Prove that function $f$ is injective if its Jacobian matrix is positive definite. Assume that $\Omega\in\mathbb{R}^m$ is an open convex set and the vector-valued function $f:\Omega\rightarrow\mathbb{R}^m$ is differentiable. If Jacobian matrix $J_f(x)$ is positive definite for all $x\in\Omega$, prove that $f$ is an injective function on $\Omega$.
I have no way of dealing with it. Is there a theorem to do with it?

Hint: suppose $x\neq y$ but $f(x)=f(y).\ $ On $\Omega$, define for $0\le t\le 1,\  \gamma (t)=ty+(1-t)x,\ $ which is well-defined because $\Omega$  is convex. 
Now, compute $D\langle ((f\circ \gamma)(t)-f(x)), (y-x)\rangle$ and use the Mean Value Theorem and your hypothesis on the Jacobian, to arrive at a contradiction. 
