
#1
May812, 09:06 PM

P: 2

I am a high school student who has finished calculus, multivariable calculus, and differential equations, but I feel as though the treatment I received of those topics was rather inadequate and shallow  lacking mathematical rigor, so to speak. Can anyone comment on the quality of those two textbooks for selfstudy (as well as for teaching other students calculus over the summer)? I would buy Apostol or Spivak, but they are horrendously expensive and probably impractical given my current financial situation.




#2
May812, 10:37 PM

P: 628





#3
May912, 12:59 PM

Sci Advisor
HW Helper
P: 9,422

David Widder was a Harvard professor of mathematics when I was an undergraduate there in the 1960's. His book was regarded as a standard of high quality rigorous treatment of calculus. I do not recall the coverage. I myself only used it to learn the basic underlying results of one variable calculus like the existence of extreme values for continuous funcions on a real closed bounded interval. It served me well.
C.H. Edwards was a topologist at the University of Georgia 10 or 20 years ago, and was renowned as one of the best teachers and expositors there for decades. He won every teaching award conceivable at the university and more broadly in the state and maybe more widely. Edwards book can be scanned at Amazon and covers several variables rigorously, including proofs of implicit and inverse function theorems, and stokes type theorems. Both books should be excellent, with Edwards possibly seeming slightly more modern, since it was published in 1973, as opposed to Widder's, published in 1961. It is possible e.g. that Edwards may treat differential forms, and not Widder, but I don't know for sure. Take a look at them in a library and see which one is easier to understand. 



#4
May1012, 04:10 PM

Sci Advisor
P: 3,173

Quality of Widder's Advanced Calc. and Edwards's Adv. Calc. of Several Variables
Few mathematicians go through all the topics of advanced calculus and master the proofs of all the technicalities. They do enough of such work so that if the need arises they can go back to the material to understand any proofs they need. Most mathematicians deal with the proofs in advanced calculus by regarding them as special cases of results in a more advanced field called "functional analysis".
I have Widder's "Advanced Calculus" 2nd Edition. In my opinion, trying to go through that book pagebypage would not be the most effective use of your time. I doubt the "average bright person" would make much progress in a single summer. Widder is rigorous relative to the standards of a typical introductory calculus text, but his subject matter is not very general or abstract. I would recommend Widder's book to engineers. I'd recommend that a person pursing mathematics spend their time on a topic more general than "Advanced Calculus". (So I probably wouldn't recommend Edward's book either, but I haven't seen it.) If your goal is to pursue math, try something in the field of "Calculus on Manifolds" or "Introductory Real Analysis". Advanced Calculus by Widder Table of Contents 1. Partial Differentiation (includes Taylor's Theorem in sevveral variables and Jacobians) 2. Vectors (includes gradient, inner product, outer product) 3. Differential Geometry (arc length, osculating plane, curvature and torsion, FrenetSerret formulas, differential forms, Mercator maps) 4. Applications of Partial Differentiation (includes Lagrange multipliers) 5. Stieljes Integral (includes HeineBorel theorem, uniform continuity) 6. Multiple Integrals (includes Duhamel's theorem, center of gravity, moments of inertia) 7. Line and Surface Integrals (includes Green's theorem, exact differentials, Gauss's theorem, Stoke's theorem) 8. Limits and Indeterminate Forms (includes l'Hospital's rule, limits points of a sequence, Cauchy's criterion) 9. Infinite Series (includes the usual tests for convergence, differentiation of series, Cesaro summability, Cauchy's inequality, uniform convergence) 10. Convergence of Improper Integrals (includes the usual tests, uniform convergence, divergent integrals, Schwarz inequality, Minkowski inequality) 11. The Gamma Function. Evaluation Of Definite Integrals (includes Stirling's formula, the Beta function, Euler's constant) 12. Fourier Series 13. The Lapace Transform 14. Applications of the Laplace Transform If you are tutoring students who are taking Calculus I and II, most of those topics won't be relevant. 


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