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...
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...
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)...
Say I have two bodies, idealized as points with mass, in Galilean spacetime A^4. When thinking about the 2-body problem (just two bodies with interaction forces in the entire universe), one usually goes from the 3-dim. to the 2-dimensional problem using some special idea. I read the following...
In physics one often uses the following: If the rotation of a vector field A vanishes, one can write A as the gradient of some scalar field, i.e. rot(A)=0 \Rightarrow A=\bigtriangledown \Phi.
Is this true without further restrictions? If yes: Why?
Thanks in advance...Cliowa
I'm looking for a book on Quantum Mechanics on an introductory level (concerning the physics), which is fairly advanced concerning the mathematics (i.e. some book that does not praise as a mathematical revolution that there actually is something called a dual space to some vector space). Do you...
Say I have to vector spaces V,W and a linear transformation \Phi:V\rightarrow W. I know that (given v,p\in V) if I interpret a tangent vector v_p as the initial velocity of the curve \alpha(t)=p+tv I have, relative to a linear coordinates system on V, v_p=x^i(v)\partial_{i(p)}.
The thing I don't...
Let M be a diff. manifold, X a complete vectorfield on M generating the 1-parameter group of diffeomorphisms \phi_t. If I now define the Lie Derivative of a real-valued function f on M by
\mathscr{L}_Xf=\lim_{t\rightarrow 0}\left(\frac{\phi_t^*f-f}{t}\right)=\frac{d}{dt}\phi_{t}^{*}f |_{t=0}...
Let E, F be Banach spaces, and let L(E;F) denote the space of linear, bounded maps between E and F. My goal is to understand better higher order derivatives.
Let's take E=\mathbb{R}^2, F=\mathbb{R}. Consider a function f:U\subset\mathbb{R}^2\rightarrow\mathbb{R}, where U is an open subset of...
Let L(A;B) be the space of linear maps l:A\rightarrow B.
My goal is to derive the Leibniz (Product) Rule using the chain rule. Let f_i:U\subset E\rightarrow F_i, i=1,2 be differentiable maps and let B\in L(F_1,F_2;G). Then the mapping B(f_1,f_2)=B\circ (f_1\times f_2):U\subset E\rightarrow G...
Let's say I'm given a DEQ: (1) y^{(n)}+a_{n-1}\cdot y^{(n-1)}+\ldots + a_{0}\cdot y=0, where y is a real function of the real variable t, for example. I could now rewrite this as a system of DEQ in matrix form (let's not discuss why I would do that): (2) x'=Ax,\quad x=(y,\ldots,y^{(n-1)}). If I...
How could the universe possibly expand??
My understanding concerning black holes is this: If there is enough mass (which means quite alot, in this case) with only very little extension, i.e. lots of mass in a tiny little region of spacetime, spacetime will curve according to this and eventually...
The situation is this: Two space vehicles A and B are doing a race in space. A is in front of B and they are both in orbit around the earth. For simplicity let this orbit be a circle (i.e. neglectable eccentricity). Now, B wants to get past A, that is, B wants to cross the "line" connecting the...
Dear community
I'm trying to get a grip on this integral:
\int \frac{\sqrt{1-x}}{\sqrt{x}-1} dx.
I tried substituting x=\sin^{2}(u), which leaves me (standing) with
\int \frac{\sin(u)\cos^{2}(u)}{\sin(u)-1} du.
But I just can't solve it, no matter which way I try.
I would be thankful for every...
Physicists do it all the time: playing around with those symbols like \frac{dx}{dt}. They just treat them like ordinary variables, which they certainly are not. Let me give you an example: If you want to change an integration variable from, say, x to r and you know that r^{2}=a^{2}+R^{2}-2Rx...
Hello everybody
I'm given a continuous function f (from the real numbers to the real numbers) which I know obeys the following functional equation:
f(x)=f(x^{2})
How can I proof that this function is constant?
I started out like this: Looking at a number x in [0,1[ I said to myself that...
How can I calculate the value of this series as a function of k?
\sum_{n=k}^{\infty} n\cdot \big(\frac{1}{2}\big)^{n}
Plugging in numbers and trying to see a pattern doesn't work for me, I can't see it. I haven't found a way to decompose this sum into several ones where I can see...
This should be a proof of the fact that exp(x)*exp(y)=exp(x+y). Have a look at it:
\begin{align*}
\exp(x)\cdot\exp(y)&=\left(\sum_{k=0}^{\infty}\frac{x^k}{k!}\right)\cdot\left(\sum_{\ell=0}^{\infty}\frac{y^{\ell}}{\ell!}\right)\\...
Hello everybody
I'm very much interested in the thread about "Feynmans Calculus" (having read the books, too). The problem is I don't understand quite some of the stuff, because I don't have the necessary fundamental knowledge.
So I thought to confront you with some lower level questions...