# Tagged Questions

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### Integral over compact boundary is finite in the context of potential function

Consider some bounded domain $\Omega\subset \mathbb{R}^n$, s.t. the boundary $\partial\Omega$ is a smooth submanifold of dimension $(n-1)$ and fix some point $x_0\in \partial\Omega$. Now I want to ...
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### Newtonian potential is harmonic

I got this problem in page 321 from Advaced Calculus, written by Friedman: "Let $\sigma(x,y,z)$ be a continuous function, and let $S$ be a continuously differentiable surface. The Newtonian potential ...
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### Laplace transform $y''''+37y''+36y=g(t)$

Hey this problem is making me insane so have at it and let me know what I keep screwing up. Express the solution of the initial value problem in terms of a convolution integral: ...
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### Laplace's equation, integral, tends to steady state?

If $v(x,y)$ solves Laplace's equation $v_{xx} + v_{yy} = 0$ on a bounded domain $S$, and $u(x,y,t)$ solves $u_t = u_{xx} + u_{yy}$ on $S$, with $u=v$ on $\partial S$ for all $t$, one can show that ...
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I'm looking at the following problem. Prove that if $h$ is harmonic on an open neighborhood of the disc $B(w,\rho)$, then for $0 \leq r < \rho, 0 \leq t < 2\pi$, $$h ... 0answers 107 views ### Green first identity and harmonic function I proved the first Green identity$$\int_{\partial\Omega}f\cdot(D_ng)d\partial\Omega=\int_\Omega \bigtriangledown f \cdot \bigtriangledown g + f\bigtriangledown ^2g d\Omega and now I need to prove ...
My question refers to this answer. I was hoping someone could explain in more detail the following reasoning. It remains to observe that $\Delta v$ is the distribution composed of the ...
I have shown that the following function is harmonic and am attempting to find it's harmonic conjugate: $u=e^{-2xy}\sin(x^2-y^2)$ I know that to find the harmonic conjugate I need to use the ...