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    Solving a vector equation which seems to be indeterminate.

    I have a vector equation: \vec{A} \times \vec{B} = \vec{C} . \vec{A} and \vec{C} are known, and \vec{B} must be determined. However, upon trying to use Cramer's rule to solve the system of three equations, I find that the determinant we need is zero. I know now that I need to choose a...
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    Vector calculus identity

    Can someone help me prove the identity \ u \times (\nabla \times u) = \nabla(u^2 /2) - (u.\nabla)u without having to write it out in components?
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    If the integral is zero, when is the integrand also zero?

    If \int_{-\infty}^{\infty} f(k) e^{i\mathbf{k}.\mathbf{r}} d\mathbf{k} =0 then is \ f(k)=0 ? Is it correct to say that this is an expansion in an orthonormal basis, \ e^{ik.r} , and so linear independence demands that f(k) be zero for all k?
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    Coupled differential equations

    I have the equations \frac{l}{u^{2}} \frac{du}{dx}=constant and \frac{1}{u} \frac{dl}{dx}=constant. By "eyeball", I can say the solution is l \propto x^{n} \ and \ u \propto x^{n-1}. I can't see how I could arrive at these solutions 'properly', if you know what I mean
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    2-D Poisson's equation - Green's function

    In the x-y plane, we have the equation \nabla^{2} \Psi = - 4\pi \delta(x- x_{0}) \delta (y- y_{0}) with \Psi = 0 at the rectangular boundaries, of size L. A paper I'm looking at says that for R^{2} = (x-x_{0})^{2} + (y-y_{0})^{2} << L^{2} , that is, for points...
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    Commutation of differentiation and averaging operations

    I've been studying Turbulence, and there's a lot of averaging of differential equations involved. The books I've seen remark offhandedly that differentiation and averaging commute for eg. < \frac{df}{dt} > = \frac{d<f>}{dt} Here < > is temporal averaging. If...