Recent content by DeIdeal

I'm honestly not sure what you mean by that? In the standard simple linearly polarised plane wave, both electric and magnetic fields reach their minima (and maxima) simultaneously. EDIT: Are you talking about something like this? Note that the amplitudes of both electric and magnetic fields...

It's just Fourier analysis: By writing ##\mathbf{E}(\mathbf{x},t)=\intop_{\mathbb{R}}\intop_{\mathbb{R}^3}\mathrm{d}^3\mathbf{q}\mathrm{d}\omega\mathbf{E}(\mathbf{q},\omega)e^{i\mathbf{q}\cdot\mathbf{x}-it\omega}## and similarly for the magnetic field, one has, for example, MIII as (with usual...

Well, almost. Remember that 1) contractions are always between upper and lower indices (at least when you're not just working on ##\mathbb{R}^n##) and 2) You shouldn't use the same index as a dummy index and a free index (essentially, same index can appear only twice in an expression, and if it...

Wait, what's your T? By Levi-Civita tensor I mean the antisymmetric tensor density ##\varepsilon_{\mu\nu\sigma\rho}##. On an orientable manifold, it is a "true" tensor when multiplied by a suitable coefficient (##\sqrt{|\det(g)|}##, or ##\sqrt{-g}## in typical notation on Lorentzian manifolds)...

By hand? Not in general, but in that specific case the solutions seem to be logarithms of modified Bessel functions, so I'd instinctively let y(x(t))=exp(x(t)) and try to solve the resulting equation with the Frobenius method (or just try to get the modified Bessel equation out). Haven't tried...

For a practical example, one can solve second-order ODEs (with sufficiently nice properties) by using the Frobenius method. There's no general way of constructing a series solution to an arbitrary DE, the procedure depends on the type of the equation. There are obviously DEs that have no...

You should be fine with starting with all lower indices (after all, that's what you want to end up with), as you can always use the metric to lower and raise the indices of the L–C tensor. And yes, your initial expression should include the most general possible linear combination of these...

What definition of sinh are you using? I'm asking because that's usually taken to be the definition, so unless you're using an alternative one, there are not really any more "whys" involved.

I'm not sure what you're trying to say, but this doesn't really sound correct, unless you mean this in some specific sense. For example, O(n,R) has two connected components, U(n) is connected.

The summation rule applies, in the last expression c is being summed over, but in delta-expression it is not. If you contract the indices in it, you get trace of the identity matrix, which is 3 in three dimensions: ##3=\delta^c_c=g^{ac}g_{ac}\neq\delta^c_d=g^{ac}g_{ad}## The first is a scalar...

You could in principle use other tensors, it's just that you have to use "proper" tensors, and that form is the only one that has the correct (anti)symmetry properties. A "lighter" proof can be found by going to a locally flat coordinate system, and noting that in it, your Riemann tensor should...

No problem! For future reference, other common LaTeX fonts include Bold \mathbf \mathbf{a},\mathbf{A} Roman (not italicized unlike the regular math text) \mathrm \mathrm{a},\mathrm{A} Fraktur \mathfrak \mathfrak{a},\mathfrak{A} Calligraphic \mathcal \mathcal{A} Script \mathscr \mathscr{A}...

\mathbb{C}? (The ? is not a part of it)

It certainly belongs to the set of Real numbers.

More about why ice cream parlors were (apparently) considered a bad thing. (ctrl+f for ice cream).