TITLE: An application $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ and $C^{1}$ such that $f(x)=0$ for $x>r$ implies the value of jacobian integral is zero
QUESTION [9 upvotes]: Let $f \colon \mathbb{R}^n \to \mathbb{R}^n$ of class $C^{1}$. Suppose that exists $r>0$ such that $f(x)=0$ if $|x|\geq r$ .Prove that exists $k>0$ such that:
$\displaystyle \int_{B[0,k]}$ det$Jf(x)=0$
I have yet to see Stokes theorem which i think is related to this question and i'm trying to adapt the proof of "change of variable" theorem without much success,any hint is appreciated.

REPLY [3 votes]: This integral has a similar form to the change of variables formula, with one crucial difference: There is no absolute value of the Jacobian here. I think it is unllikely you could find a nice proof by trying to change coordinates. (I think any such proof would essentially boil down to using Stoke's without explicitly saying so.)
Stoke's Theorem simply states that the integral of the Jacobian determinant in the ball is the same the integral of $f(x)$ over the boundary of the ball. If we pick k to be larger than r, then this integral is 0 since $f(x)$ is zero outside the ball of radius r.
Stoke's theorem is essentially a generalization of the Fundamental Theorem of Calculus to higher dimensions. and  $detJf(x)$ is what we use in place of the derivative of $f(x)$.