Last updated Thu Apr 25 03:23 EDT 2019
Due-date (virtual): The day of the final exam. There are no hand-in problems.
- C. Read Rosenlicht Section VII.4.
- A.
- Rosenlicht Chap. VII/ 35, 36
- 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.
- Rosenlicht Chap. IX/ 1
- B.
- In the handout "Some notes on multivariable derivatives", do exercises 4.1 and 4.2, and read Example 5.2.
- Click here for other non-book problems.
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