A Fundamental Theorem of Calculus for Second-order Directional Derivative
Classical Analysis and ODEs
2023-08-28 v2 Differential Geometry
Abstract
Given a two-variable function without critical points and a compact region bounded by two level curves of , this note proves that the integral over of the second-order directional derivative of in the tangential directions of the interceding level curves is proportional to the rise in -value over . Also discussed are variations on this result when critical points are present or becomes unbounded.
Keywords
Cite
@article{arxiv.2107.11038,
title = {A Fundamental Theorem of Calculus for Second-order Directional Derivative},
author = {Pisheng Ding},
journal= {arXiv preprint arXiv:2107.11038},
year = {2023}
}