Under the polar coordinate, the unit sphere is $$ S^2=\{(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)\in \mathbb R^3:\theta\in[0,\pi],\varphi\in[0,2\pi] \} $$ the induced metric is $$ g=\begin{pmatrix} 1 &0 \\ 0 &\sin^2\theta \end{pmatrix} $$ Now, consider a part of $S^2$ $$ S=\{(\cos\varphi,\sin\varphi\cos\delta,\sin\varphi\sin\delta)\in \mathbb R^3:\varphi\in[0,\pi], \delta\in[0,K]\} $$ And consider a non-induced metric, for example, $$ \tilde g =\begin{pmatrix} f(\theta,\varphi) &0 \\ 0 & h(\theta,\varphi) \end{pmatrix} $$ Then, how to calculate the area of $S$ under the metric $\tilde g$ ?
Rough drawing of $S$.
The $S^2$
