A Fundamental Theorem of Calculus for Second-order Directional Derivative

TL;DR

This paper establishes a fundamental relationship between the integral of second-order directional derivatives over a region and the change in the function's value, extending classical calculus results to second derivatives.

Contribution

It proves a new theorem linking second-order directional derivatives and function value changes for functions without critical points, with extensions to more complex cases.

Findings

Integral of second-order directional derivatives relates to function value change.

The theorem applies to regions bounded by level curves without critical points.

Extensions discussed for regions with critical points or unbounded regions.

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.