Image of unit ball under twice differentiable function Let $f: \mathbb{R^n} \to \mathbb{R^n}$ be a twice differentiable function such that $f(x) = 0$ outside of the unit ball $B \in \mathbb{R^n}$. I have to show that (the measure is Lebesgue): $$\int_B \det Df = 0$$
I tried to prove that $|f(B)| = 0$ (which should be equivalent to the above) but I've had no luck. I'm quite stuck since I have no further theorem that I can think of, and honestly the equality doesn't even intuitively seem to hold. 

Let $f=(f_1,\ldots,f_n)$. Consider the form
$$
w=f_1df_2\wedge\ldots \wedge df_n-f_2df_1\wedge df_3\ldots \wedge df_n+\ldots+(-1)^nf_ndf_1\wedge \ldots \wedge df_{n-1}
$$
and apply the Stokes theorem:
$$
\int_{\partial B}w=\int_{B}dw.
$$
