In standard multivariable analysis texts, the change of variables for multivariable integration in Euclidean space is almost always stated for a $C^1$ diffeomorphism $\phi$, giving the familiar equation (for continuous $f$, say)

$$\int_{\phi(U)}f=\int_U(f\circ\phi)\cdot|\det D\phi|$$

Of course, this result by itself is not very useful in practice because a diffeomorphism is usually hard to come by. The better advanced calculus and multivariable analysis texts explain explicitly how the hypothesis that $\phi$ is injective with $\det D\phi\neq0$ can be relaxed to handle problems along sets of measure zero -- a result which is necessary for almost all practical applications of the theorem, starting with polar coordinates.

Despite offering this slight generalization, very few of the standard texts state that the situation can be improved further still: there is an analogous theorem for arbitrary $C^1$ mappings $\phi$, not just those that are injective everywhere except on a set of measure zero. We simply account for how many times a point in the image gets hit by $\phi$, giving

$$\int_{\phi(U)}f\cdot\,\text{card}(\phi^{-1})=\int_U(f\circ\phi)\cdot|\det D\phi|$$

where $\text{card}(\phi^{-1})$ measures the cardinality of $\phi^{-1}(x)$.

I think this theorem is a lot more natural and satisfying than -- and surely just as heuristically plausible as -- the first. For one thing, it removes a huge restriction, bringing the theorem closer to the standard one-variable change of variables for which injectivity is not required (though of course the one-variable theorem is really a theorem about differential forms). In particular, it emphasizes that regularity is what's important, not injectivity. For another thing, it's not a big step from here to the geometric intuition for degree theory or for the "area formula" in geometric measure theory. (Indeed, the factor $\text{card}(\phi^{-1})$ is a special case of what old references in geometric measure theory called the "multiplicity function" or the "Banach indicatrix.") It's also used in multivariate probability to write down densities of non-injective transformations of random variables. And last, it's in the spirit of modern approaches to gesture at the most general possible result. The traditional statement is really just a special case; injectivity only becomes essential when we define the integral over a manifold (rather than a parametrized manifold), which we want to be independent of parametrization. I think teaching the more general result would greatly clarify these matters, which are a constant source of confusion to beginners.

Yet many of the standard multivariable analysis texts (Spivak, Rudin PMA and RCA, Folland, Loomis/Sternberg, Munkres, Duistermaat/Kolk, Burkill) don't mention this result, even in passing, as far as I can tell. The impression a typical undergraduate gets is that the traditional statement is the final word on the matter, not to be improved upon; after all, the possibility of improvement isn't even hinted at, even when the multivariable result is compared to the single variable result. So I've had to hunt for discussions of the extension; I've found it here:

I'm also confident I've seen it in some multivariable probability books, but I can't remember which. But none of these is a standard textbook, except perhaps for Zorich.

My question: are there standard analysis references with nice discussions of this extension of the more familiar result? Probability references are fine, but I'm especially curious whether I've missed some definitive treatment in one of the classic analysis texts.

Also feel free to speculate, or explain, why so few texts mention it. (Is there really any good reason for not mentioning it, when failing to do so implicitly trains students to think injectivity is an essential ingredient for this kind of result?) I'm hoping there's a more interesting answer than "most authors don't mention it because the texts they learned from didn't either" or "even an extra sentence alluding to the possibility of a more general result is too much to ask for since the traditional theorem is hard enough to prove on its own."

(Cross-posted on MSE.)

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    $\begingroup$ For another extension note that in your second formula one can replace card$(\phi^{-1})$ with deg($\phi, U$) and |det(D$\phi$)| with det(D$\phi$). In general, what you want is a wide subject in Geometric Measure Theory; see area and coarea formulae. $\endgroup$ – Pietro Majer Jan 1 '16 at 8:19
  • $\begingroup$ @PietroMajer: Hi Pietro, thanks. Yes, you'll see I allude to the "area formula" in the post. My question is whether this shows up in any standard multivariable analysis texts, and why it's not more regularly mentioned at this more elementary level, when it's an easy and significant generalization. $\endgroup$ – symplectomorphic Jan 1 '16 at 8:29
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    $\begingroup$ What you want is the coarea formula. Proving e.g. that the function $p\mapsto {\rm card}\;\phi^{-1}(p)$ is measurable is not worth the pain in a standard analysis text addressed to people that don't know the standard change-in-variables formula. $\endgroup$ – Liviu Nicolaescu Jan 1 '16 at 12:31
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    $\begingroup$ Is it super-important to have those formulas boxed? I feel the urge to unbox them. $\endgroup$ – Andrej Bauer Jan 1 '16 at 13:43
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    $\begingroup$ @symplectomorphic The coarea formula is a significant generalization, but it is not easy. $\endgroup$ – Liviu Nicolaescu Jan 1 '16 at 18:08

The better advanced calculus and multivariable analysis texts explain explicitly how the hypothesis that $\varphi$ is injective with $\det D \varphi \neq 0$ can be relaxed to handle problems along sets of measure zero -- a result which is necessary for almost all practical applications of the theorem, starting with polar coordinates.

I speculate that most authors don't go beyond the injective immersion condition because it is sufficient for most practical applications of the theorem, polar and spherical coordinates being among the most important examples. The difficulty of formulating and proving the change of variables formula for integration is out of proportion to the rest of the content of an advanced calculus course, so if you are writing a textbook on the subject then there is a strong temptation not to stray too far from what you need to handle the basic examples and applications. This also explains why some authors are happy to live with the assumption that $\phi$ is a diffeomorphism - if your goal is just to prove Stokes' theorem, then why make it harder?

It wouldn't hurt, I suppose, to allude to more general versions of the theorem in a parenthetical remark or an extended exercise, but I don't think the stakes are very high. Undergraduates are usually accustomed to the fact that they aren't getting the most general possible theorems in their classes.

  • $\begingroup$ The lack of reasonable generality in this case is what made math feel not a Turing complete subject to beginners. $\endgroup$ – John Jiang Jan 1 '16 at 16:12
  • $\begingroup$ Thanks for the answer, Paul. I buy this for run-of-the-mill advanced calculus books aimed at engineers or non-mathematicians, so it doesn't bother me for, e.g., Marsden-Tromba's book. But surely the goal of Rudin's, Folland's, Loomis/Sternberg's books, etc., is not simply "to prove Stokes's theorem." I asked this question because it shocked me to discover the regularity with which all the classic references for advanced undergraduate analysis straight-up ignore this theorem, yet they elsewhere relish in pointing out the possibility of a much more general result without giving a proof. $\endgroup$ – symplectomorphic Jan 1 '16 at 16:21
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    $\begingroup$ @symplectomorphic A good principle for mathematical writing (not just math-for-engineering writing) is: don't generalize past your examples. I have found that when Folland, for instance, alludes to a more general version of a theorem it is often so that he can say something interesting about an example. Can you think of a specific computation or theorem which both belongs in an advanced calculus textbook and requires a more sophisticated version of the change of variables formula? $\endgroup$ – Paul Siegel Jan 1 '16 at 17:46
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    $\begingroup$ @symplectomorphic I've heard many students ask "Why is the proof of the change of variables formula for integrals so complicated?" I've never heard a student ask about the injectivity hypothesis. $\endgroup$ – Paul Siegel Jan 3 '16 at 3:30
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    $\begingroup$ @PaulSiegel: I won't get into a battle over who's heard who say what. There's plenty of questions on MSE about when injectivity is required. The conservative responses I've heard here I think are telling. $\endgroup$ – symplectomorphic Jan 3 '16 at 3:39

As Paul Siegel says in his answer: The usual formula is sufficient for most practical applications. I would go further and say:

The plain form of the change of variables theorem makes it much more clear that the main motivation for this theorem is just to compute integrals.

The change of variables theorem is really a workhorse-theorem to work with and (as far as I see) is not something that is structurally important. Check Christian Blatter's answer at MSE to see a mathematician with years of experience telling you how often he really used the non-injective form.

Also, the plain form really shows that the Jacobian is the crucial thing here and also the proofs of the plain form (at least the ones I know) makes that pretty clear. However, if you want to prove the more general form, I don't know anything else than to start from the plain result and add on top of that.

And my last point: As I said above, the change of variables theorem is a workhorse to do something, namely, to compute integrals. If you would ever calculate an integral of the form

$$\int_{\phi(U)}f\cdot\,\text{card}(\phi^{-1})=\int_U(f\circ\phi)\cdot|\det D\phi|$$

what would you do? You would check for each point how often it is reached by $\phi$ and would patch the results together (neglecting the issues with null-sets) using the plain form on each patch. This is something that a student would came come up with by himself. Hence, the general form is not at all helpful to do the very thing for which the plain form of the theorem in intended. Balancing how complicated the proof of the more general result is and how intuitive and (most importantly) practically not so useful the result is, it seems clear what you should do when writing a textbook.

  • $\begingroup$ Dirk, I don't disagree the plain form is all that is needed for most "practical applications." But theoretical analysis books, such as Spivak, Rudin, Folland, and Loomis/Sternberg, are not primarily interested in practical applications. My claim is that the more general result is more satisfying theoretically; the plain form is a special case, and the general form leads directly to other important ideas in degree theory and geometric measure theory. So from a theoretical and pedagogical point of view, I just don't see why it's not mentioned. $\endgroup$ – symplectomorphic Jan 2 '16 at 8:49
  • $\begingroup$ @symplectomorphic A "practical application" in this context means a reason why the formula is useful in advanced calculus. In that sense, the authors you mentioned are very interested in practical applications - that's one of the reasons their books are so popular. You asked why a certain formula doesn't appear in very many advanced calculus textbooks, and the reason seems to be: the formula doesn't have any applications to advanced calculus (even though it may be useful elsewhere). What is unsatisfying about this answer? $\endgroup$ – Paul Siegel Jan 3 '16 at 3:28

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