Understanding how a function relates to Chapter 9 of baby Rudin

Question

In exercise 17 of chapter 9, Rudin lets a function from $\mathbb{R}^2$ to itself be defined by $f_1(x,y)=e^x\cos(y)$, $f_2(x,y)=e^x\sin(y)$. Rather than a solution to the exercise, I am interested in how the questions posed in the exercise (range of f, Jacobian is nonzero but f is not injective, finding the inverse, images of f under lines parallel to the coordinate axes) relate to the topics in chapter 9 (I know it's an example of the inverse function theorem but other parts seem irrelevant) or perhaps other topics in mathematics. Sorry if this is too broad but some guidance would be appreciated!

Answer 1

The exercise is intended to build up your knowledge of the behavior of maps between higher dimensional spaces, including showing you that some things about functions on $\Bbb R$ do not carry over.

Part (b) in particular is to show you that the rule for $f : \Bbb R \to \Bbb R$ that "$f' \ne 0$ anywhere $\implies f$ is injective everywhere" does not hold in higher dimensions. Here is an $f : \Bbb R^2 \to \Bbb R^2$ for which not only is the derivative not $0$, but the Jacobian is never $0$. Yet this $f$ is clearly not injective.

Parts (a) and (d) give you a feel for how the function looks. Part (c) wants you to develop a specific solid example of what the inverse function theorem tells you in general.

Such exercises are common in textbooks, because they ground your understanding of the knowledge you've just obtained so that you can build on it.