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An Introduction to Higher Mathematics, Volume I

Hua Loo-Keng
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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Hua Loo-Keng is generally counted with S. S. Chern as being responsible for opening China up to the mathematical west in the dark days of the totalitarian regimes of the twentieth century (and, yes, plus ça change, plus c'est la même chose …), But unlike Chern, who operated very visibly, first from Chicago and then from his headquarters at Berkeley, Hua spent most of his years in communist China, condemned to obscurity (and invisibility) and suppression at the hands of Mao’s regime and the madness of the cultural revolution.

One day in the middle-to-late 1970s Hua resurfaced. I was an undergraduate at UCLA’s mathematics department at that time, and the number theory circle stirred with excitement at the news of this great event. As I recall, the scuttlebutt was that Hua had proved something wonderful concerning the Goldbach conjecture, and this certainly would fit the bill, given Hua’s biographical facts. In any case it was clear even to us rookies that the return of Hua was a big deal, on both the humanitarian and the mathematical fronts, and it certainly made a deep impression.

Thus, it is exciting for me to be able to review a classic work by Hua, viz. his big, two-volume Introduction to Higher Mathematics. An additional dimension is added due to the fact that Hua was of course largely self-educated, although, as it says on the book’s dust-flap, he “spent time at Cambridge in the 1930s, when he made notable contributions to number theory.” It is indeed the case that Hua came to England as an already mature mathematician, and his activities there count largely as collaborations with scholars like Davenport and Heilbronn.

The book under review, or rather the two-volume set, is, in a way, consonant with this British connection: it reminds me of another truly wonderful Oxbridge book on foundational but genuine mathematics, viz., G. H. Hardy’s A Course of Pure Mathematics — both works are gems of mathematical pedagogy and richly repay self-study, provided the student is bright, committed, and persistent. I am reminded, also, of tales of British mathematicians like Freeman Dyson and Hardy himself, stories in which an adolescent finds his way into the irresistibilities of pure mathematics by working his way though a particular book. For Hardy (at Winchester) it was the Cours d’Analyse by Camille Jordan, and, later, for Dyson (also at Winchester, more than a generation later) this same work played a similar role (he worked through it in the company of his friend James Lighthill) and he also covered Elementary Treatise on Differential Equations, by H. T. H. Piaggio, solving all its 700+ problems (cf. p.479 of the fabulous book, QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga, by S. S. Schweber). Indeed, the British tradition of self-study and proto-research is eminently represented by these cases in point, and Hua and the books now under review obviously fit beautifully into this autodidactic scheme.

The flyleaf to An Introduction to Higher Mathematics contains a paragraph stating that “the course [that gave rise to the books] reflects Hua’s instinctive technique, using the simplest tools to tackle even the most difficult problems, and contains both pure and applied mathematics, emphasizing the interdependent relationships between different branches of the discipline” (again a prominent feature of the British approach). The first book then takes the reader from a discussion of R and C and vector spaces, through calculus and elementary analysis, to elements of differential geometry, Fourier analysis, and ordinary differential equations — all this in about 800 pages: no stinting on attention to detail here! The second book (at nearly 600 pages) starts with the geometry of C and non-Euclidean geometry, goes on to a lot of complex analysis (including very thorough coverage of harmonic functions, residue theory, and the maximum modulus principle and its relatives), to summability (how could it be otherwise for some one who was at Cambridge when Hardy ruled there?) and (yes!) a great deal of work on elliptic functions, more linear algebra, difference and differential equations, and a finishing crescendo of applications of (broadly) linear algebra to the theory of quadratic forms, finding volumes in manifolds (and ultimately algebraic groups), and non-negative square matrices. The sweep of Hua’s coverage is manifestly Homeric.

His presentation is limpid and exceedingly elegant. The books are very easy and pleasant to read, even though they’re anything but facile or trivial; no, the coverage is outstanding and thorough, and the mathematics is beautiful: there is nothing here that is inaccessible to a novice who in fact will proceed all the way to real mastery of a good portion of the undergraduate curriculum — and more. I cannot think of any book that matches Hua’s set for this purpose, and an ideal self-study curriculum would consist of Hardy’s Pure Mathematics, Hua’s books presently under review (modulo the proviso that there is overlap: but it’s like having to choose between Vladimir Horowitz and Arthur Rubinstein — there’s no way to go wrong …), and then, if I may be parochial, Hardy and Wright, An Introduction to the Theory of Numbers. Of course, if one is not called to number theory, the latter could be replaced by books in one’s own area of choice. Regardless, however, at this stage of the game, a solid foundation has been built and mathematical maturity is no longer an issue.

An Introduction to Higher Mathematics is not equipped with problem sets or chatty asides, indeed its style is not oriented in the direction of motivation: this is not really necessary at the present level, of course, and Hua’s goal is to teach his audience a lot of mathematical technique in a correct, complete, and concise manner. (To an extent the style of the books’ prose reminds me of Edmund Landau: one of my favorite authors, but not every one’s cup of tea). The books have wide margins, however, and they beg to be filled. But as one sets out to do so, it may be wise to heed the following warning given by Wang Yuan, who over a half century ago was Hua’s assistant tasked, among other things, to aid him with the production of An Introduction to Higher Mathematics: “… there are … places at which the author [writes] ‘Similarly it is not difficult to prove that …’ or words to that effect. This is acceptable when writing for someone who is as knowledgeable as the author. However, when it comes to students coming across such material for the first time, or even for myself when I was working closely with him as his assistant, it was not at all easy. For this reason, when studying the material, special attention must be given to such places where one needs to put in the extra work …” But that is how the game is played after all, and An Introduction to Higher Mathematics is clearly a gem in this enterprise.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Volume I: 1. Real and complex numbers
2. Vector algebra
3. Functions and graphs
4. Limits
5. The differential calculus
6. Applications of the derivative
7. Taylor expansions
8. Approximate solutions to equations
9. Indefinite integrals
10. Definite integrals
11. Applications of integral calculus
12. Functions of several variables
13. Sequences, series and integrals with variables
14. Differential properties of curves
15. Multiple integral
16. Curvilinear integral and surface integral
17. Scalar field and vector field
18. Differential properties of curved surfaces
19. Fourier series
20. System of ordinary differential equations.