Baby Rudin is the standard reference for people who are up for the challenge.Ĥ) Multivariable analysis - A rigorous generalization of the material from (3).
Ross's book "Elementary Analysis" is pretty compact, covers everything important, in my opinion, if a little basic. The idea is to figure out what makes the theorems tick, rather than proving the theorems, as by now, students have seen the intermediate value theorem before, but since they've probably never heard of a "supremum" they are not in a position to prove it. The bulk of the course is in these "foundational" topics in analysis. What follows is material that is more likely to be "advanced calculus."ģ) Elementary analysis - limits of sequences, series, topology of the reals, functions, continuity, etc. I suppose Larson is the standard reference. I don't know a book that I'm really happy with that's "naive" in the sense of (1) but covers multivariable other than Simmons book above. They are likely aimed at engineers or physicists rather than aspiring mathematicians. There is some amount of emphasis on geometry in the sense of vectors, surfaces, and maybe some introductory linear algebra, but the bulk of the theory is left out.īoth these classes are the kind of things new students take in their early years in mathematics. You do more complicated integrals, learn about partial derivatives and a chain rule for several variables. Proofs of key theorems are in appendices, which is nice.Ģ) Multivariable calculus: This is calculus where all the stuff from the "naive" section is generalized to several variables. The former is aimed much more at mathematicians, while the latter is a giant book that is also useful in multivariable. Some standard books for naive calculus that I like are Calculus Deconstructed, by Nitecki (Hope I spelled that right), the book by Simmons, Calculus with Analytic Geometry. These are the "easy" applications of calculus that you can do without much theory. You see things like related rates, applications to physics, arc length, volumes of revolution, etc. Students see limits in terms of tables of values, and the idea of "go close but don't touch." You compute so many derivatives and integrals your eyes bleed. The exact order of the approaches varies from place to place, but these different trips through the theory of calculus all have a few things in common.ġ) Naive calculus: This is calculus which is highly computation and application based. It's a prefect choice for students who feel boredom watching long lectures and wants things to finish them quickly with the maximum knowledge gain. So join me here and do it in a quick and easy way.Edit: It occurs to me I did not fully answer the question in the OP, so I have included some more details now.Ĭalculus is taught in America (and maybe in other parts of the world) in several "passes," with increasing rigor and scope. The lectures' videos are appealing, attractive, fancy (with some nice graphic designing), fast and take less time to walk you through the whole lecture.
At the end it carries plenty of solved numerical problems with the relevant examples.
#ADVANCED CALCULUS VERIFICATION#
This course includes videos explanation starting right from introduction and basics, then takes graphical and numerical phase with formulas, verification and proofs both graphically and mathematically. Learn Advanced Calculus of Higher Mathematics through animation.
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