Recent content by summerwind

  1. S

    Magnetostatics problem: find B and H

    Homework Statement A long wire carries a current I and is centered in a long hollow cylinder of inner radius a and outer radius b. The cylinder is made of a linear material with permeability \mu. Find \mathbf{B} and \mathbf{H} everywhere. Homework Equations The Attempt at a...
  2. S

    Prove two non-degenerate inner product spaces (Rn and R(p, n-p)) are isomorphic

    I think you have the right idea for showing that 0 is not an eigenvalue of A, though you're not using the same names for the matrices as I used. The idea is that an invertible matrix remains invertible when you rotate it to a different basis because of the property of determinants that you...
  3. S

    Prove two non-degenerate inner product spaces (Rn and R(p, n-p)) are isomorphic

    I tried to edit the Latex, but it doesn't seem to have changed. I should have written, 'There's some rotation U s.t. U\ A'\ U^T\ is diagonal. Set B\ =\ U\ A'\ U^T\ .'
  4. S

    Prove two non-degenerate inner product spaces (Rn and R(p, n-p)) are isomorphic

    You're right - we can't assume that the eigenvalues of A are {x, -x}. I thought it might be possible to prove this, but I realize now that it's not true in general. We do know, however, that 0 is not an eigenvalue of A (by non-degeneracy). So the problem is reduced to the following...
  5. S

    Prove two non-degenerate inner product spaces (Rn and R(p, n-p)) are isomorphic

    I don't know for sure that this works, but here's an idea. We know that A can be diagonalized. Suppose that the set of eigenvalues of A is {x, -x} for some positive real number x. Then the P we need is just some orthogonal matrix (ie, an improper rotation, det P = +1 or -1) multiplied by a...
  6. S

    Prove two non-degenerate inner product spaces (Rn and R(p, n-p)) are isomorphic

    What are the matrices A and A'? I can help you if they are what I think they are.
  7. S

    Question on initial value problem

    I'm happy to help out if you correct a few errors first. In (1): you have written 'A = 1.' But in fact your final answer has A = 0. Are you using 'A' to mean different things? In (2): we know that y(0) = 1, so it can't be the case that 1 + A = 0. Finally, your answer y(x) = 1 + sin (x) does...
  8. S

    Rewrite polar equation as Cartesian (trig identities)

    I'm not sure I understand. Are you trying to write \theta\ and a in terms of x and y? If so, take a look at x/y or y/x.
  9. S

    Why Does the Modulus Only Take Positive Values?

    The modulus of a complex number z is defined to be the positive square root of z z'. That's it - it's just a definition. The motivation for this definition is that the modulus is supposed to be the length of z if you think of z as a vector in the x-y plane. In the case of z = 4 + 3i, the...