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  1. K

    Implicit function theorem for several complex variables

    This is the statement, in case you're not familiar with it. Let ## f_j(w,x), \; j=1, \ldots, m ## be analytic functions of ## (w,z) = (w_1, \ldots, w_m,z_1,\ldots,z_n) ## in a neighborhood of ##w^0,z^0## in ##\mathbb{C}^m \times \mathbb{C}^n ## and assume that ##f_j(w^0,z^0)=0, \...
  2. K

    Question about Michelson Morley experiment

    Hi everyone. I'd like you to help me make sense of a question I found about the Michelson Morley experiment. This is it: "Explain how and why the uncollimated light rays form interference fringes? Can you say what would happen if the rays were collimated?" Does this question even make...
  3. K

    Sense of guilt for leaving my girlfriend.

    I have been with this girl for 7 years, and known her for 12. Things hadn't been completely right for a long time but lately, say in the last year, they got worse. We argued frequently because we both have kind of strong personalities. Ultimately though, I decided that some of her personality...
  4. K

    Radioactivity - quantum tunnelling

    Can anyone link to a synthetic and understandable explanation of radioactivity through basic quantum mechanics? It does not need to be a comprehensive explanation at all, examples or partial explanations are fine. Online class notes are welcome but so are books suggestions. Thanks in advance
  5. K

    Normal to graph of a function

    Suppose you have a single variable differentiable function r: R -> R restricted to an interval, like [0,1] for simplicity, if you want. Consider the surface of revolution obtained by rotating the graph of r around the x axis. How do I find the NORMAL VECTOR to the surface at each of its points...
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    Parabolic mean value formula

    A million dollar to anyone who can tell me where that 1/r^(n+1) comes from in the second equality on page 3 of this pdf...
  7. K

    Quantum Harmonic oscillator

    Consider the usual 1D quantum harmonic oscillator with the typical hamiltonian in P and X and with the usual ladder operators defined. i) I have to prove that given a generic wave function \psi , \partial_t < \psi (t) |a| \psi (t)> is proportional to < \psi (t) | a | \psi (t) > and...
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    Time evolution operator - Confusion

    Hi everyone. I am given a somewhat common potential well V(x)=0 for ¦x¦<a and infinite elsewhere. I am told that at t = 0 my particle is in a state represented by the wavefunction \psi(x,0)= A(\sin{(\frac{\pi x}{a})}+ \sqrt{2} \cos{(\frac{3 \pi x}{2 a})}) where A is a constant use for...
  9. K

    Uniform convergence

    Is it true that a cauchy sequence of continuous functions defined on the whole real line converges uniformly to a continuous function? I thought this was only true for functions defined on a compact subset of the real line. Am I wrong?
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    Manifestly covariant

    In my relativity notes, I have several remarks like the following one: "The Lorentz condition on the potentials can be written in manifestly covariant form in this way: \partial_i A^i = 0 , where the A^i are the components of the 4-potential." This made me realize I probably have not...
  11. K

    Can someone state clearly a sufficient condition for a 4-tuple of

    Can someone state clearly a sufficient condition for a 4-tuple of quantities to be a four vector? For instance I saw the naive definition of speed four vector is not but I couldn't really understand when a 4-tuple is or is NOT a four vector.
  12. K

    Spacetime interval

    I know that the spacetime interval is the same in coordinate system moving wrt each other at constant speed. But is it true that the spacetime interval is invariant under rotations? If so can you suggest a proof or post a link to one?
  13. K

    Griffiths Harris

    Hi everyone I'm a math student trying to go through Griffiths Harris Principles of algebraic geometry. I'm especially interested in complex tori and K3 surfaces but I confess I'm having a hard time. Can anyone suggest a reference (online notes or books, preferably a single book) that covers the...
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    Trivial (?) alg. geometry problem

    Trivial (!?) alg. geometry problem 1. Homework Statement Consider Y=Q_1,Q_2,\ldots,Q_r \subset \mathbb{A}^n , a finite set of r different points. What are the generators of the ideal I(Y) 3. The Attempt at a Solution Knowing that I(Q_i)=(X_1-Q_{i,1},\ldots,X_n-Q_{i,n}) and so on...
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    Keep making the same mistakes over and over

    A recurrent problem I have when facing written examinations (I'm a math major) is calculation mistakes. They often have a devastating effect e.g. I just badly failed a quantum mechanics examination because I found the wrong eigenstates for a certain Hamiltonian which led to the rest of the...
  16. K

    Distributional derivatives

    When do derivatives in the sense of distributions and classical derivative coincide? Of course f needs to be differentiable. What else? Any reference?
  17. K

    Domain of solution to Cauchy prob.

    Prove that the solution of the CP y'=-(x+1)y^2+x y(-1)=1 is globally defined on all of \mathbb{R} How would you go about this? I thought about studying the sign of the right member if the equation. But what would I do next?
  18. K

    Cauchy prob.

    y'=\sin (x+y+3) y(0)=-3 I tried substituting x+y+3=u and solving I get \tan (u(x)) - \sec (u(x)) = x but what the heck can I do now?
  19. K

    Easy ODE

    y'= \frac{y}{1+e^x}+e^{-x} It's an easy first order linear inhomogenous eq. I solved it by hand with the formula that one can find anywhere AND with Mathematica, but when I take the derivative to check the solution it comes out wrong and it's freaking me out. Can anyone here post just the...
  20. K

    Dirichlet problem

    1. Homework Statement Let B_R = \{ x \in \mathbb{R}^n: |x| < R \}. Calculate the solution of the following Dirichlet problem: -\Delta v = 1 in B_R u = 0 on \partial B_R Calculate the solution of the problem. 2. Homework Equations 3. The Attempt at a Solution I know that the...
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