Recent content by psholtz

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    Vector Potentials

    It's not hard to show that the function: g = \frac{1}{2} (c \times r) is a "vector potential" function for the constant vector "c". That is, that: \nabla \times g = c The calculation is straightforward to carry out in Cartesian coordinates, and I won't reproduce it here. However...
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    Basel Problem

    Yes, thanks... This was something along the lines of the intuition I was going by, but didn't quite get it to this point. Thanks..
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    Basel Problem

    Sure, a Fourier series would be straightforward. I'm familiar w/ how Fourier analysis can be used to sum the first series, but it's not immediately clear to me how to proceed from that solution, to the sum for the second series. Could you give me a pointer/hint?
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    Basel Problem

    The Basel Problem is a well known result in analysis which basically states: \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... = \frac{\pi^2}{6} There are various well-known ways to prove this. I was wondering if there is a similar, simple way to calculate the value of the...
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    Eigenvalues of a symmetric operator

    Thanks, micromass... excellent explanation.
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    Eigenvalues of a symmetric operator

    Do you think that's what they were getting at in the Wikipedia article? If we suppose the existence of complex numbers, or allow them at any rate, is it safe to say that a square matrix of size n will always have n eigenvalues (counting multiplicities)?
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    Eigenvalues of a symmetric operator

    Reading more from Wikipedia: To me, it would seem that there must be n roots (counting multiplicities) for the characteristic polynomial for every square matrix of size n. In other words, every square matrix of size n must have n eigenvalues (counting multiplicities, i.e., eigenvalues are...
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    Eigenvalues of a symmetric operator

    That's true, but a 2D rotation matrix still has eigenvalues, they just aren't real eigenvalues. But the eigenvalues still exist. Moreover, the 2D rotation matrix isn't symmetric/Hermitian. It's usually of the form: T = \left(\begin{array}{cc} \cos\phi & \sin\phi \\ -\sin\phi & \cos\phi...
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    Eigenvalues of a symmetric operator

    I'm reading from Wikipedia: I thought linear operators always had eigenvalues, since you could always form a characteristic equation for the corresponding matrix and solve it? Is that not the case? Are there linear operators that don't have eigenvalues?
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    Affine Functions

    Homework Statement I'm trying to show that every affine function f can be expressed as: f(x) = Ax + b where b is a constant vector, and A a linear transformation. Here an "affine" function is one defined as possessing the property: f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)...
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    Matrix Factorization

    The matrix giving the relation between spherical (unit) vectors and cartesian (unit) vectors can be expressed as: \left( \begin{array}{c} \hat{r} \\ \hat{\phi} \\ \hat{\theta} \end{array} \right) = \left( \begin{array}{ccc} \sin\theta \cos\phi & \sin\theta \sin\phi & \cos\theta \\ -\sin\phi &...
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    Blackbody radiation text

    So what exactly is the difference between (a) a hot gold ingot; and (b) the Sun? Why is one a blackbody, but the other isn't? How can they "measure" or "determine" that the Sun absorbs 100% of the radiation incident upon it? What kind of experiments do they do to confirm this?
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    Blackbody radiation text

    OK, but that's the part I don't get.. The Sun isn't black.. Nor is a red hot oven black. Does the use of the term "black" here have nothing to do w/ the actual color of the object?
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    Blackbody radiation text

    So this is something that had always been a bit of a stumbling block for me, but I think I'm starting to grasp it .. at least partially. What we really have going on is two distinct bodies: (a) the "radiator" itself, which might be the walls of an oven (i.e., the actual "solid" object that...
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    Blackbody radiation text

    So "black body" radiation really means "cavity" radiation? That was my next question.. Why is the glowing red heat from an oven called "black" body radiation? :-)
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