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Another Useful Lemma

Suppose that $ g(\theta,\phi)$ and $ f(\theta,\phi)$ are two well-behaved functions that satisfy either

$\displaystyle \frac{1}{\sin\theta}\,\frac{\partial f(\theta,\phi_\pm)}{\partial \phi}$ $\displaystyle = 0,$ (12.225)
$\displaystyle \frac{1}{\sin\theta}\,\frac{\partial g(\theta,\phi_\pm)}{\partial \phi}$ $\displaystyle = 0,$ (12.226)

or

$\displaystyle f(\theta,\phi_\pm)$ $\displaystyle = 0,$ (12.227)
$\displaystyle g(\theta,\phi_\pm)$ $\displaystyle =0.$ (12.228)

It is easily demonstrated that

$\displaystyle \int_{\mit\Omega}g\,D\,f\,d{\mit\Omega} =-\int_{\mit\Omega}\left(...
...f}{\partial\phi}\right)d{\mit\Omega}= \int_{\mit\Omega} f\,D\,g\,d{\mit\Omega}.$ (12.229)



Richard Fitzpatrick 2016-01-22