The missing link (Q1361130)

The author's objectives are: (1) to show that the conventional treatment of the Cauchy-Goursat Theorem is more complicated than necessary, and that the theorem really does follow from Green's Theorem without the assumption that the integrand is a \(C^1\) function; (2) to show a generalization of the Mean Value Theorem to higher dimensions as an equality. The resulting generalized MVT is used to prove Green's Theorem using only the assumption of integrability of \((\partial Q/\partial x-\partial P/\partial y)\) rather than that \(P\) and \(Q\) are \(C^1\). This version of Green's Theorem then implies the Cauchy-Goursat Theorem.