Jacobian and Stokes Theorem

Question

Let $f:U \to\mathbb{R}^{n}$ a $C^{2}$ function in the open set $U \subset \mathbb{R}^{n}$. Suppose $D \subset U$ is a compact domain with boundary $\partial D$ of $C^{2}$ class. If $f(x)=0$ for all $x\in \partial D$, prove that $\int_{D} det f'(x)dx=0$.

I know to prove only the bidimensional case...

Answer 1

If I remember correctly, $$ \det f' \, dx_1 \wedge \dots \wedge dx_n = f^*(dx_1 \wedge \dots \wedge dx_n) = d f^*(x_1) \wedge \dots \wedge d f^*(x_n) = d(f_1 \wedge df_2 \wedge \dots \wedge df_n). $$

By Stoke's theorem, $$ \int_D \det f'(x) \, dx = \int_{\partial D} f_1 \, df_2 \wedge \dots \wedge df_n = 0 $$ since $f_1 = 0$ on $\partial D$.