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I know this question is probably not adequate to this SE either, but let me explain my situation: I'm civil engineering's college, so, there isn't a SE for civil engineering, and my doubts about integration in engineering are pratically pure physics. So, as I said, I'm graduating, but I'm from Brazil, education here is the same thing as nothing (belive me, it really sucks, I see people come out of physics's college without knowing who Maxwell was..), and I read my calculus books and see they all use Riemann integrals (of course, they don't say that..); but, in my searchs, I see a lot of Lebesgue integration, especially concerning problems of calculating the center of mass of an object, in continuum mechanics, etc. So, here's my question:

  1. Lebesgue integration isn't the most easy thing in the world, I don't have a lot time, and education here sucks, should I spent my time studying Lebesgue integration instead of Riemann's?

  2. Which one do you guys use more in physics with applications in engineering?; of course, the last one it's a little trouble, because engineering just "borrow" from physics. But, in general, which compensates more?

Ricardo
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  • I was elaborating more on an answer, but the basic point is that in the examples you mention, Lebesgue is only a bit more useful than Riemann. And you never use Riemann, you just use it to construct the Fundamental Theorem of Calculus linking (anti-)derivatives and integrals (and to put at ease the minds of mathematicians, proving the area exists). On a practical setting, one never does the approaching rectangles. – Davidmh Apr 22 '14 at 21:21
  • I think this is a good question, but might be more suitable for math.stackexchange. – Bombyx mori Apr 22 '14 at 21:23
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    There's a good Richard Hamming quote that I think is quote appropriate for this situation: "Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane." – DumpsterDoofus Apr 22 '14 at 21:35
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    Thank you. @Bombyx, I think about math SE, but since I will only use integration applied to physics, I thought you guys would be the best to answer.. – Ricardo Apr 22 '14 at 21:37
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    @user5462: In short, I probably wouldn't waste too much time on Lebesgue integration, unless it's really absolutely necessary. Instead, try to focus on what's practical and useful for the work that you are studying for, and don't worry too much about slippery mathematical details like Lebesgue integration. Most engineers I know have never even heard of Lebesgue integrals, and it was never covered in any of my physics classes either. – DumpsterDoofus Apr 22 '14 at 21:39
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    @user5462: I would encourage you to learn it any way, it is abstract but is useful for many purposes. Once you know real analysis and functional analysis well, you are prepared to study more advanced mathematical physics, which often involve a fair amount of PDE. – Bombyx mori Apr 22 '14 at 21:42
  • Thank you, I think I will. The only problem is that here I don't have a lot incentive to that, but I can always find time..You know, @DumpsterDoofus made a nice suggestion, and I tried to do that, but I just didn't learn nothing, you guys have no idea... I don't want to learn the basics, I really want to know; solid knowledge. And I like math a lot, if I don't get through engineering, I could do something else.. – Ricardo Apr 22 '14 at 21:51
  • Related: http://physics.stackexchange.com/q/38761/2451 – Qmechanic Apr 22 '14 at 22:16
  • IMO you likely won't come across the need for Lebesgue integration in engineering unless you study signal processing or stochastic control systems. In those fields, the related idea of measure theory is essential for a good grasp of stochastic processes. Where you might find benefit is that, once you've gotten your head around it, conceptually the Lebesgue integral is simpler, cleaner, and the various calculus theorems are more general, more readily proven but they do take a bit of machinery to get your head into. The Wiki page on Lebesgue integration has a delightful one paragraph summary ... – Selene Routley Apr 23 '14 at 06:16
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    ... by Lebesgue himself of his idea to his friend Paul Montel. Ponder this and study carefully how the Lebesgue integral takles the integral of the function $\phi:[0,1]\to{0,,1}$ where $\phi(x)=1$ if $x\in\mathbb{Q}$ ($x$ is rational) and $\phi(x)=0$ if $x$ is irrational ($\int,\phi,dx=0$, BTW) and once you've done that, you'll have a good grasp of the gist of the idea and you may be in a better position to know whether you need to study it further. – Selene Routley Apr 23 '14 at 06:20
  • BTW A good text that I found highly readable and interesting (although it's pretty old now, but then so am I and I first read it 30 years ago) is H. Royden, "Real Analysis". PS: I notice Amazon recommends you buy Munkres, "Topology" with it: the latter is a good reference to have on your shelf once you've grasped topology but thoroughly nightmarish to learn from. But one thing at a time! It's a shame some of these books don't come out in kindle. – Selene Routley Apr 23 '14 at 06:28
  • Thank you, I will take a look at the book. In the moment I'm reading Rudin's "real analysis", I thought about reading Bourbaki's integration and functions of real variable, but they're too big and requires to much of the reader. – Ricardo Apr 23 '14 at 13:46
  • @user5462 Bourbaki is meant to be a grand summary of relevant mathematics for professional mathematicians - it's more like a very thorough encyclopoedia than a textbook. Do you know it's a pseudonym for a group of leading mathematicians? – Selene Routley Apr 24 '14 at 01:25
  • Yes, first time I see the name was in Sautoy's "The music of prime numbers" and he said right away it was a group of mathematicians. I found their books very good, and I think the general criticism about them it's overreacted. I mean, they have the best mathematicians of the time, Andre Weil's works inspired the Langlands program, and his conjectures in algebraic geometry are just a masterpiece. They ware mathematicians, not philosophers, and some people criticize them for not worrying too much about the foundation. But I don't think that's a reason for not read his books. I mean, pratically – Ricardo Apr 24 '14 at 19:49
  • every mathematician in history doesn't worry too much about the foundation. The ones who do, are more philosophers than mathematicians. Betrand Russell for example, didn't make a single result in advanced mathematics, but all the modern mathematical logic is based on his Principia Mathetimca ( definite descriptions , theory of types,etc.), minus the class calculus and with, of course, Gödel's incompleteness theorem. – Ricardo Apr 24 '14 at 19:52

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