Divergence operator in cylindrical & sherical

[sedaw]

look for some proof for the formula of the divergence operator in cylindrical & spherical coordinate
is there any on the net ?
TNX !

the formula here:
http://hyperphysics.phy-astr.gsu.edu/hbase/diverg.html

 

[Raskolnikov]

Try http://mathworld.wolfram.com/CylindricalCoordinates.html and http://mathworld.wolfram.com/SphericalCoordinates.html (just Ctr-F "divergence").

 

[gomunkul51]

The "Del" operator in any orthogonal curvilinear coordinates is:

$$\widetilde{\bigtriangledown} =\left ( \frac{1}{h_{1}}\frac{\partial }{\partial u_{1}}, \frac{1}{h_{2}}\frac{\partial }{\partial u_{2}},\frac{1}{h_{3}}\frac{\partial }{\partial u_{3}} \right )$$

where:

$$h_{1},h_{2},h_{3}$$

are the "scaling factors"

and

$$u_{1},u_{2},u_{3}$$

are the parametrization variables.Example in Spherical Coordinates:

$$h_{r}=\left \| \frac{\partial \vec{r}}{\partial r} \right \|=\left \| \frac{1}{\bigtriangledown r} \right \|= \left \| \frac{1}{\left ( \frac{\partial r}{\partial x}, \frac{\partial r}{\partial y},\frac{\partial r}{\partial z} \right )} \right \|$$
*same idea for Theta and Phi...

$$\widetilde{\bigtriangledown} =\left ( \frac{1}{1}\frac{\partial }{\partial r}, \frac{1}{r}\frac{\partial }{\partial \theta },\frac{1}{r sin(\theta) }\frac{\partial }{\partial \varphi } \right )$$

makes sense? :)

 

[sedaw]

thank you !

 

[vela]

Wikipedia also usually a good resource to find these kinds of formulas collected in one place.