Another Useful Lemma

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

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

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

or

$$f(\theta,\phi_\pm) = 0,$$ (12.227)

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

It is easily demonstrated that

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