Double Calculus

TL;DR

This paper introduces a simplified version of Bögel's 1930s double calculus, extending classical single-variable calculus results to two variables, providing a new framework for higher-dimensional analysis.

Contribution

It presents a streamlined, modified double calculus theory for two variables, including extensions of key classical calculus theorems, which was previously not well known.

Findings

Includes two-variable versions of Rolle's theorem and Lagrange's mean value theorem.

Extends Cauchy's mean value theorem and Fermat's extremum theorem to double calculus.

Provides a foundation for higher-dimensional calculus analysis.

Abstract

We present a streamlined, slightly modified version, in the two-variable situation, of a beautiful, but not so well known, theory by B\"{o}gel, already from the 1930s, on an alternative higher dimensional calculus of real functions, a double calculus, which includes two-variable extensions of many classical results from single variable calculus, such as Rolle's theorem, Lagrange's mean value theorem, Cauchy's mean value theorem, Fermat's extremum theorem, the first derivative test, and the first and second fundamental theorems of calculus.