On the integral formula of the Jacobian determinant
Shibo Liu
TL;DR
The paper proves that the integral of the Jacobian determinant over a bounded domain depends only on the boundary values of a pair of maps that agree on the boundary, yielding an analytic route to the Brouwer fixed point theorem. It provides two proofs: a classical analysis approach that constructs a divergence-free cofactors-based vector field and applies the divergence theorem, and a concise differential-forms argument using d and Stokes theorem. A corollary for linear perturbations is presented, and the work situates the result within prior literature on null Lagrangians and boundary-data dependence. Overall, it offers alternative, boundary-data–driven proofs of a fundamental fixed-point result with clear connections to geometric and topological methods.
Abstract
It is known that the integral of the Jacobian determinant of a smooth map $f: \barΩ \rightarrow \mathbb{R}^n$ depends only on $f |_{\partial Ω} $ 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.