Discovering Math in Physics: From Vector Calculus to Group Theory and Beyond

[patrickd]

A year of college calculus gets you into, maybe, the early 1800's in terms of offering some mathematical insight into physics. The literature attempting to educate us non-scientists on developments thereafter tends to rely on words alone, the authors apparently agreeing with their editors that each equation knocks off about 10% of the potential readership.

From vector calculus to understand Maxwell's equations, to the variational calculus of the Lagrangian formulation (which apparently supplants f=ma once one has passed bachelor's level), to the differential geometry of General Relativity (often represented by pictures of a sphere and a saddle), to group theory and symmetry considerations (an equilateral triangle and a snowflake), the pop-science authors have little to offer but hand-waving.

My question is, are the Forum readers aware of any resources designed to offer some insight into these and similar topics somewhere between the saddle or snowflake level and a graduate math course?

 

[PeterDonis]

are the Forum readers aware of any resources designed to offer some insight into these and similar topics somewhere between the saddle or snowflake level and a graduate math course?

In my experience, physics textbooks do at least a decent job (and often much better than decent) of introducing the math that is needed for the physics they are covering, at a physicist's level of rigor. For example, the two classic GR textbooks that I am most familiar with, MTW and Wald, both cover enough differential geometry for the physics, without (at least IMO) getting bogged down in mathematics.

 

[Drakkith]

I've done a mass cleanup of this thread to get rid of a number of off-topic posts. Please pay attention to the OP's original question. The original question is:

My question is, are the Forum readers aware of any resources designed to offer some insight into these and similar topics somewhere between the saddle or snowflake level and a graduate math course?

 

[patrickd]

... at a physicist's level of rigor.

Peter, thanks for replying. "A physicist's level of rigor" could be a laughable oxymoron (if you are a mathematician), just the right amount (if you are an undergraduate physics major), or a level well above one's head (if you are me.) I'm not a physicist (I'm an ophthalmologist), so a level of rigor somewhat lower than a physicist's is what I am seeking.
Still, I know there's more to GR than a bowling ball on a rubber sheet, I'd really like to have some intuition about what a gauge theory is, and I'd like to know what SU(3) means, and why physicists care. When physicists talk to me (their reader) about QM, they talk about cats, but when they talk to each other, they talk about fields and Hamiltonians. I'm looking for a little deeper insight into some of these things.

 

[PeterDonis]

a level of rigor somewhat lower than a physicist's is what I am seeking

You may be a bit pessimistic about how low a physicist's level of rigor actually is. Some physics texts (Wald's GR textbook comes to mind) actually do take the time to go through explicit proofs of important mathematical results (although the proofs might not completely satisfy a pure mathematician); others are content to just state key results without proof. You might try Sean Carroll's online lecture notes on GR:

http://arxiv.org/abs/gr-qc/9712019

He explains the key mathematical concepts (such as a manifold) without spending much time on rigorous proofs, just focusing on how they are used in physics.

I'd really like to have some intuition about what a gauge theory is, and I'd like to know what SU(3) means, and why physicists care.

Unfortunately I don't know if there is any "in-between" treatment of these things; if you want an answer to "why physicists care" that goes beyond a quick sound bite ("because it turns out that gauge theories can be used to describe lots of physics"), you will need to dig into group theory, Lie groups (of which SU(3) is one), Lagrangians and Hamiltonians, Noether's theorem, etc., and I don't know of any treatment of all these subjects that ties them all together without considerable preliminary work required.