Learning Real Analysis -> Calculus

[joecoz88]

Learning Real Analysis --> Calculus

Hello,

Just a quick question I am wondering about. I am going to take my first real analysis course next semester using Rudin. Obviously I have already gone through the usual calculus sequence. I am wondering if learning real analysis will help refresh/solidify my skills in computational/applied calculus, or is real analysis so theoretical that the two will remain separate?

 

[jostpuur]

It often happens in calculus, that one is able to perform some computation, but at some steps one is unable to properly justify some tricks. This is where people usually say something like "we assume $f$ sufficiently smooth", without been able to really tell what the "sufficiently smooth" means. These are kind of steps where theory of real analysis can be used to complete the computations into a rigorous form.

I am wondering if learning real analysis will help refresh/solidify my skills in computational/applied calculus,

My answer to this is yes, although this is not about computational tricks. Real analysis will help you understand better some tricks, you might have already used before without fully understanding them.

or is real analysis so theoretical that the two will remain separate?

My opinion is that this actually depends on the student him/herself. Real analysis is sometimes presented in a way that it might appear to be separate from concrete calculations, which is unfortunate IMO. I have not understood the purpose of all results I've encountered so far either. But the usefulness of at least some results in real analysis should become clear to a student who is genuinely interested in understanding concrete calculations.

 

[lurflurf]

Baby Rudin is not very computational or applied. If you desire further development of computational or applied aspect, you should read a different book in addition.

 

[NoMoreExams]

Yeah go get Big Boy Rudin :) (kidding). But why are you taking analysis? Just for fun or ... Like lurflurf said, if you are seeking to develop your computational skills further, analysis might not be for you. Maybe first semester will give you insights that jostpuur mentioned but anything beyond that, once you get into measure theory, probably will not.

 

[daytripper]

I'm also concerned about this, as I get into higher mathematics. I'm about to take vector analysis and algebra and I'm not entirely sure what that entails, but is it going to at least be tangential to calc? I love calculus and I'd like to build on my knowledge and not forget it in a month like I did with Taylor Expansion Series. Will this happen or will I actually use calculus in these classes and subsequent classes (thereby refreshing and cementing my understanding)?

 

[NoMoreExams]

Not sure what you mean by vector analysis and algebra. Is that linear algebra or vector calc?

 

[jostpuur]

I'm also concerned about this, as I get into higher mathematics. I'm about to take vector analysis and algebra and I'm not entirely sure what that entails, but is it going to at least be tangential to calc? I love calculus and I'd like to build on my knowledge and not forget it in a month like I did with Taylor Expansion Series. Will this happen or will I actually use calculus in these classes and subsequent classes (thereby refreshing and cementing my understanding)?

The names of courses with certain material vary a little bit, but my guess is that you are now familiar with single-variable calculus, and you are about to take a course on multi-variable calculus. If so, then your worries are not quite the same as the ones in the original post, because by real analysis, we usually mean analysis that is one step further from multi-variable calculus.

(please mention, if I guessed wrong)

 

[adriank]

In my first analysis course, there was no multivariable stuff at all. So in that sense, analysis (in my experience) isn't beyond multivariable calculus; it's really more a rigorous treatment of calculus from the foundations. I'd say that taking analysis will give a deeper understanding of the calculus you already know (you probably didn't see a proof of the Mean Value Theorem or Taylor's theorem in your calculus course, or even a definition of a real number), but it probably won't help in actually doing computations in calculus, or in applications of calculus. They're not separate at all; after all, calculus is mathematics, and everything in mathematics must have a solid foundation; that foundation for calculus is analysis.

If you like working with logic and proofs and such, then analysis is probably an interesting course to take; you get to build everything up just from the natural numbers. :)

 

[joecoz88]

Thanks for the replies. I am a pure math major and I have already taken linear and abstract algebra, and I am going on to real analysis and Galois theory next semester. The reason I asked the question is because I have noticed that as my skills in upper division (proof based) mathematics improve, I am forgetting everything else, like how to solve differential equations or how to compute a line integral. Is it necessary to brush up on these subjects if, for example, I chose to study differential geometry or even topology? Algebra and number theory are my main interests, though.

 

[jostpuur]

In my first analysis course, there was no multivariable stuff at all. So in that sense, analysis (in my experience) isn't beyond multivariable calculus

Agreed, but a first analysis course, and a course on real analysis are totally different things. You might wonder why totally different things have such similar names, but that's the way it is anyway

First course on analysis is something like derivatives, mean value theorem, l'Hospital's rule, and so on.

A course on real analysis is something like Lebesgue integration, dominated convergence theorem, $\sigma$-algebras, and so on.

The naming could have been more descriptive, of course...

 

[adriank]

Well, to me "real analysis" includes all of those, and the OP mentioned it would be his first real analysis course. :)

 

[jostpuur]

I am going to take my first real analysis course next semester using Rudin.

What book by Rudin are you talking about? Other one of these?

https://www.amazon.com/dp/007054235X/?tag=pfamazon01-20

https://www.amazon.com/dp/0071002766/?tag=pfamazon01-20

 

[joecoz88]

Yeah it is the first one, "Principles of Mathematical Analysis."


Here is the course description, though its probably not completely accurate:

The real number system. Sequences, limits, and continuous functions in R and 'Rn'. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.

 

[jostpuur]

whoops. I thought (or guessed) that you would have meant the second one, because that's what I've got used to calling the Rudin's book. It could be that my answers weren't best possible then.