Here $F:\Bbb R^3\rightarrow \Bbb R^3$ is given by $F(x,y,z)=(e^{x}\cos(y),e^{x}\sin(y),z^3)$.
The question is to find all $(x,y,z)$ such that $dF(x,y,z)$ is invertible.
I think it is a question about the inverse function theorem(because it comes from the part of exercise on the inverse function theorem). But I cannot find out how to use it.
I think from linear algebra, what I need to do is just finding out all the points that the matrix of $dF$ is invertible, may I please ask if it is correct?
If it is incorrect, could someone please tell me why? May I please ask how can I find all $(x,y,z)$ such that $dF(x,y,z)$ is invertible? Thanks so much!
The Jacobian of $F$ computes to $$[dF(x,y,z)]=\left[\matrix{e^x\cos y&-e^x\sin y&0\cr e^x\sin y&e^x\cos y&0\cr 0&0&3z^2\cr}\right]$$ and has $${\rm det}\big(dF(x,y,z)\bigr)=3e^{2x}z^2\ .$$ It follows that $dF(x,y,z)$ is invertible at all points $(x,y,z)\in{\mathbb R}^3$ with $z\ne0$.