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Is the solution to Laplace's equation harmonic over a path in space

  1. May 2, 2005 #1


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    I was hesistant wheter to post this in the physics of math section but it's much of math problem I think.

    Suppose I have a function V(x,y,z) which obeys Laplace's equation over some path in space. That is to say, for some path parametrized by [itex]\vec{r}(t) = x(t)\hat{x} + y(t)\hat{y} + z(t)\hat{z}[/itex], [itex]a\leq t \leq b[/itex], [itex]\nabla ^2 V(\vec{r}(t))=0[/itex].

    Is it true that the extreme values of V along that path are located at the end? (I.e. at [itex]V(\vec{r}(a))[/itex] and [itex]V(\vec{r}(b))[/itex])?
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  3. May 2, 2005 #2


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    I dunno too much math,but i'll say that your problem is 1D,so that V should satisfy some ODE...

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