What are some practical applications of real analysis and partial derivatives?

[gummz]

I'm going so deep into the theory of things on my second semester. I'm getting into real analysis and partial derivatives, and it all seems so conceptual. The practicality of integration and differentiation are apparent, but less so in real analysis and partial derivatives, especially when the proofs pertaining to partial differentiation seem to me a computational miracle than anything else.

I need good textbooks that touch on the usage of real analysis and partial derivatives (more is welcome, of course). Do you have any suggestions? I want to feel that the proof I'm reading has applications in software companies. That this rule is used in calculating interest at Fortune 500 companies... something like that.

 

[jedishrfu]

Partial derivatives come up a lot in Vector Calculus which is immensely useful so Id stick with it. As an example, the gradient, the divergence and the curl are all partial derivatives of one sort or another.

http://en.m.wikipedia.org/wiki/Vector_calculus

 

[STEMucator]

See the avatar.

In all seriousness though, you really shouldn't feel that way about the most fundamental subject in your education. Math has plenty of real life application, but you must first acquire the knowledge before you can use it.

Try focusing more on visualizing the mathematics, rather than getting caught up in the semantics. All of the symbols have meanings, and if you can learn to see the meaning, then your experience with mathematics will surely improve. I'm sure it will make the equations easier to memorize and understand as well.

Let your imagination take charge when you look at all of those crazy symbols.

I would recommend Stewart's most recent multi-variable calculus text to give you good conceptual, visual and computational background.

 

[vanhees71]

Be sure that vector calculus is one of the most important topics to study for physicists. Nearly all physics uses it, most dominantly of course in electrodynamics and quantum (field) theory. It's pretty tough in the beginning, but you must go through to be able to follow any serious lecture on almost all topics in physics!

 

[MidgetDwarf]

I'm going so deep into the theory of things on my second semester. I'm getting into real analysis and partial derivatives, and it all seems so conceptual. The practicality of integration and differentiation are apparent, but less so in real analysis and partial derivatives, especially when the proofs pertaining to partial differentiation seem to me a computational miracle than anything else.

I need good textbooks that touch on the usage of real analysis and partial derivatives (more is welcome, of course). Do you have any suggestions? I want to feel that the proof I'm reading has applications in software companies. That this rule is used in calculating interest at Fortune 500 companies... something like that.

Maybe up until now you went through math just plug and chugging? Is it the proof writing that is bugging you? If so, the book called How To Prove It, will help you.

 

[Svein]

I'm going so deep into the theory of things on my second semester. I'm getting into real analysis and partial derivatives, and it all seems so conceptual. The practicality of integration and differentiation are apparent, but less so in real analysis and partial derivatives, especially when the proofs pertaining to partial differentiation seem to me a computational miracle than anything else.

Remember: Mathematics are a collection of tools. If you know your tools, you are able to use them in whatever type of work you are going to do. Some of us - the mathematicians - are dedicated toolmakers and do not care if the tools are useful or not.

Number theory was for a long time regarded as pristine mathematics - nobody could imagine any possible use for it. But then wave mechanics happened...

 

[homeomorphic]

Try A Radical Approach to Real Analysis, by David Bressoud.

Another book that might help is Discourse on Fourier Series by Cornelius Lanczos.

These are better at explaining what the point of real analysis is. Still, it's not that applied. But it's closer to being applied than most real analysis books.

As far as partial derivatives go, I'm not really sure what it is that you are talking about, but you probably need to study more physics, especially electromagnetism. And as far as the proofs, there's probably a way to think of them that makes them seem less of a computational miracle, but it might be challenging to find.

I have a very applied mindset, too, but sometimes, you have to be a little bit more patient and not expect proof that everything is useful right away. I think more applied approaches to things are possible, but if you want to learn math, you sort of have to put up with the way people do things right now. Even if things were a little bit more application-friendly in terms of pedagogy, as they could be, I don't think it's the way that math works that everything has to have an immediate use.

Number theory was for a long time regarded as pristine mathematics - nobody could imagine any possible use for it. But then wave mechanics happened...

I think you want to say cryptography, although I hear there are connections between quantum field theory and number theory. As far as quantum mechanics goes, it seems to be the case that it was more physically-motivated math that was important, although a lot of it was in terms of less immediately applicable theoretical questions. Physicists probably just look at the Dirichlet problem as just something that's evident from experiment, and they might not care about the theoretical existence of solutions. However, the study of integral equations relating to such things is part of what lead to the creation of functional analysis, and it comes full circle because even physicists now use Hilbert spaces and operators and all that.