Chapter 9 of Principles of Mathematical Analysis

March 22nd, 2005 ~ Posted in: Mathematics

Chapter 9 of Principles of Mathematical Analysis is on functions of several variables, from a vector space approach. My major problem with it– the same problem I have with almost every chapter in this book– is that Rudin seems to go out of his way to make his statements obscure. Is that what I have to look forward to in graduate literature?

Here’s a typical example:

Suppose \mathbf{f} maps an open set E \subset \R^n into \R^m, and \mathbf{f} is differentiable at a point \mathbf{x} \in E. Then the partial derivatives (D_j f_i)(\mathbf{x}) exist, and
\displaystyle \mathbf{f}^\prime(\mathbf{x})\mathbf{e}_j = \sum_{i=1}^m (D_j f_i)(\mathbf{x})\mathbf{u}_i \quad (1 \leq j \leq n).

Oh, okay… I see that is explained in the form I was about to suggest later on. I was going to complain that he didn’t simply state

\displaystyle \mathbf{f^\prime(x)} = \begin{pmatrix} (D_1 f_1)(\mathbf{x}) & \cdots & (D_n f_1) (\mathbf{x}) \\ \vdots & \cdots & \vdots \\ (D_1 f_m)(\mathbf{x}) & \cdots & (D_n f_m) (\mathbf{x}) \end{pmatrix},

but he does end up doing that, after the end of the proof. I say he should have put that at the start, because that is the form that a student who took, say, Multivariable Calculus would have the most familiarity with.

In general, there are a lot of places in this chapter where at least mentioning the matrix notation equivalent of the purely algebraic statements he makes would save a lot of time in understanding what he’s trying to say. And let’s not even start on the topic of illustrations.

Maybe someone should make a project of making illustrations for this horrible book, since so many schools seem to use it. Then we could hand out a little booklet to students that has illustrations for each chapter, with references.

Mathematics ~

2 Responses to “Chapter 9 of Principles of Mathematical Analysis”

  • 1. tpc
    March 23rd, 2005 at 5:07 am

    I understand your frustrations, I’m reading chapter 10 now and face similar problems. But the guy wrote the book 50 years ago and I guess things were different back then. Moreover, the way linear algebra was taught has also changed considerably. If you read older algebra texts, you find that the standard was to teach groups, rings, modules and then linear algebra. Nowadays, we work with matrices way before we know what’s a group.

    There are tons of analysis books around, so we can always look for alternatives. You can try this http://www.amazon.com/exec/obidos/tg/detail/-/0387946144/102-8571848-9332100?v=glance

  • 2. Alex
    March 23rd, 2005 at 6:22 pm

    I own several already that I’m not reading (one by Fomin and Kolmogorov, and another one in the yellow book), but the table of contents in that one looks intriguing enough for investigation. Mostly, I don’t refer to outside resources for that class, because I’m too lazy to switch up notations.

    The old way of teaching linear algebra sounds like it might be fun. There are a couple of books in the UH library that take that approach, but they end up being huge, so I stay away from them.

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