For a real-valued function of three or more real variables, this expression extends easily using appropriate rotation matrices.
For a function which maps elements from a subset X of the real line
In physics, if a system behaves the same regardless of how it is oriented in space, then its Lagrangian is rotationally invariant.
According to Noether's theorem, if the action (the integral over time of its Lagrangian) of a physical system is invariant under rotation, then angular momentum is conserved.
For infinitesimal rotations (in the xy-plane for this example; it may be done likewise for any plane) by an angle dθ the (infinitesimal) rotation operator is then thus in other words angular momentum is conserved.