Intro to Differential Geometry or in-depth PDE Course?

[BillyBones]

Hello,

I am currently a High School Senior who has completed Multivariable Calc (up to stokes theorem), basic Linear Algebra ( up to eigenvalues/vectors) and non-theory based ODE (up to Laplace transforms) at my local University. (All with A's) I am hell bent on taking either one of the courses mentioned above, but am bewildered by the contrasting answers I have seen regarding the prereqs. Am I adequately prepared, or will some other classes like topology be necessary?

Thanks

 

[jedishrfu]

I would think you should be prepared enough to take either course. The best suggestion though would be to check with the profs teaching these courses to see if they are a fit to what you know to date.

Another way to get an answer to your question is to see what courses are required for a given degree. You could also look at the course numbers and the course directory to see what courses occur between the ones you've taken and the ones you want to take. Normally they are in a roughly increasing order of difficulty.

Right now I can only think of Advanced Calculus or Real Analysis or Boundary Value problems as courses taken before these courses.

You could also check the books used in the courses and from the table of contents see if there's any gap in your math knowledge.

Years ago at my college, I took Tensor Analysis which amounted to an extension of Vector Analysis. It seems that you've studied that somewhat with the Divergence theorem and Stoke's theorem sans the tensor notation so that might be a good fit. Are you familiar with coordinate transformations, the Jacobian of transformation and tensor notation? These would come up in Differential Geometry and possibly the PDE course.

Here's a summary/introduction to Differential Geometry:

https://en.wikipedia.org/wiki/Differential_geometry_of_surfaces

and to PDEs:

https://en.wikipedia.org/wiki/Partial_differential_equation

I would also caution you about moving too fast through your math courses. You should keep pace with the teacher and dig deeper into the material so that you have a solid understanding of the material. Things get progressively faster and more focused as you move up the academic chain where 1 year of college is about 3 years of high school and 1 year of grad school is like 3 years of undergrad.

I made this mistake testing out of Calculus and began to hit a brick wall in my Junior year as things got a lot tougher and my math skills weren't at the level of the seniors in the courses I was taking. The end result is poorer grades and shallow understanding. I didn't care at the time as I felt grades weren't that important but in the private sector when you go for jobs these grades can pop up in interviews. And when applying to grad school the grades are one indicator of academic success so you want them to be the best possible.