Prove the mapping $(x,y,z)\mapsto (x+e^y,y+e^z,z+e^x)$ is locally invertible. 
Show that the mapping $\mathbb{R}^3\to \mathbb{R}^3$, $(x,y,z)\mapsto (u,v,w)$ which is defined by $$\begin{align*}
u&=x+e^y\\
v&= y+e^z\\
 w &=z+e^x
\end{align*}$$is locally invertible  everywhere.

Any assistance will be appreciated. Thanks

This mapping is clearly smooth. By the inverse function theorem, all we have to check is that the Jacobian determinant is nonzero everywhere:

$$\left|\matrix{1 & e^y & 0\\0&1&e^z\\e^x&0&1}\right|=1+e^xe^ye^z>0.$$

