Last updated Fri Apr 24 16:13 EDT 2020 (note (2) on Rosenlicht VII.36 was added)
Due-date: Wednesday, 4/22/20
You are required to do all of the problems below. You will not be required to hand in any of them.
A. Rosenlicht Chap. VII/ 9, 10, 12, 25 (do non-book problem B4 before #25, in order to make sense of the sum), 35, 36. Notes on #36: (1) 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.
(2) To proceed from the conclusion of Rosenlicht's hint for #36 to the conclusion of the problem itself, prove the following lemma: Let \(U\subset {\bf R}^2\) be an open set, \(h:U\to {\bf R}\) a continuous function. Assume that \(\int_a^b \big(\int_c^d h(x,y)\,dy\big)\,dx =0\) for every closed rectangle \([a,b]\times [c,d] \subset U\). Then \(h\) is identically zero.
B. Click here for non-book problems.
Note: Problem B3 is unlikely to be a hand-in problem. I included it in the assignment to make you aware of the results, particularly part (c), without my taking class time to prove them. Writing out careful proofs for this problem would consume time that would be better-spent on other problems. I'll be content with you having a general idea of why the results of B3 are true.C. Read this handout on improper integrals. Exercises in the handout should be read, since some results are stated in the exercises. However, doing these exercises is not part of this assignment (though some might be part of a future assignment). The due-date for this reading will be later than the due-date for the rest of the assignment; it's okay if you don't do the reading all at once.
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