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  1. M

    Brand new physics animation site

    Hi, I just wanted to let you know that there is a brand new Physics Flash animation web site out there. It contains interactive content that can be used in a classroom setting (used by professors or teachers) or by students themselves. The level of the material presented is an introductory level...
  2. M

    Average age problem

    Yes, the key is in the long run. I'm saying that the "net effect" would be like having that situation I described. You can have a tree older than 73. I'm trying to argue that if you cut 25 trees a year ranging from very young to very old (older than 73), their average age would be around the age...
  3. M

    Average age problem

    I was thinking the following: Because you plant 25 new trees each year, over time, the average age of the tree stock would be equivalent of that having 25 trees of each age ranging from age 0 to age (1850/25) - 1 = 73. Because sometimes you remove a young tree and sometimes a very old tree, over...
  4. M

    Average age problem

    Yes. I think the tricky part to this problem is mathematically describing the removal of the 25 trees -- there's some of probability involved here. A tree that is removed could be one that was planted the year before or it could be 100 years old.
  5. M

    Average age problem

    Hi, I was curious how one would solve this problem: In a town, there are 1850 trees along public roads. Each year, the town has to remove on average 25 trees of random age because of various reasons (natural death, fungus infection, insects, hit by cars, roots damaged by construction, etc.)...
  6. M

    Outgassing, vacuum question

    I'm trying to calculate the outgassing rate for the following problem, but I seem to be missing something (maybe the like the pumping speed or time). The base pressure of a particular vacuum system is 10^-9 torr. A sample with a fingerprint of area ~1 cm^2 is introduced into the system...
  7. M

    Exact differential problem

    I have trouble doing a problem involving exact differentials: Consider a uniform wire of length L and cross-section area A. A force F is applied to the wire. We can write the relationship: dL = (L/YA)dF + (aL)dT where Y is the Young's modulus, a the coefficient of thermal expansion, and...
  8. M

    Initial-value problem

    Ok I see. Thanks.
  9. M

    Initial-value problem

    Oh sorry, did you say that it only needs to satisfy the Lipschitz condition for y=1? How do you know that? My theorem for uniqueness says that f needs to satisfy a Lipschitz condition on D, where D is the convex set {(t,y)}, where 0 <= t <= 1 and -infinity < y < +infinity. Thanks.
  10. M

    Initial-value problem

    Yes, that's what I'm trying to show. For each pair of points (t,y1) and (t,y2) where t is in [0,1], we have |F(t,y1) - F(t,y2)| <= L |y1 - y2| where L is a Lipschitz constant for F. And y1 and y2 can be anything between positive and negative infinity. But it seems that it doesn't satisfy the...
  11. M

    Initial-value problem

    Hello, I have trouble showing that the following initial-value problem has a unique solution. I also need to find this unique solution. y' = e^(t-y), where 0 <= t <= 1, and y(0) = 1. How can I test the Lipschitz condition on this? Thanks in advance.
  12. M

    Dual Tensor and Field Tensor

    Ok, thanks for your help. Since \frac{\partial G^{\mu\nu}}{\partial x^\nu} = 0 produces divB=0 and Faraday's Law, I ended up trying to get them from \frac{\partial F_{\mu\nu}}{\partial x^\lambda} + \frac{\partial F_{\nu\lambda}}{\partial x^\mu} + \frac{\partial F_{\lambda\mu}}{\partial...
  13. M

    Dual Tensor and Field Tensor

    Thanks, but I'm having trouble understanding the notation. I don't really know much about relativity. We are just doing a chapter on special relativity in our E&M class, and all we have learned about field and dual tensors are that F^{\mu\nu} = \left( \begin{array}{cccc} 0 & E_x/c & E_y/c...
  14. M

    Dual Tensor and Field Tensor

    Hi, could someone show me how to express \frac{\partial G^{\mu\nu}}{\partial x^\nu} = 0 which are Maxwell's equations, G is the dual tensor, in terms of the field tensor F: \frac{\partial F_{\mu\nu}}{\partial x^\lambda} + \frac{\partial F_{\nu\lambda}}{\partial x^\mu} + \frac{\partial...
  15. M

    Fourier series problem

    Ok, actually for part a, I'm pretty sure I should do an even extension of the function at x=0 so that it runs from -pi to pi. And I can then determine the B_n's. But for part b, it looks like I need to somehow extend to -2pi to 2pi. I looked in books, and does an odd extension at x=pi so that...
  16. M

    Fourier series problem

    Hi, I need help on the following problem on Fourier series: Let phi(x)=1 for 0<x<pi. Expand 1 = \sum\limits_{n = 0}^\infty B_n cos[(n+ \frac{1}{2})x] a) Find B_n. b) Let -2pi < x < 2pi. For which such x does this series converge? For each such x, what is the sum of the series? c) Apply...
  17. M

    E&M fields of a moving point charge

    Hi, I need help on the following question: Suppose a point charge q is constrained to move along the x-axis. Show that the fields at points on the axis to the RIGHT of the charge are given by \vec{E} = \frac{q}{4\pi\epsilon_0}\frac{1}{r^2}(\frac{c+v}{c-v})\hat{x} and \vec{B} = 0 What...
  18. M

    Taylor series expansion question

    Hi, I have a question about Taylor series: I know that for a function f(x), you can expand it about a point x=a, which is given by: f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + ... but I would like to do it for f(x+a) instead of f(x), and expand it about the very same point...
  19. M

    Expectation value of an anti-Hermitian operator

    Ok thanks Daniel and Seratend.
  20. M

    Expectation value of an anti-Hermitian operator

    Are you referring to this post: https://www.physicsforums.com/showthread.php?t=68937 I have trouble understanding what you wrote. Could you please explain it in detail? Thanks.
  21. M

    Expectation value of an anti-Hermitian operator

    Hi, could anyone tell me how one would show that the expectation value of a anti-Hermitian operator is a pure imaginary number? Thanks.
  22. M

    Magnetic force on coil problem

    I'm stuck on the following problem: A long thin coil of length l, cross-sectional area S, and n turns per unit length carries a current I. It is placed along the axis of a large circular ring of radius a, which is carrying a current I'. If d is the displacement of the center of the coil from...
  23. M

    Dynamical systems proof - nonwandering set

    Thank you very much, xanaduet.
  24. M

    Uniqueness of cubic interpolating polynomial

    Ok that makes sense. So I guess what you are using is the Fundamental Theorem of Linear Algebra -- determinant of the coefficients is not 0 iff the coefficients of the polynomial are unique? Thanks for your help.
  25. M

    Uniqueness of cubic interpolating polynomial

    This is a numerical analysis question, and I am trying to prove that the p(0), p'(0), p(1), p'(1) define a unique cubic polynomial, p. More precisely, given four real numbers, p00, p01, p10, p11, there is one and only one polynomial, p, of degree at most 3 such that p(0) = p00, p'(0) = p01...
  26. M

    Fourth-order eigenvalue problem

    Ok I got it now using the equations you got. \lambda = -n^4 where n = 1, 2, 3, ... This answer looks much more reasonable. And I think the lambda > 0 case shouldn't even be considered since lambda is a real number, and you can't take the fourth root of negative real number. Thanks...
  27. M

    Fourth-order eigenvalue problem

    All right thanks no problem. Could you look at the second part for lambda > 0? I tested the solution and it does not work somehow. :( I'm thinking now that lambda > 0 does not yield eigenvalues?
  28. M

    Fourth-order eigenvalue problem

    I haven't tried the determinant method. Did you get something different? Anyway, for lambda > 0, The roots of the characteristic polynomial were: \sqrt[4]{\lambda}e^{\frac{i\pi}{4}}, \sqrt[4]{\lambda}e^{\frac{3i\pi}{4}}, \sqrt[4]{\lambda}e^{\frac{5i\pi}{4}}...
  29. M

    Fourth-order eigenvalue problem

    Hi saltydog, here is what I have. Sorry Daniel, I didn't have time last week to post everything step by step. Please correct me where I'm making a mistake: Since the boundary conditions are: u(0) = u''(0) = u(\pi) = u''(\pi) = 0 For lambda < 0, my solution is: u(x) =...
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