Last updated Thurs Apr 13 19:22 EDT 2017
Due-date: Wednesday, 4/19/17
You are required to do all of the problems below. You will not be required to hand them all in. I have indicated below which ones you do have to hand in on the due-date. Don't make the mistake of thinking that I'm collecting only the problems I think are important.
The "due date" above is the date that your written-up problems should be handed in, but don't wait to get started on the assignment. You should always get started on problems as soon as we cover the relevant material in class.
- A. Rosenlicht Chap. VII (pp. 160–167)/ 25 (do non-book problem B2 before this, in order to make sense of the sum), 26 (Problem 15, referred to here, is B2(d)), 28–30, 35, 36. Of these, hand in only 25, 30, 35. In your writeup for #30 you may assume the formula for the derivative of \(\tan^{-1}\) that you should have derived in #29 and that is implicit in the statement of #30.
- Note on #28: This problem, which assumes you've read Section VII.4, is intended just as review (in case you're forgotten any of the trig functions) and as preparation for #29. For the derivative computations, it's okay if you just compute the derivative of "tan".
- Note on #30: the book's "and therefore" is misleading. In the first part of the problem, you are showing only that the indicated power series converges to \(\tan^{-1} x\) if \(|x|\) is strictly less than 1. An extra argument is needed to show that the series also converges to \(\tan^{-1} 1\) when \(x=1\).
- Note on #36: There are ways to prove this result ("the equality of cross-partials [or mixed partials]") without using integrals, for example by making good use of the Mean Value Theorem. The last time I tried to do this problem using the book's hint, I found that I needed extra assumptions: that for each \(y\), the function \(x\mapsto \frac{\partial f}{\partial x}(x,y)\) is continuous, and that for each \(x\), the function \(y\mapsto \frac{\partial f}{\partial y}(x,y)\) is continuous. (Both of these conditions are guaranteed if \(\frac{\partial^2 f}{\partial x^2}\) and \(\frac{\partial^2 f}{\partial y^2}\) exist at every point of the given open subset.) If you find that you need to make these assumptions, make them (just say that you're making them). I might give some extra credit if you're able to do the problem the way the book suggests without using any extra assumptions.
The fact that "\(\frac{\partial^2 f}{\partial x\partial y} = \frac{\partial^2 f}{\partial y\partial x} \)" can be proven without any assumptions other than the continuity of \(\frac{\partial^2 f}{\partial x\partial y}\) and \(\frac{\partial^2 f}{\partial y\partial x} \)—in particular, without assuming even the existence of \(\frac{\partial^2 f}{\partial x^2}\) and \(\frac{\partial^2 f}{\partial y^2}\)—is a curiosity, rather than an important fact. The space of real-valued functions, on a given open subset \(U\)in \({\bf E}^2\), for which the only second partial derivatives assumed to exist (or to be continuous) are the two mixed partials, never arises in practice. It is a geometrically unnatural space, in the sense that this space's defining condition is not preserved if we rotate coordinate axes. A space that is important \(C^2(U)\), the space of functions \(U\to {\bf R}\) for which all four second partials exist and are continuous. The condition that all second partials exist, or that all second partials are continuous, is invariant under rotation of coordinate axes.
- B.
- Click here for (some) non-book problems. Of these, hand in only B2, B3ac.
- Do problems 17–18 in the Improper Integrals handout. Do not hand in either of these.
- C. Read Rosenlicht Section VII.4.
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