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Mar
23
comment Explicit proof of the derivative of a matrix logarithm
Hmm, in that case I'll probably have to ask another question because I'm trying to prove $\delta \det{X} = (\det{X}) \mathrm{Tr}\,(\delta M M^{-1})$. A friend asked me about this and I told him I had proved it in the context of a course on general relativity. But when I went back and looked at that proof, I noticed some of these subtleties that I seem to have brushed over when I originally wrote down the proof.
Mar
23
awarded  Commentator
Mar
23
comment Explicit proof of the derivative of a matrix logarithm
Interesting, would $\text{d}\log{X} = \text{d}X X^{-1}$ hold if $X$ were a diagonal matrix? If not, is there any other particular property that $X$ must have for this to hold? And would I be right to say that the definition in terms of a Taylor series is the fundamental one for the matrix exponential and the matrix logarithm?
Mar
23
revised Explicit proof of the derivative of a matrix logarithm
Qualified the kind of matrix $X$ is in my particular case
Mar
23
comment Explicit proof of the derivative of a matrix logarithm
@JasonZimba Thanks for the references! In my particular case $X(x)$ is a general (square) diagonalizable matrix.
Mar
23
asked Explicit proof of the derivative of a matrix logarithm
Nov
30
answered Relative probability
Nov
22
awarded  Editor
Nov
22
revised Two-layer Perceptron for XOR
Added idea of using -b instead of 1-b for bias
Nov
21
asked Two-layer Perceptron for XOR
Aug
14
comment Find value of a line integral using Stokes' theorem
For future reference: mathematical physics doesn't mean "any mathematics in the context of a physics problem". And I would also suggest migrating this to Math.SE since there is no physics to be explained here and that is what Physics.SE is all about. On Math.SE, they will still ask you to show your own work but the question would be on topic there.
Mar
11
awarded  Supporter
Feb
16
answered Question about interpretation of the divergence of a vector field
Jan
29
comment Cylindrical waves
That's right. On a sidenote: in this case there's no problem applying the boundary conditions because they are zero. However in general, if the BC's are e.g. $u(R,t) = f(t)$, you'd have to fourier transform those as well before applying them. (or inverse-fourier transform your solution and apply the original BC's)
Jan
29
comment Cylindrical waves
To see that this is a Bessel DE, try rescaling your $r$. (i.e. substitute $r' = kr$ where $k = \omega/c$)
Jan
19
awarded  Scholar
Jan
19
accepted Radius of convergence for the exponential function
Jan
19
comment Radius of convergence for the exponential function
Ahh, beautiful :) Thank you for this clear explanation, @Tunococ was right that I was looking for one sequence of $a_k$ for all $R$.
Jan
19
asked Radius of convergence for the exponential function
Dec
8
comment Solving the vector Laplace equation in cylindrical coordinates
@Fabian: I'll take a look at Jackson's book, thanks for the reference. But perhaps I should have said the external magnetic field is along the z-axis and the cylinder lies along the z-axis as well, so it's not really a waveguide problem. The physical context is that of a superconducting nanowire along z in an external magnetic field along z, which I am describing using the Ginzburg-Landau equations. This Laplace equation is derived from the second GL equation with the assumption that the order parameter is zero outside and constant inside the nanowire (and the regime is stationary).