This problem has me stumped:(adsbygoogle = window.adsbygoogle || []).push({});

If [tex]r = (x^2 + y^2)^{1/2}[/tex], show that

[tex] div \left( \frac{h(r)}{r^2}(x \vec{i} + y \vec{j}) \right) = \frac{h'(r)}{r}[/tex]

My trouble is with mixing the polar coordinates with the position vector. If I write the above as

[tex] div \left( \frac{h((x^2 + y^2)^{1/2})}{(x^2 + y^2)}(x \vec{i} + y \vec{j}) \right) = \frac{h'((x^2 + y^2)^{1/2})}{(x^2 + y^2)^{1/2}}[/tex]

I can try to get from the left side to the right side by computing the partial derivatives, but when I started this it involved so much messy differentiation (double chain rule, quotient rule) that it just seemed like it wasn't the right approach, so I didn't even try it.

Buteven if I did, what would h' be if h is a function of two variables? I've never seent the notation f '(x,y), just directional derivatives and partials...

What I'm saying is, even if I did do all the messy differentiation and kept all my stuff in order, I don't see how I would be able to write an h'(r) in the end.

Where am I going off track?

Thanks.

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# Divergence Problem

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