- #1

Spectre5

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The following is in my textbook (V = vector):

[tex] \nabla \cdot V = \frac {1}{r} \frac {\partial{(rV_{r}})}{\partial{r}} + \frac {1}{r} \frac {\partial{V_{\theta}}}{\partial{\theta}} + \frac {\partial{V_{z}}}{\partial{z}}[/tex]

where:

[tex] \nabla = \hat {r} \frac {\partial}{\partial {r}} + \hat {\theta} \frac {1}{r} \frac {\partial}{\partial {\theta}} + \hat {k} \frac {\partial}{\partial {z}} [/tex]

and

[tex] V = \hat {r}V_{r} + \hat {\theta}V_{\theta} + \hat {k}V_{z} [/tex]

This is, obviously, in cylindrical coordinates.

However, I would expect the result to be as follows:

[tex]\nabla \cdot V = \frac {\partial{V_{r}}}{\partial{r}} + \frac {1}{r} \frac {\partial{V_{\theta}}}{\partial{\theta}} + \frac {\partial{V_{z}}}{\partial{z}}[/tex]

Where did I go wrong? The second two terms I get the same thing, but I am confused on where that first term comes from in the given formula. Thanks in advance.