On the integral formula of the Jacobian determinant

It is known that the integral of the Jacobian determinant of a smooth map $f:\bar{\mathrm{\Omega }}\to {\mathbb{R}}^n$ depends only on $f{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}$ 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.