Symmetry of Second Derivatives - Clairaut's Theorem

In mathematical analysis, Clairaut's theorem (or Schwarz's theorem) named after Alexis Clairaut and Hermann Schwarz, states that if

has continuous second partial derivatives at any given point in, say, then for

In words, the partial derivations of this function are commutative at that point. One easy way to establish this theorem (in the case where n = 2, i = 1, and j = 2, which readily entails the result in general) is by applying Green's theorem to the gradient of f.