# Multivariable calculus: hard problems with solutions

I'm practicing for my multivariable calculus exam and I'm having some trouble mostly because I have no way of knowing if my solutions are correct or not.

For example, a typical problem goes like this:

Let $f:\mathbb{R^2}\longrightarrow\mathbb{R}$ defined by:

$$f(x,y)=\begin{cases} \sin(y-x) & \text{for} & y>|x| \\ \\ 0 & \text{for} & y=|x| \\ \\ \frac{x-y}{\sqrt{x^2 + y^2}} & \text{for} & y<|x| \end{cases}$$

1. Study $f$ with respect to continuity on its domain.
2. Study $f$ with respect to differentiability on its domain.

I think I know how to solve this, but I have no way to verify my answer and I might be unaware of some subtleties. Moreover, I did some browsing, but I was unable to find examples containing functions defined with branches such as this one. As you probably know, branches are precisely what make this a non-trivial problem (at least for me!).

So, I came here to ask for recommendations on books or online resources with solutions (don't need all the details, just the results) to problems like this one.

Thanks!

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There are problems with solutions on this web-site: cramster.com – HbCwiRoJDp Jan 26 '12 at 11:14
Try Marsden and Tromba's Vector Calculus. It has answers (actually, fairly complete solutions) to all odd-numbered problems. – ItsNotObvious Jan 26 '12 at 14:53
For this particular problem I suggest introducing new euclidean coordinates $(u,v)$: Put $$x:={1\over\sqrt{2}}(u-v)\ ,\quad y:={1\over\sqrt{2}}(u+v)\ .$$ – Christian Blatter Jan 26 '12 at 18:53