A Fundamental Theorem of Calculus for Second-order Directional Derivative

Abstract

Given a two-variable function $f$ without critical points and a compact region $R$ bounded by two level curves of $f$, this note proves that the integral over $R$ of the second-order directional derivative of $f$ in the tangential directions of the interceding level curves is proportional to the rise in $f$-value over $R$. Also discussed are variations on this result when critical points are present or $R$ becomes unbounded.