Let's Learn Physics: Good Vibrations from Wave Equations

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@narfwhals7843 3 жыл бұрын

"The result will get very angry at you" "some sort of momentumy type thing" This is the mathematical rigor I'm here for! This was quite a lot but very interesting to see the results emerge. Sorry I couldn't see it live, 1am is not the greatest time for a 2 hour math class :) You spent a lot of time showing that the basis functions are orthonormal, but is that generally true? Can the function space be curved? Is there a point in trying to apply a metric tensor? (I'm recently trying to apply metric tensors to everything. I may have a problem...)

@zapphysics 3 жыл бұрын

No worries on not seeing it live! Perhaps in the future I will have to start trying a more European-friendly time. The answer to your first question as to whether or not the basis functions have to be orthonormal is certainly no. One example that comes to mind immediately is the case of polynomials: we can expand (suitably well-behaved) functions out into a linear combination of polynomials just like we expanded out functions into sines/cosines and exponentials. Clearly, though, polynomials are not going to be orthonormal in the same way that the functions used here are, unless we choose the polynomials to explicitly satisfy orthonormality conditions (this is what is done for e.g. the Legendre polynomials). I'm not sure that the notion of "curved" can be applied to a space of functions (I don't know that it can't, I just haven't seen it done), but the same idea of a metric tensor can certainly be applied in the sense that one can define a non-trivial inner product between two functions. For example, this is the case of the Laguerre polynomials: these are only orthonormal functions with respect to the inner product \int_0^\infty dx f(x) g(x) exp(-x) (sorry for the terrible Latex notation). In this case, the exp(-x) is serving a similar purpose to the metric tensor we see in finite-dimensional spaces. I'm sure this can be/has been generalized to cases where the different functions are weighted differently as well, e.g. \int_a^b dx f_m(x) f_n(x) g_{mn}(x), but I'm not sure what the utility of this is (again, not to say it doesn't have utility, I just have haven't seen it used before). Hope that answers your questions!

@narfwhals7843 3 жыл бұрын

@@zapphysics thanks. It's really interesting to see how tightly connected calculus and geometry are. I guess it's not called "differential geometry" for nothing. And seeing the connection of algebra and geometry on Euler's identity as well... It's all geometry all the way down. I think this is why geometric approaches to physical theories are so attractive.