How do I show a harmonic function has partial derivatives of all orders? All the proofs suggested earlier somehow assume that the partials exist without actually proving it.
@AkivaWeinberger Ok, that is correct. But your proof only shows that u and v have first and second order partials which is obvious by definition of harmonic functions.
For n greater than 3, you have somehow assumed that the partial exists without actually proving it.
@AkivaWeinberger You said "To get $u_{xyx}$ differentiate $f$ horizontally then vertically then horizontally" without actually showing $u_{xyx}$ exists.
Does every automorphism of the linear algebraic $\mathbb{R}$-group $\mathbf{PGL_2}$ come from an $\mathbb{R}$-algebra automorphism of $M_2(\mathbb{R})$?
I'm unsure what to do with this question. I generally object to answering questions I think shouldn't have been posed. But there are also interesting things to say (that are only vaguely related).
asymptotically what happens is the solutions converge to one of the two minima locally uniformaly and the interface is a surface moving by a curvature flow
gotta throw in Planck's constant in order to talk about asymptotics eventually
If you just had a single well $V\propto x^2$, then the eigenvalues would be of the form $\lambda_n = \hbar(a n+b)$ where $a,b$ are constants that I'd have to work out more carefully
and the eigenfunctions would be stuff like Gaussians * polynomials
for instance, if you contract both sides with $A_j,B_k,C_l,D_m,$ you get $$(\mathbf{A}\times\mathbf{B})\cdot(\mathbf{C}\times \mathbf{D})=(\mathbf{A}\cdot \mathbf{C})(\mathbf{B}\cdot\mathbf{D})-(\mathbf{A}\cdot \mathbf{D})(\mathbf{B}\cdot\mathbf{C})$$
> So what’s missing? Full formality demands that we justify that the first de Rham cohomology of our space vanish. Now, I’m not suggesting that we make physics undergrads learn about homology — it might not be a terrible idea, though — but we can satisfy this in the context of a course just by admitting that we are...
I mean, in physics you'll typically go from there to writing the potential as $$V(\vec{r})-V(\vec{a})=-\int_{\vec{a}}^{\vec{r}} \vec{E}\cdot d\vec{l}$$
in which case the point is largely that the RHS only makes sense so long as that integral is path-independent.
And that, in turn, means that you require $\oint \vec{E}\cdot d\vec{l}=0$ for any closed path
The real point, imo, is that an intro text takes for granted that $\nabla\vec{E}=0$ is valid everywhere in space, i.e. that that's true by definition in electrostatics
and once you assume that, then your domain is just R^3 and therefore contractible
Suppose we are given that every real harmonic function is locally the real part of a analytic function. Then how do we deduce that every harmonic function has continuous partial derivatives of all orders?
@MikeMiller I don't know why the second should follow from the first. But I think it should involve that if the function whose real part is our harmonic, then it has continuous derivatives of all orders.
I learned about the Hairy ball Theorem today. I understand the basic explanation, but I didn't really get the math part. I need to learn what a vector field is.
If $(a_n)$ is a sequence of positive numbers such that $\lim_{n \to \infty} \frac{a_{n+1}}{a_n}=0$, what can I say about $\lim_{n \to \infty} a_n$? Is it $0$?
