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There are two points on the same sphere with coordinates ${R, \theta_1, \phi_1}$ and ${R, \theta_2, \phi_2}$. Also I have the operator $\displaystyle { \nabla _{{\Omega _1}}^2 + \nabla _{{\Omega _2}}^2}$, where

$ \nabla _{{\Omega _i}}^2 = \frac{1}{\sin \theta_i} \frac{\partial} {\partial \theta_i} \left( \sin \theta_i \frac{\partial}{\partial \theta_i} \right) + \frac{1}{\sin ^ 2 \theta_i} \frac{\partial ^ 2} {\partial \varphi^2 _i}$

(the angular part of delta operator) and I know that:

$ \cos \vartheta = \frac{(\vec {r_1} \cdot \vec {r_2})}{R^2} = \cos \theta_1 \cos \theta_2 + \sin \theta_1 \sin \theta_2 \cos (\varphi_1 - \varphi_2).$

I need to prove that

$ { \nabla _{{\Omega _1}}^2 + \nabla _{{\Omega _2}}^2} = 2\left( {\frac{{{d^2}}}{{d{\vartheta ^2}}} + \cot\vartheta \frac{d}{{d\vartheta }}} \right).$

Please help me to do this.

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    @anon yes, sure! and the right side operator acts on $f(\vartheta)$2012-02-14

0 Answers 0