Thanks for that clarification. When you say
do you really mean that the process of thought is that 1) when you apply an electric field to a medium bound charges (dipole moments) will be "created" (locally) and 2) you then construct a polarization field that has divergence equal to the density...
Hey all,
I'm studying laser-matter interactions and was wondering: Is there any physical meaning to a non-vanishing polarization field with non-trivial constitutive relation but vanishing divergence? (By non-trivial I mean the constitutive equation does not stipulate that the polarization and...
I agree completely. What puzzles me is rather the way the electrostatic potential can be "acquired" through contact. Example: A bird that first sat on some tree (which has the same potential as the earth) and then flies off to a power line, where he (suddenly?) "acquires" the potential of the...
Thinking in terms of electric circuits and electrostatic potential I understand how an electric current arises as manifestation of a difference in potential. How does this work at a more microscopic level? I.e. how does an electron know what potential it's environment is at?
E.g.: If I...
Attention: This is completely wrong in full generality! By the very definition of a Banach space, this is only true in a Banach space.
This is misleading. "The other direction" simply means you want to prove that the real numbers are a Banach space. Even if you forget about the construction...
Hey all,
I was working a little on parabolic pde, and came across this (comes up in regularity theory). Consider a Hilbert triple V\subset H\subset V^* (continuous embeddings) and a linear operator A(t) from V to V*, where t ranges in some interval [0,T]. Now let w\in H^1(0,T;V^*)\cap L^2(0,T;V)...
Notice that the function f in \sum_{n=0}^{\infty} 2^{n}f(2^{n}) need not be well-defined for all arguments in the real numbers, but only for the natural numbers (including the zero).
What you have is a positive monotone decreasing sequence (which is denoted here by f(n) but could just as well be...
Are you given any special topology on the sets X,Y?
You could start by assuming that f(x) is not a limit point of f(A), i.e. there exists a neighborhood U of f(x) in Y, such that...
You don't even need Lagrange for that. Say you have a cyclic group G of order m, then for any a in G the order of a is at most n, right? Do you know what to do next?
regards...Cliowa
So it should be clear to you that in your case (where n=12) you don't have a group with respect to the usual mod operation. If you consider an additive group {0,...,n-1}, where 0 is the identity, taking mod n works. Notice that the crucial point is that for 2 numbers to be identified mod n can...
Your mod-calculations are a bit odd. When you talk about taking {1,...,11} with multiplication mod 12, you identify all mupltiples of 12 with 1, etc, i.e. you identify 24 and 1, 25 and 2, 26 and 3, 27 and ...
Do you know what the cotangent bundle is? One can view the cotangent bundle as the tangent bundle with the tangent spaces replaced by their dual spaces. So an element of a cotangent space acts on the tangent space, i.e. you feed it a tangent vector and out comes a number. That's precisely what...
Say your function f is in C^k. Let f_n denote the n-th fourier coefficient. Then one can prove that n^k f_n\rightarrow 0 as n\rightarrow \infty.
Is that enough for your problem?
I guess the only resolution would be to have the product of two distributions depending on two different variables, a double integral kind of thing. But physicist manage to suffocate every clue about the real meaning of what they're doing using their fancy integral notation anyway.
It now seems pretty clear to me where our opinions diverge. You accept "inifitesimals" and their manipulation as a mathematically sound concept and I don't. I'll try to explain my point of view in the following, so you can choose to ignore it or comment on it.
As far as I understand, all this...
If you multiply by y, your answer will be the same as long as you stick with your initial domain, which excludes y=0. So surely the two results must agree on the level sets of your first equation. But, that's what I was trying to say, if you multiply by anything you might get a new level set...
I'm sorry to hear that you feel I'm playing games, which clearly was not what I was intending to do. As you observed, there are some issues on which we'll disagree. Let's leave it that way. There is however one thing on which I would like to elaborate:
1st problem: If you take a section of a...
Your eqns aren't texing because the end command is [ / tex] instead of your backslash.
The point is that you define a function y implicitly. What makes you think you're allowed to multiply by y and obtain the same result? The statement you get surely is true (multiplying 0 by anything will...
The analytic thing is just not rigorous math. If you knew where you were going, you could have done it yourself, as a student, but if you didn't, there's no chance you would have ever come up with the ds stuff. My point simply is that one introduces here objects which don't have any rigorous...
My overall impression of your sites is okay, but there are a few things I'd like to point out:
1) You give 2 different definitions of tensors. Wouldn't it be nice to show that they actually amount to the same thing? I feel that's quite important.
2) Concerning the "analytic" approach: I find...