The two cultures of integration theory

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

On densities

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

Nothing but a little marrow

From Rabelais’s prologue to the first book of Gargantua and Pantagruel, translated in 1653 by Sir Thomas Urquhart:

Most noble and illustrious drinkers, and you thrice precious pockified blades (for to you, and none else, do I dedicate my writings), Alcibiades, in that dialogue of Plato’s, which is entitled The Banquet, whilst he was setting forth the praises of his schoolmaster Socrates (without all question the prince of philosophers), amongst other discourses to that purpose, said that he resembled the Silenes. Silenes of old were little boxes, like those we now may see in the shops of apothecaries, painted on the outside with wanton toyish figures, as harpies, satyrs, bridled geese, horned hares, saddled ducks, flying goats, thiller harts, and other such-like counterfeited pictures at discretion, to excite people unto laughter, as Silenus himself, who was the foster-father of good Bacchus, was wont to do; but within those capricious caskets were carefully preserved and kept many rich jewels and fine drugs, such as balm, ambergris, amomon, musk, civet, with several kinds of precious stones, and other things of great price. Just such another thing was Socrates. For to have eyed his outside, and esteemed of him by his exterior appearance, you would not have given the peel of an onion for him, so deformed he was in body, and ridiculous in his gesture. He had a sharp pointed nose, with the look of a bull, and countenance of a fool: he was in his carriage simple, boorish in his apparel, in fortune poor, unhappy in his wives, unfit for all offices in the commonwealth, always laughing, tippling, and merrily carousing to everyone, with continual gibes and jeers, the better by those means to conceal his divine knowledge. Now, opening this box you would have found within it a heavenly and inestimable drug, a more than human understanding, an admirable virtue, matchless learning, invincible courage, unimitable sobriety, certain contentment of mind, perfect assurance, and an incredible misregard of all that for which men commonly do so much watch, run, sail, fight, travel, toil and turmoil themselves.

Whereunto (in your opinion) doth this little flourish of a preamble tend? For so much as you, my good disciples, and some other jolly fools of ease and leisure, reading the pleasant titles of some books of our invention, as Gargantua, Pantagruel, Whippot (Fessepinte.), the Dignity of Codpieces, of Pease and Bacon with a Commentary, &c., are too ready to judge that there is nothing in them but jests, mockeries, lascivious discourse, and recreative lies; because the outside (which is the title) is usually, without any farther inquiry, entertained with scoffing and derision. But truly it is very unbeseeming to make so slight account of the works of men, seeing yourselves avouch that it is not the habit makes the monk, many being monasterially accoutred, who inwardly are nothing less than monachal, and that there are of those that wear Spanish capes, who have but little of the valour of Spaniards in them. Therefore is it, that you must open the book, and seriously consider of the matter treated in it. Then shall you find that it containeth things of far higher value than the box did promise; that is to say, that the subject thereof is not so foolish as by the title at the first sight it would appear to be.

And put the case, that in the literal sense you meet with purposes merry and solacious enough, and consequently very correspondent to their inscriptions, yet must not you stop there as at the melody of the charming syrens, but endeavour to interpret that in a sublimer sense which possibly you intended to have spoken in the jollity of your heart. Did you ever pick the lock of a cupboard to steal a bottle of wine out of it? Tell me truly, and, if you did, call to mind the countenance which then you had. Or, did you ever see a dog with a marrowbone in his mouth,—the beast of all other, says Plato, lib. 2, de Republica, the most philosophical? If you have seen him, you might have remarked with what devotion and circumspectness he wards and watcheth it: with what care he keeps it: how fervently he holds it: how prudently he gobbets it: with what affection he breaks it: and with what diligence he sucks it. To what end all this? What moveth him to take all these pains? What are the hopes of his labour? What doth he expect to reap thereby? Nothing but a little marrow. True it is, that this little is more savoury and delicious than the great quantities of other sorts of meat, because the marrow (as Galen testifieth, 5. facult. nat. & 11. de usu partium) is a nourishment most perfectly elaboured by nature.

In imitation of this dog, it becomes you to be wise, to smell, feel and have in estimation these fair goodly books, stuffed with high conceptions, which, though seemingly easy in the pursuit, are in the cope and encounter somewhat difficult. And then, like him, you must, by a sedulous lecture, and frequent meditation, break the bone, and suck out the marrow,—that is, my allegorical sense, or the things I to myself propose to be signified by these Pythagorical symbols, with assured hope, that in so doing you will at last attain to be both well-advised and valiant by the reading of them: for in the perusal of this treatise you shall find another kind of taste, and a doctrine of a more profound and abstruse consideration, which will disclose unto you the most glorious sacraments and dreadful mysteries, as well in what concerneth your religion, as matters of the public state, and life economical.

Yes, but proceed as though I did not know it

From Moliere’s Le Bourgeois gentilhomme, translated by A. R. Waller:

Teacher of Philosophy: […] What do you wish to learn?

Monsieur Jourdain: Everything I can, for I am intensely anxious to be learned; it troubles me that my father and my mother did not see that I was thoroughly grounded in all knowledge when I was young.

Teacher of Philosophy: An admirable sentiment: Nam sine doctrina vita est quasi mortis imago. Doubtless you know Latin and understanding that.

Monsieur Jourdain: Yes, but proceed as thought I did not know it: explain to me what it means.

Teacher of Philosophy: It means, Without knowledge, life is little more than the reflection of death.

Monsieur Jourdain: That Latin is right.

Teacher of Philosophy: Do you not know some of the principles, some of the rudiments of knowledge?

Monsieur Jourdain: Oh! Yes, I know how to read and write.

Teacher of Philosophy: Where would you like us to begin? Would you like me to teach you logic?

Monsieur Jourdain: What is logic?

Teacher of Philosophy: It teaches that which educates the three operations of the mind.

Monsieur Jourdain: What are the three operations of the mind?

Teacher of Philosophy: The first, the second and the third. The first is to have a proper conception of things, by means of universals; the second is to judge accurately, by means of categories; and the third is to draw a conclusion accurately by means of the figures Barbara, Celarent, Durii, Ferio, Baralipton, etc.

Monsieur Jourdain: Those words are regular jawbreakers. That logic does not appeal to me. Let us learn something nicer.

I must set it to music

From Berlioz’s Evenings in the Orchestra (1854), translated by Charles E. Roche, pp. 314-315:

Beethoven, when carried away by the subject of Leonora, or
Conjugal Love, saw in it only the sentiments it gave him the
opportunity of expressing, and never took into account the sombre
monotony of the spectacle It presents. The libretto, of French
origin, was first set to music in Paris by Gavaux. Later an Italian
opera was made out of it for Paer, and it was after having heard
in Vienna the music of the latter’s Leonora that Beethoven had
the naive cruelty to say to him: “The subject of your opera pleases
me; I must set it to music.”

It was not the only quip Ludwig had in him. There’s also this, from Anton Schindler’s biography Beethoven as I Knew Him:

On New Year’s Day, 1823, Beethoven, his nephew, and the author were sitting at their noon meal, when the master was handed a New Year’s card from his brother who lived close by. The card was signed, “Johann van Beethoven, Land-owner.” Immediately the master wrote on the back of the card, “Ludwig van Beethoven, Brain-owner,” and returned it to him.

The notebook as intellectual form

Benjamin, Arcades Project

Leopardi, Zibaldone

Valéry, Cahiers

Weil, The Notebooks of Simone Weil

These are acts of creation, not just acts of compilation (though they the work of the latter can inspire the former), and they depart from the diary, from the personal journal, in their analytical and intellectual focus. They are more records of thinking than records of personal life. On the act of compilation, see also: commonplace book, formulary, gnomologium, hypomnema (include Erasmus’s Adages, Francis Bacon’s Promus, Locke’s A New Method of Making Common-Place-Books).

Hysterical misery into common unhappiness

From the end of Freud’s “Psychotherapy of Hysteria” in Studies on Hysteria (1895), with Josef Breuer:

When I have promised my patients help or improvement by means of a cathartic treatment I have been faced by this objection: “Why, you tell me yourself that my illness is probably connected with my circumstances of my life. You cannot alter these in any way. How do you propose to help me then?” And I have been able to make this reply: “No doubt fate would find it easier than I to relieve you of your illness. But you will able to convince yourself that much will be gained if we succeed in transforming your hysterical misery into common unhappiness. With mental life that has been restored to health you will be better armed against that unhappiness.”

Committing philosophy upon him

From the opening of Nozick’s Philosophical Explanations:

Children think an argument involves raised voices, anger, negative emotion. To argue with someone is to attempt to push him around verbally. But a philosophical argument isn’t like that—is it?

The terminology of philosophical art is coercive: arguments are powerful and best when they are knockdown, arguments force you to a conclusion, if you believe the premisses you have to or must believe the conclusion, some arguments do not carry much punch, and so forth. A philosophical argument is an attempt to get someone to believe something, whether he wants to believe it or not. A successful philosophical argument, a strong argument, forces someone to a belief.

Though philosophy is carried on as a coercive activity, the penalty philosophers wield is, after all, rather weak. If the other person is willing to bear the label of “irrational” or “having the worse arguments,” he can skip away happily maintaining his previous belief. He will be trailed, of course, by the philosopher furiously hurling philosophical imprecations: “What do you mean, you’re willing to be irrational? You shouldn’t be irrational because…” And although the philosopher is embarrassed by his inability to complete this sentence in a noncircular fashion—he can only produce reasons for accepting reasons—still, he is unwilling to let his adversary go.

Wouldn’t it be better if philosophical arguments left the person no possible answer at all, reducing him to impotent silence? Even then, he might sit there silently, smiling, Buddhalike. Perhaps philosophers need arguments so powerful they set up reverberations in the brain: if the person refuses to accept the conclusion, he dies. How’s that for a powerful argument? Yes, as with other physical threats (“your money or your life”), he can choose defiance. A “perfect” philosophical argument would leave no choice.

What useful purpose do philosophical arguments serve? Do we, trained in finding flaws in history’s great arguers, really believe arguments a promising route to the truth? Does either the likelihood or arriving at a true view (as opposed to a consistent and coherent one) or a view’s closeness to the truth vary directly with the strength of the philosophical arguments? Philosophical arguments can serve to elaborate a view, to delineate its content. Considering objections, hypothetical situations, and so on, does help to sharpen a view. But need all this be done in an attempt to prove, or in arguing?

Why are philosophers intent on forcing others to believe things? Is that a nice way to behave toward someone? I think we cannot improve people that way—the means frustrate the ends. Just as dependence is not eliminated by treating a person dependently, and someone cannot be forced to be free, a person is not most improved by being forced to believe something against his will, whether he wants to or not. The valuable person cannot be fashioned by committing philosophy upon him.

With virtually no patina

Listening to WHRB yesterday in the car, I was floored by a recording of Perahia playing Handel’s Suite No. 3 in D minor, HWV 428. The final two movements—air with five variations, and presto—are thrilling. I’d never heard the piece, and I still don’t know much about the seventeen keyboard suites published in two collections—first eight (HWV 426–433), known as the Great Eight, in 1720 (at age thirty-five), then another nine (HWV 434–442) around 1733—beyond the one that gives us the “Harmonious Blacksmith” (HWV 430, in E major) and another (HWV 434, in B-flat major) from which Brahms borrowed the theme for his Handel variations. (Schiff has a splendid recording of the latter on a disc with the Brahms.)

My ignorance is not totally my fault: very little seems to have been written about the suites—they do not command the same attention as Bach’s French and English suites or the partitas—and the only near-complete recording I can find of all seventeen on piano is by Richter and Gavrilov, but inexplicably they leave out the resplendent 434. Here is what Hungarian-American musicologist Paul Henry Lang writes in George Frideric Handel (he speaks of eight suites in the second volume, but there were multiple printings, apparently, and ultimately a ninth was added):

Of all of Handel’s works it is the many harpsichord pieces that may provide a glimpse of his creative youth. This is natural, because keyboard music was the German cantor’s native soil. Chrysander published a collection of these pieces, to which he gave the title Klavierbuch aus der Jugendzeit. Here we can find many prototypes and original versions of some of the pieces reworked and published later. “Reworking” is the key to the uneven quality to Handel’s output in this area, for the keyboard pieces show a wide range in quality, from the slight and insignificant to the magnificent and highly artistic. When an old piece was used in its original shape—that is, when Walsh or a Continental pirate published it without Handel’s permission—the result was unworthy of the great composer. When Handel had a chance to “correct” a youthful piece we are dealing with an altogether different kind of music, and, of course, the new pieces added to the collection by the mature master are almost all first-rate Handel. The music is no longer that of a young provincial German composer but of an elegant, experienced, and knowledgeable international composer intimately acquainted with Italian and French music.

The success of these pieces was phenomenal; they were the most popular compositions of their sort in all of Europe. Published by John Cluer and Walsh as independent volumes of “Lessons,” selections often appeared both in London anthologies and in the pirated publications of Dutch, Swiss, French, and German printers. In sales the harpsichord volumes outdid by far Couperin’s, Rameau’s, and Bach’s similar collections. As usual when the business methods of the estimable publishing house of Walsh are combined with Handel’s own ways with his musical hoard, things become hazy as to time, place, and even the identity of the composer. The first volume of suites, of 1720, was not yet within Walsh’s grasp; it was published by John Cluer “for the Author.” These suites could not have been composed before the Italian journey. Perhaps some of them were written in Hanover, but, at any rate, they surely were thoroughly gone over for the “corrected” edition. The second set, published by Walsh in 1733, without Handel’s permission, also contains eight suites, but this music is considerably weaker than the 1720 collection, undoubtedly because the material, somehow filched by Walsh, was not subjected to Handel’s usual reconditioning treatment. Among other reasons that indicate an arbitrary collection is the neglect of tonal order. The scheme in the first book of suites is carefully arranged and contrasted: A major, F major, D minor, E minor, E major, F-sharp minor, G minor, F minor. In the second book there is no orderly succession, and it is most unlikely that Handel would have agreed to pairs of consecutive suites in the same key. Of the third set, published later, not only the date is uncertain: one wonders whether these “suites” were not put together by the publisher from single, unrelated pieces. Indeed, we are not even sure Handel had anything to do with this largely insignificant music.

It is a shame pianists do not play these more often. As Richter writes in his notebooks, “these Suites are veritable miracles, laminated in gold but with virtually no patina.”

Fortunately there are several recordings of 428, in D minor. Here’s a smattering of the ones I like, timestamped to the final two movements. Richter’s is the least exciting, even dull; he makes up for it with his verbal endorsement. As the other piano versions make clear, these pieces should not be relegated to the harpsichord repertoire—or the dustbin of musical history.

  • Gould, on harpsichord, played as if on piano



  • Daria van den Bercken, on piano



  • Richter, on piano



  • Perahia, on piano



  • Shura Cherkassky, on piano



  • Éric Heidsieck



  • Ottavio Dantone, on harpsichord

Air, with a blow out at the end of the fifth variation:



Burrows, Handel

Dean, The New Grove Handel

Lang, George Frideric Handel

The heaven of legal concepts

The opening of Felix S. Cohen, “Transcendental Nonsense and the Functional Approach” (1935):

Some fifty years ago a great German jurist had a curious dream. He dreamed that he died and was taken to a special heaven reserved for the theoreticians of the law. In this heaven one met, face to face, the many concepts of jurisprudence in their absolute purity, freed from all entangling alliances with human life. Here were the disembodied spirits of good faith and bad faith, property, possession, laches, and rights in rem. Here were all the logical instruments needed to manipulate and transform these legal concepts and thus to create and to solve the most beautiful of legal problems. Here one found a dialectic-hydraulic-interpretation press, which could press an indefinite number of meanings out of any text or statute, an apparatus for constructing fictions, and a hair-splitting machine that could divide a single hair into 999,999 equal parts and, when operated by the most expert jurists, could split each of these parts again into 999,999 equal parts. The boundless opportunities of this heaven of legal concepts were open to all properly qualified jurists, provided only they drank the Lethean draught which induced forgetfulness of terrestrial human affairs. But for the most accomplished jurists the Lethean draught was entirely superfluous. They had nothing to forget.

Von Jhering’s dream has been retold, in recent years, in the chapels of sociological, functional, institutional, scientific, experimental, realistic, and neo-realistic jurisprudence. The question is raised, “How much of contemporary legal thought moves in the pure ether of Von Jhering’s heaven of legal concepts?” One turns to our leading legal textbooks and to the opinions of our courts for answer. May the Shade of Von Jhering be our guide.

Almost unable to despise

The thirty-second of Leopardi’s Pensieri (Thoughts), written in 1837, translated by J.G. Nichols:

As he advances every day in his practical knowledge of life, a man loses some of that severity which makes it difficult for young people, always looking for perfection, and expecting to find it, and judging everything by that idea of it which they have in their minds, to pardon defects and concede that there is some value in virtues that are poor and inadequate, and in good qualities that are unimportant, when they happen to find them in people. Then, seeing how everything is imperfect, and being convinced that there is nothing better in the world than that small good which they despise, and that practically nothing or no one is truly estimable, little by little, altering their standards and comparing what they come across not with perfection any more, but with reality, they grow accustomed to pardoning freely and valuing every mediocre virtue, every shadow of worth, every least ability that they find. So much so that, ultimately, many things and many people seem to them praiseworthy that at first would have seemed to them scarcely endurable. This goes so far that, whereas initially they hardly had the ability to feel esteem, in the course of time they become almost unable to despise. And this to a greater extent the more intelligent they are. Because in fact to be very contemptuous and discontented, once our first youth is past, is not a good sign, and those who are such cannot, either because of the poverty of their intellects or because they have little experience, have been much acquainted with the world. Or else they are among those fools who despise others because of the great esteem in which they hold themselves. In short, it seems hardly probable, but it is true, and it indicates only the extreme baseness of human affairs to say it, that experience of the world teaches us to appreciate rather than to depreciate.

I think of the opening of Marianne Moore’s poem “Poetry”:

I, too, dislike it: there are things that are important
beyond all this fiddle.
Reading it, however, with a perfect contempt for it,
one discovers that there is in
it after all, a place for the genuine.