I don't know much more about Analysis than what I've read about it on Wikipedia, although I have just begun reading Introduction to Calculus and Analysis I, by Richard Courant.

My understanding is that Analysis is mostly concerned with Calculus topics on a much more rigorous level: mainly the study of proofs for the theorems and concepts that comprise Calculus.

If that is true, how are proofs applied to real-world, practical problems? I guess I'm either confused in my understanding of what Analysis is, or don't understand how it is useful in a practical sense.

I've heard people say that "Analysis just covers virtually the same topics as Calculus, but at a more in-depth level," but what I've read about it on Wikipedia makes me skeptical about these claims. Is it true that Analysis is just a sort of more in-depth/thorough version of Calculus?

  • $\begingroup$ I don't think the kind of proofs you find in analysis are necessarily about "real-world, practical problems"--you can certainly evaluate definite integrals, solve ODEs, etc. with no knowlege of the dominated convergence theorem or the Bolzano-Weierstrass theorem. These things are rather about understanding the abstract structure of real numbers, sequences, series, and functions. $\endgroup$ – David Zhang Jul 1 '15 at 3:09
  • $\begingroup$ Undergraduate real analysis is no different from calculus. Standard limits, derivatives, integral stuff but with emphasis on the foundations and logic as opposed to computation. In calculus, you use the first derivative test to find extrema; in analysis, you prove the first derivative test (it's a cute proof -- look it up). More advanced analysis is not much like calculus, but rather deals with a potpourri of topics like dynamical systems, differential geometry, measure theory, and the like. Even if you're not a math major, it's would be a nice class to learn about how math works nicely. $\endgroup$ – user217285 Jul 1 '15 at 3:10

I guess it depends on what you mean by ``real world application." In a typical undergraduate real analysis course one learns the precise definitions of limit, continuous, etc. and proves theorems like the Mean Value Theorem, so that you know why the theorems are true. I would agree that an undergraduate real analysis course is very much like calculus in greater depth.

  • $\begingroup$ By "real-world, practical problems," I meant: In what ways are the proofs and theorems one learns in an Analysis course more useful than the techniques of Calculus course for solving problems of a practical nature, such as those encountered in everyday experiences. These may not necessarily represent "real-world" problems in each instance, but my favorite applications so far of Calculus are Related Rates and Optimization. They're simplified versions of the type of problems I'd to solve in future as a professional engineer. Is Analysis useful for these types of problems, as well? More useful? $\endgroup$ – tommytwoeyes Jul 4 '15 at 16:08

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