On the integral formula of the Jacobian determinant

TL;DR

This paper presents two new proofs, using classical analysis and differential forms, for the known result that the integral of a Jacobian determinant depends only on the boundary values of a smooth map, with implications for fixed point theorems.

Contribution

It introduces two novel proofs of a fundamental integral property of Jacobian determinants, enhancing understanding through classical analysis and differential forms methods.

Findings

The integral of the Jacobian determinant depends only on boundary values.

New proofs provide alternative approaches to a classical result.

Implications for analytic proofs of fixed point theorems.

Abstract

It is known that the integral of the Jacobian determinant of a smooth map $f: \bar{\Omega} \rightarrow \mathbb{R}^n$ depends only on $f |_{\partial \Omega} $ and this result leads to an analytic proof of the Brouwer fixed point theorem. In this note we provide two new proofs of this result, one by classical analysis and one by differential forms and Stokes formula.