# Recent content by psholtz

1. ### Vector Potentials

It's not hard to show that the function: g = \frac{1}{2} (c \times r) is a "vector potential" function for the constant vector "c". That is, that: \nabla \times g = c The calculation is straightforward to carry out in Cartesian coordinates, and I won't reproduce it here. However...
2. ### Basel Problem

Yes, thanks... This was something along the lines of the intuition I was going by, but didn't quite get it to this point. Thanks..
3. ### Basel Problem

Sure, a Fourier series would be straightforward. I'm familiar w/ how Fourier analysis can be used to sum the first series, but it's not immediately clear to me how to proceed from that solution, to the sum for the second series. Could you give me a pointer/hint?
4. ### Basel Problem

The Basel Problem is a well known result in analysis which basically states: \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... = \frac{\pi^2}{6} There are various well-known ways to prove this. I was wondering if there is a similar, simple way to calculate the value of the...
5. ### Eigenvalues of a symmetric operator

Thanks, micromass... excellent explanation.
6. ### Eigenvalues of a symmetric operator

Do you think that's what they were getting at in the Wikipedia article? If we suppose the existence of complex numbers, or allow them at any rate, is it safe to say that a square matrix of size n will always have n eigenvalues (counting multiplicities)?
7. ### Eigenvalues of a symmetric operator

Reading more from Wikipedia: To me, it would seem that there must be n roots (counting multiplicities) for the characteristic polynomial for every square matrix of size n. In other words, every square matrix of size n must have n eigenvalues (counting multiplicities, i.e., eigenvalues are...
8. ### Eigenvalues of a symmetric operator

That's true, but a 2D rotation matrix still has eigenvalues, they just aren't real eigenvalues. But the eigenvalues still exist. Moreover, the 2D rotation matrix isn't symmetric/Hermitian. It's usually of the form: T = \left(\begin{array}{cc} \cos\phi & \sin\phi \\ -\sin\phi & \cos\phi...
9. ### Eigenvalues of a symmetric operator

I'm reading from Wikipedia: I thought linear operators always had eigenvalues, since you could always form a characteristic equation for the corresponding matrix and solve it? Is that not the case? Are there linear operators that don't have eigenvalues?
10. ### Affine Functions

Homework Statement I'm trying to show that every affine function f can be expressed as: f(x) = Ax + b where b is a constant vector, and A a linear transformation. Here an "affine" function is one defined as possessing the property: f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)...
11. ### Matrix Factorization

The matrix giving the relation between spherical (unit) vectors and cartesian (unit) vectors can be expressed as: \left( \begin{array}{c} \hat{r} \\ \hat{\phi} \\ \hat{\theta} \end{array} \right) = \left( \begin{array}{ccc} \sin\theta \cos\phi & \sin\theta \sin\phi & \cos\theta \\ -\sin\phi &...