Math Needed for Physics: How Much is Enough?

[Helical]

From what I understand the math required for physics majors isn't extremely rigorous (except for the most theoretical of branches). I have also gathered from this forum that the reason for this is physicists use math as sort of "tool" and don't study it as a science (this may not be the appropriate term but I'm not sure of a better one) as a mathematician would. That being said, is a deep understanding of math required to get a deep understanding of physics? Or is it truly just a "tool"? For instance, can someone have a deep understanding of General Relativity with only a "working knowledge" of tensor calculus?

Thanks.

 

[Asphodel]

Hmm, define rigorous.

Seriously.

It really depends on how you do your physics. The level of mathematical sophistication required to study physics is much higher than > 90% of undergrads will ever learn (numbers shamelessly made up, do your own research, bwahaha!), but it is not "rigorous" from the perspective of someone taking 400-level mathematics.

Physicists are like anyone else, we specialize. Outside of that area of specialization, they may know nothing at all, or only a little more than most people would that they picked up out of curiosity. The math you work with frequently, you're likely to learn with a higher degree of sophistication as it becomes necessary / useful. The famous story about Einstein's correspondence with Levi-Civita comes to mind.

So yes, it's largely a tool...but sometimes knowing your tools better allows you to use them more masterfully.

 

[wunderboy]

rigor?

i will be soon graduating with a double bs in math and physics, so i have done quite a bit in each. until you get a little ways along in math, it is difficult to truly see the differences in the math used in physics and the math used purely for the sake of using math. let me just say that the math i use for physics is quite complicated, but i never have to prove any of it... yes, physics students may claim to proove, but it is more of a rigorous derivation rather than a rigorous proof. NOT TO DEGRADE THE MATH USED BY PHYSICISTS... but, as far as physics is concerned, math is a tool (albeit an often complicated one). for example, i can utilize group theory to exploit symmetries in a physical system, but when i took group theory as a math class, it was so complicated and so abstract that i did not think that i would be able to apply it... proof after proof insanity! however, knowing the theoretical mumbo-jumbo offers me more flexibility in solving problems presented in a quantum class.

 

[mgiddy911]

I think this goes both ways. There are things I learn in math (doing nearly a dual math and physics degree) that seem to have nearly no application to physics at all. There are times in Physics where it seems we learn math only pertaining to specific problem solving skills, such as classes like Math Methods in Physics. But there are also classes that seem abstract but slowly take form into multiple applications (Linear Algebra comes to mind here)

For instance and correct me if I'm wrong, but not much I have learned in Real Analysis has had any applications in my physics classes, unless you consider a rigorous proof of why derivatives and integrals work to be an application. As was said earlier, sure we use them as tools, and we might think (from a physics aspect) we can prove things, but really we just do derivations. We utilize derivatives and Integrals, but the underlying properties are left for a Real Analysis Class.

 

[Poop-Loops]

i will be soon graduating with a double bs in math and physics, so i have done quite a bit in each. until you get a little ways along in math, it is difficult to truly see the differences in the math used in physics and the math used purely for the sake of using math. let me just say that the math i use for physics is quite complicated, but i never have to prove any of it... yes, physics students may claim to proove, but it is more of a rigorous derivation rather than a rigorous proof. NOT TO DEGRADE THE MATH USED BY PHYSICISTS... but, as far as physics is concerned, math is a tool (albeit an often complicated one). for example, i can utilize group theory to exploit symmetries in a physical system, but when i took group theory as a math class, it was so complicated and so abstract that i did not think that i would be able to apply it... proof after proof insanity! however, knowing the theoretical mumbo-jumbo offers me more flexibility in solving problems presented in a quantum class.

Well the "proofs" we do in physics don't prove the math, but derive the physics. Usually they only involve one or two clever tricks and grinding through the algebra and lo! it all cancels out and you get something elegant in the end.

I've never done a math proof, but my Diff EQ's professor brought one he did in grad school for some method of solving Diff EQ's (don't remember which method, or even if it had to do with solving it, actually) and it was 10 pages long.

Pretty much none of the math that's proved in Boas's Mathematical Methods book has any complicated proofs, either.

Really, a physicist needs more skills in modeling a problem than proving that it can be modeled.

 

[physicspupil]

As a person with an undergrad degree in math who is now studying physics, here's my take:

The math required in physics is different than the math required for math's sake (I'm not too far along in physics - just started the upper division stuff, but that's my take so far). For example, in physics we use a lot of vector calculus, gradients, have to integrate funky looking exponential functions, etc. Believe it or not, we do NOT deal with these issues much while obtaining an undergraduate degree in math (maybe the applied math people do - I don't know), but if you're general or pure math, you don't. And when you do, you deal with simplier functions than what you find in physics (i.e. the type from your math book idealized to work out to yield a simple expression - not a semi-realistic result from nature that might be 'ugly'. In fact, it is my opinion that physics people are actually BETTER at performing and using many mathematical techniques even if they do not have a deep understanding of the theory underlying said techniques. They seem to be able to deal with more cumbersome mathematics with more efficiency.

As somebody else said, you really don't do real proofs in physics like you would in say modern algebra class. You just do rigorous derivations (and if you try to do a real proof), your physics prof. often marks you down (why, I don't know - maybe they don't understand it/not what they were wanting, etc.). Anyhow, taking a course in mathematical logic can actually hamper your ability to do physics "proofs" (cause it gives you an idea of what a proof should be which is NOT what your physics professor is wanting). Physics people really should stop misusing the word!

I actually like pure stuff like abstract algebra (I think its easier than upper division mechanics) - you're playing with ideas and structures; not trying to integrate ugly differential equations that you've forgot how to deal with since its been so long since you took Diff Eqs (not too mention, your diff eqs book didn't have anything that complicated in it anyhow to begin with. By the way, believe it or not, most math majors do not have to take Diff Eqs (its an elective), but if you want to go into physics, you darn well better take it - at least ordinary, but I think I'm going to end up having to take partial too just to do the math)...

 

[Crosson]

Mathematics is the study of our intuitions of space and time.

Physics is what you get when you consider substance, along with space and time.

Engineering is what you get when you add functionality, along with substance and space and time.

 

[Shackleford]

Mathematics is the study of our intuitions of space and time.

Physics is what you get when you consider substance, along with space and time.

Engineering is what you get when you add functionality, along with substance and space and time.

I like that.

 

[muppet]

Mathematics is the study of our intuitions of space and time.

A few thoughts on this:
-Complex analysis (for example) is full of mind-numbingly counterintuitive results (the euler formula for a start- exponentials just don't do that! ). Is intuitions really the right word? (Technically, the study of our intuitions about anything would be the remit of psychology anyway )
-Some areas of maths are fairly obviously grounded in spatial intuition (our notion of what a vector in R^3 "looks like" for example). Time though... I'm curious about what you're thinking of here. You can describe the time evolution of a system by parametrizing functions of t, sure. But the asymmetry of "time's arrow" is something we currently always have to put in 'by hand', as it were.
-How does number theory fit into this picture?

No argument with the rest of it though

 

[Shackleford]

Yeah, I'm not quite sure I agree with the implication that physics is a derivation of mathematics as well.

 

[viet_jon]

Mathematics is the study of our intuitions of space and time.

Physics is what you get when you consider substance, along with space and time.

Engineering is what you get when you add functionality, along with substance and space and time.

to be an Engineer, you don't need to know the math works, rather you just need to know how to apply it?