As a Math Stack Exchange question recently reminded me, every good calculus student eventually asks what the dx in integration really means—and every good calculus teacher has her own answer. But the answers are often confused.
The mess arises in part because the question can be interpreted syntactically or semantically. As a piece of notation, it is essentially a bookkeeping device, indicating the variable with respect to which the integration is carried out—as Leibniz, who gave us the symbol, well understood. In symbolic computation the formal string y dx thus means something quite different than x dx. The semantic question is even more interesting. As a matter of mathematical substance, the dx generalizes in two directions: in one direction it denotes a measure, and in another direction it denotes a differential form. These are the two cultures of integration theory. Experts speak both languages—and can translate fluently between them—but we do a very bad job at putting the two in conversation in the undergraduate curriculum.
The difference shows up not just in the sort of object we integrate (a function in the former case, a form in the latter), but also in the sort of object we integrate over. Measure-theoretic integration works over (certain) bare sets, but form-theoretic integration requires more structure: a set plus an orientation. Integrating a function over the unit disc is measure theory; integrating a form over the unit disc oriented counterclockwise is exterior calculus. The one theory is static: it is a theory of content, how much stuff is in a set. The other theory is dynamic: it is a theory of flux, how stuff is flowing through a set. The difference is also visible in how the two integrals transform. For measures, the change of variables theorem uses the absolute value of the Jacobian. For forms, the theorem drops the absolute value.
This distinction is important both conceptually (it leads to very different theories) and historically (the ideas emerged quite independently). But it tends to be glossed over in the typical undergraduate calculus sequence, where the various types of integrals are run together rather than carefully distinguished. (Witness the ambiguity of the phrase “surface integral.”) And it is all but lost once manifolds become the central object of study. In most cases, given the goal of generalizing vector calculus or developing de Rham cohomology, the tendency is to shortchange the measure-theoretic perspective, reducing the meaning of “calculus on manifolds” simply to the exterior calculus. This trajectory makes it look like Stokes’s theorem is the ultimate culmination of the basic ideas of derivative and integral, the Whiggish consummation the concepts have been building to all along.
Not that the theory of forms isn’t deep and important. But it is easy to leave the typical undergraduate sequence having forgotten entirely about measure, and we may come away thinking the only integration possible on a manifold is integration of forms. Indeed one could be forgiven for asking whether a differential geometer even needs to know the name Lebesgue. The impression is only reinforced when the diligent student of calculus is dropped into a first course in measure. There one encounters a totally new language, beginning with sigma algebras. Eventually connections come into view, but at first blush the subject appears to have little to do with calculus, and nothing to do with manifolds.
This state of affairs is unfortunate, but also easy to remedy. The distinction is already apparent in the two basic types of surface integrals discussed in any multivariable calculus class: the integral of a function with respect to surface area (which does not require an orientation), and the flux integral of a vector field through a surface (which does require an orientation). Even more basically, it’s apparent in the two basic types of line integrals: the integral of a function with respect to arc length (which, again, does not require an orientation), and the work integral of a vector field along a curve (which does require an orientation). Emphasizing the differences between these two types of integrals would not only help resolve conceptual confusion; it would convey a fuller awareness of what “calculus on curved spaces” can mean and anticipate a wider range of mathematical territory.
Inside Euclidean space, the measure-theoretic ideas of arc length and surface area lead to notions such as surface measure (using Jacobians in a chart) and Hausdorff measure (for more complicated sets), generalizing Lebesgue measure to handle sets of lower dimension. This is the beginning of geometric measure theory, motivated historically by problems in the calculus of variations. Outside Euclidean space, where tools can’t be imported from the ambient space, these idea lead to the (intrinsic) notion of a density on a smooth manifold. (A related notion is what Ted Frankel calls a pseudoform in The Geometry of Physics.) Such densities always exist (by a partition of unity argument), but on a Riemannian manifold, a natural density is determined by compatibility with the metric: the Riemannian density, also known as the Riemannian volume element. It thus makes sense to talk about volume—and the integrals of functions with respect to volume—on any Riemannian manifold, orientable or not. (As it should: the Möbius band has an area, of course, even though it is not orientable.) Arc length, area, and volume elements aren’t forms; they aren’t even linear! They’re densities.
This understanding is not impossible to acquire, but it is harder than it needs to be. The few references that discuss these matters clearly and explicitly are all relatively advanced. The easiest fix, I think, is to linger a little longer over the special status of arclength and surface area integrals in the calculus sequence, taking time to preview these two rich cultures of integration.
On the two types of integral
Terry Tao, “Differential Forms and Integration”
“Integration of forms and integration on a measure space”, Math Stack Exchange
On surface measure and its relation to Hausdorff measure
John Benedetto and Wojciech Czaja, Integration and Modern Analysis
Gerald Folland, Real Analysis: Modern Techniques and Their Applications, sections 11.2-11.4 (theorem 11.25 gives the relationship between surface measure and Hausdorff measure)
Francesco Maggi, Sets of Finite Perimeter and Geometric Variational Problems, chapter 11 (theorem 11.3 gives the relationship between surface measure and Hausdorff measure)
Kennan Smith, Primer of Modern Analysis, chapter 15 (section 6 gives the relationship between surface measure and Hausdorff measure)
Daniel Stroock, Essentials of Integration Theory for Analysis (section 5.2 constructs surface measure, and section 8.3.4 shows the relationship with Hausdorff measure)
Michael E. Taylor, Measure Theory and Integration
John Hubbard and Barbara Hubbard, Vector Calculus, Linear Algebra, and Differential Forms, Chapter 5 (like Munkres, defines Riemannian density and surface measure for submanifolds of Euclidean space, without identifying them as such)
Folland Real Analysis, section 11.4
Serge Lang, Differential and Riemannian Manifolds
John M. Lee, Introduction to Smooth Manifolds, 2nd edition, pp. 427-434 (includes a discussion of the divergence theorem)
Lynn Loomis and Shlomo Sternberg, Advanced Calculus, revised edition, chapter 10 (includes the divergence theorem and an implicit discussion of the surface area problem)
James Munkres, Analysis on Manifolds, chapter 5 (defines Riemannian density and surface measure for submanifolds of Euclidean space, without identifying them as such)
Liviu I. Nicolaescu, Lectures on the Geometry of Manifolds, section 3.4
“The ds which appears in an integral with respect to arclength is not a 1-form. What is it?”, MathOverflow
“Why do I need densities to integrate on a non-orientable manifold?”, MathOverflow
“Lebesgue measure theory vs. differential forms?”, Math Stack Exchange