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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 \vec{r}(t) = x(t)\hat{x} + y(t)\hat{y} + z(t)\hat{z}, a\leq t \leq b, \nabla ^2 V(\vec{r}(t))=0.
Is it true that the extreme values of V along that path are located at the end? (I.e. at V(\vec{r}(a)) and V(\vec{r}(b)))?
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 \vec{r}(t) = x(t)\hat{x} + y(t)\hat{y} + z(t)\hat{z}, a\leq t \leq b, \nabla ^2 V(\vec{r}(t))=0.
Is it true that the extreme values of V along that path are located at the end? (I.e. at V(\vec{r}(a)) and V(\vec{r}(b)))?
