Maximizing Mathematical Understanding for Advanced Physics Studies

[Mépris]

I've noticed that in quite a few universities, physics courses are to be taken again in junior/senior year and some of these are also taken again (classical mechanics, if I'm not mistaken, is one of them) in grad school! It appears to me that the main difference between the variants of the same subject is mathematical complexity.

With that in mind, would it be a good idea if one were to learn all the math first (i.e, in a more rigorous approach, the same way a math major would) and then proceed to study physics, directly with the advanced books? I understand this way would probably take much longer but I'd rather do that and understand what I'm doing with the math, than try pick up the math *while* learning some more advanced physics...

 

[Vanadium 50]

It appears to me that the main difference between the variants of the same subject is mathematical complexity.

And you'd be wrong.

Sure, that's part of the difference, but the bigger piece is that the background is different. When teaching upper division physics, you can only assume that the class has had introductory physics. When teaching 1st year grad students, you can assume they have had upper division physics. When teaching advanced grad students...well, you get the idea.

 

[Dr Transport]

... With that in mind, would it be a good idea if one were to learn all the math first (i.e, in a more rigorous approach, the same way a math major would) and then proceed to study physics, directly with the advanced books? ...

Having taken pure math courses like a math major would, theorem, proof, theorem proof, lemma, proof tetc... I found them to be of little use in my advanced physics courses, the most useful were the applied math courses. The only people that I know that benefited from your approach were my friends who went directly into Mathematical Physics.

 

[Mépris]

And you'd be wrong.

Sure, that's part of the difference, but the bigger piece is that the background is different. When teaching upper division physics, you can only assume that the class has had introductory physics. When teaching 1st year grad students, you can assume they have had upper division physics. When teaching advanced grad students...well, you get the idea.

Cool! So, it's as it says on the box; i.e, mainly a proper continuation of the courses? Would starting to study the required math during the longer breaks (I'm guessing that's Summer break in the US?) be a better idea, then?

Having taken pure math courses like a math major would, theorem, proof, theorem proof, lemma, proof tetc... I found them to be of little use in my advanced physics courses, the most useful were the applied math courses. The only people that I know that benefited from your approach were my friends who went directly into Mathematical Physics.

I wiki'd Mathematical Physics - I don't think I'll be able to appreciate this until I study higher math/physics. Anyway, is this the kind of math that came in handy in the future? It's something an e-acquaintance of mine posted on Google+, where he said he wished somebody had showed him this book as an undergrad...

 

[MathematicalPhysicist]

Regarding the book, I don't think as an undergraduate he would start to grasp it.

All the preliminaries in the book are covered at the end of Bsc/start of grad math, unless he is a child prodigy, which I guess he is.

 

[Dr Transport]

I wiki'd Mathematical Physics - I don't think I'll be able to appreciate this until I study higher math/physics. Anyway, is this the kind of math that came in handy in the future? It's something an e-acquaintance of mine posted on Google+, where he said he wished somebody had showed him this book as an undergrad...

if you want to study String Theory, it might be of some use, but as a practising theoretician in solid state, optical materials etc, I would find no use for it. Again, only my friends who went into pure mathematical physics would use it.

 

[ahsanxr]

I've noticed that in quite a few universities, physics courses are to be taken again in junior/senior year and some of these are also taken again (classical mechanics, if I'm not mistaken, is one of them) in grad school! It appears to me that the main difference between the variants of the same subject is mathematical complexity.

To give you an idea, here's what we covered in my upper-division mechanics courses:

Rushed through the ideas in introductory mechanics using more advanced math and introducing new material along the way for about the first 4-5 weeks of the course [material such as projectile motion with air resistance, energy using vector calculus, all kinds of oscillators(damped, driven etc)]. Then for the rest of the course we covered Calculus of Variations, Lagrangian Mechanics, Two-body/Central Force Problems, Non-inertial Reference Frames, Rigid Body Rotations, Coupled Oscillators. So only the first few weeks were what you described (repetition with more math), the rest was stuff I had never seen before.

With that in mind, would it be a good idea if one were to learn all the math first (i.e, in a more rigorous approach, the same way a math major would) and then proceed to study physics, directly with the advanced books? I understand this way would probably take much longer but I'd rather do that and understand what I'm doing with the math, than try pick up the math *while* learning some more advanced physics...

Advanced books assume you've already taken the introductory versions (phrases like "as you saw in your introductory physics class..." are used very often), so I don't think going directly to the advanced books would be a good idea. I would have at least some kind of familiarity, if not mastery of the introductory counterparts.