Consider $$ Q:\begin{bmatrix} \rho \\ \phi \end{bmatrix} \to \begin{bmatrix} \cosh(\rho) \cos(\phi) \\ \sinh(\rho) \sin(\phi) \end{bmatrix} $$
The task is to pick a domain as big as possible, so that $Q$ is a $C^1$-diffeomorphism.
I never proved that a function is a diffeomorphism, so I didn't know where to start. I first wrote down the Jacobian and computed the determinant. I got to $$ \det(JFQ) = \sinh(\rho)^2 + \cosh(\rho)^2 \;. $$ Then I tried to show that it's bijective; but I don't know how to show bijectivity of a vector function. I have spent like 1 hour on it and still am no step further than at the beginning. And I can't do the other questions without solving this one, I think. They want me to draw the coordinate axis for the constants $\rho$ and $\phi$ and to show that $Q$ is an orthogonal coordinate transformation and some kind of scalar factor (no idea what that is). And at last I'm supposed to describe the gradient, divergent, and laplace operator on the new coordinates.
Although now that I thnk about it, I may be able to draw the coordinate axis with the constants $\rho$ and $\phi$. Is it an ellipse with $\cosh(\rho)$ being the $x$-intercept and $\sinh(\rho)$ being the $y$-intercept?
Anyway, I hope someone could give me some hints regarding those questions.