Divergence operator in cylindrical & sherical

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  • #3
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The "Del" operator in any orthogonal curvilinear coordinates is:

[tex]
\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 )
[/tex]

where:

[tex]
h_{1},h_{2},h_{3}
[/tex]

are the "scaling factors"

and

[tex]
u_{1},u_{2},u_{3}
[/tex]

are the parametrization variables.


Example in Spherical Coordinates:

[tex]
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 \|
[/tex]
*same idea for Theta and Phi...

[tex]
\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 )
[/tex]

makes sense? :)
 
  • #4
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thank you !
 
  • #5
vela
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Wikipedia also usually a good resource to find these kinds of formulas collected in one place.
 

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