Differential Geometry in physics

[nlsherrill]

My school offers this course at the senior undergraduate/graduate level, and its only offered the semester before General Relativity is offered. Would taking this course really "help out" that much with the mathematics of GR? I am trying to select a few math courses to take that could possibly be applicable to the core physics courses. Is differential geometry applicable in other core areas of physics besides GR?

Also worth noting, there is a senior level course that is basically mathematical physics and its description of the course on the syllabus is:

"It is the purpose of this course to provide a critical mathematical analysis of the role of inertial frames in physics. It is our intent to understand how the inertial frames of Newton failed to provide a correct basis of electromagnetism and how this subsequently led to the development of special relativity. We will develop those concepts from special relativity and electromagnetism needed to meet this objective. A second objective of the course is to develop basic properties of Cartan’s exterior calculus and to show how this calculus and other concepts from modern mathematics can impact more traditional approaches to physics."

I basically have the option of taking this course, or the DiffGeo course. They both have the same prerequisites, Linear Algebra and ODE's, so I am assuming the mathematical rigor will be comparable in each course. Any suggestions?

 

[oyolasigmaz]

Would taking this course really "help out" that much with the mathematics of GR?

A professor told me it took him three weeks to learn about general relativity after learning differential geometry, and I don't think he is talking about some superficial learning. I guess it helps quite a lot. I am yet to learn though.

 

[negru]

I'd go with the diff geom course. It will be useful not only to GR, but also classical mechanics, gauge theories or string theory if you're into any of that.

 

[nlsherrill]

If it makes any difference in you guy's opinions, by the time I will have the option to take these courses, I will have already taken Modern Physics which supposedly covers basic QM and SR, and will be enrolled in the upper level E&M class. From the description of the mathematical physics course, it seems like it would possibly be just a rehash(or reinforcement?) of the mathematical techniques used in SR and E&M(possibly more advanced math?)

 

[xepma]

Any decent course on GR will spend 3-5 weeks on differential geometry. The rest of the course can also be called "applied differential geometry". A mathematics course on differential geometry will help a lot.

The mathematical physics course mentions exterior calculus, which is a sub-field of differential geometry. You will easily pick this sort of stuff up later on if you follow the math course first. I *think* the mathematical course will try to formulate E&M in the so-called covariant way. This is different from what you will learn in a regular E&M class. It's great stuff -- but it's again just an example of how differential geometry is beneficial to the field of physics.

If you want to go into theoretical physics, you will want as much exposure to math as possible. My blunt advice would be: go for the diff geo course.