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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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    Blackbody radiation text

    I'm reading from an introductory text on quantum physics, and came across this sentence: It's the second sentence that I don't understand: how can the energy in the EM field be responsible for the ability of a hollow cavity to absorb heat?
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    The Frost Line of the Solar System

    If I were a judge in a court, I would dismiss this response as being "frivolous and without merit".. (a favorite retort of judges in this country, if you've ever had the distinct pleasure). The point being, you haven't addressed the issue at hand (which is the legal definition of the word...
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    The Frost Line of the Solar System

    Is there evidence of this, or are we just guessing? Again, is there evidence of this anywhere, or is it all just random conjecture? I'm not talking sides one way or another.. but I'm just commenting on what seems to me to be a very interesting double-standard in "science".. It would...
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    The Frost Line of the Solar System

    Soooo... what exactly has stopped Jupiter or Saturn from "wandering inwards" in our Solar System? I thought planets weren't supposed to just "wander around" like that.. :-)
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    The Frost Line of the Solar System

    I haven't been following this line of research terribly closely, but aren't all these planets that astronomers are finding orbiting stars outside our Solar System always (a) Jupiter-sized planets; and (b) always orbiting the parent star in at a radius something like where Mercury is in our Solar...
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    Change of Basis + Geometric, Algebraic Multiplicities

    Making a change of basis in the matrix representation of a linear operator will not change the eigenvalues of that linear operator, but could making such a change of basis affect the geometric multiplicities of those eigenvalues? I'm thinking that the answer is "no", it cannot.. Since if...
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    Jordan canonical form

    Homework Statement Are the operators specified by the matrices: A = \left[\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{array}\right] B = \left[\begin{array}{ccc} 4 & 1 & -1 \\ -6 & -1 & -3 \\ 2 & 1 & 1 \end{array}\right] equivalent? Homework Equations See...
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    Ideals and Linear Spaces

    Is an ideal always a linear space? I'm reading a proof, where the author is essentially saying: (1) since x is in the ideal I, and (2) since y is in the ideal I; then clearly x-y is in the ideal I. In other words, if we have two elements belonging to the same ideal, is their linear...
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    Square matrix with no eigenvectors?

    It's true that in general, the rotation matrix does not have real eigenvalues.. however, for the general 2-dimensional rotation matrix: \left[\begin{array}{cc} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi\end{array}\right] it will, in general have the two (complex) eigenvalues: \lambda_1 =...
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    Square matrix with no eigenvectors?

    hkBattousai, I think HallsOfIvy is correct.. In the example he gave, the matrix has only one distinct eigenvalue (which is 1, w/ algebraic multiplicity of 2), and there is only one eigenvector corresponding to this eigenvalue (so the geometric multiplicity of the eigenvalue is 1). I...
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    Square matrix with no eigenvectors?

    Is there such a thing as a square matrix with no eigenvectors? I'm thinking not ... since even if you have: \left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right] you could just as well say that the eigenvalue(s) are 0 (w/ algebraic multiplicity 2) and the eigenvectors are: u_1 =...
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    Computational Astrophysics

    Which schools have the best programs in computational astrophysics? Judging by their websites, Princeton and Univ. of Chicago seem to have strong programs, but I was wondering if people knew of others as well??
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    Left, Right Inverses

    Suppose we have a linear transformation/matrix A, which has multiple left inverses B1, B2, etc., such that, e,g,: B_1 \cdot A = I Can we conclude from this (i.e., from the fact that A has multiple left inverses) that A has no right inverse? If so, why is this?
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    Continuous Functions, Vector Spaces

    Homework Statement Is the set of all continuous functions (defined on say, the interval (a,b) of the real line) a vector space? Homework Equations None. The Attempt at a Solution I'm inclined to say "yes", since if I have two continuous functions, say, f and g, then their sum f+g...
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    Lattice Gauge Theories

    Are Lattice Gauge Theories still considered an area of active physics research? (i.e., are people still producing PhDs in this subject?) Or has this research area become passe?
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