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I'm scheduling out the courses I will be taking at my university and I have the option to take either of these two courses. I am a physics major, specializing in cosmology. I think I will try to learn both of these courses as I have a very light summer. However I can only take one of these courses with a professor and lecture environment. Which leads to my question:

Which course is easier to learn or recommended to learn on your own and which one needs lecture supplement?

Probability theory looks like this: Elementary probability theory; modes of convergence; martingales, Gaussian, Wiener, and diffusion processes; Brownian motion; applications.

PDE looks like this: Introduction to second-order linear partial differential equations–heat, wave and Laplace equations, separation of variables in PDEs, Strum-Liouville eigenvalue problems, Fourier series analysis and Green’s functions, Laplace and Fourier transform methods.

Depending on which seems easier to self-teach, could you also point me in the right direction of how to get started?

Thank you for your time and consideration!

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    $\begingroup$ If they're of equal interest to you, I would choose based on which instructor's teaching style suits you better. $\endgroup$
    – Ben
    Oct 13, 2016 at 0:32

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It's hard to make a remote diagnosis. The mathematical theory behind partial differential equations is a lot of abstract functional analysis, which might be tough for you as a physicist. For example, I suppose you'll deal with compact operators to understand Sturm-Liouville problems. However, as PDEs occur a lot in physics, solving exercises with Fourier analysis/Green's functions might be something you're already a bit acquainted with.

Probability Theory is sometimes approached from a very abstract setting as well, in particular there are deep connections to Measure Theory, which is something that's probably hard to learn on its own. Furthermore, abstract concepts like that of a martingale might be difficult to grasp on your own.

It could go both ways here: If your course on PDEs is somewhat more applied / focused on computations than on theory, it might be easier for you to learn it on its own. If the course on Probability Theory doesn't rely heavily on Measure Theory, then I'd say it might be manageable for you too.

It all depends on your previous knowledge: If you're at ease with functional analysis, I'd take the probability course, if you're at ease with measures, I'd take the PDE course. In doubt, I'd recommend you take the PDE course, because the theory behind existence/uniqueness proofs is astoundingly abstract and highly nontrivial.

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