x, y) == D[0, 0, 1](f)(x, y))
+            True
+            sage: bool(D[1, 0, 0](f)(x, y) == D[0, 0, 1](f)(x, y))
+            True
+        """
+        if not isinstance(args, tuple):
+            args = (args,)
+        return self.DerivativeOperatorWithParameters(args)
+
+D = DerivativeOperator()