A FUNDAMENTAL THEOREM OF CALCULUS FOR SECOND-ORDER DIRECTIONAL DERIVATIVE

摘要

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 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.