- #1
Spectre5
- 182
- 0
Hey, I have a question on a derivation
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.
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.