The surface temperature of the earth is expressed by spherical coordinates:
$\begin{split} T(\phi,\theta)=-30+60\sin\phi \end{split}$
I want to calculate the temperature at the poles (north and south) and the equator and the average temperature.
Already done:
$$\overline T = \frac{1}{A}\iint_{S(R)}T\,dS $$
$x = r\sin \theta\cos\phi$
$y = r\sin\theta\sin\phi$
$0\le\phi \le 2\pi$
$0\le\theta\le\pi$
$$\frac{\delta(x,y)}{\delta(\phi,\theta)}= \begin{vmatrix} r\cos\theta\cos\phi & r\cos\theta\sin\phi \\ -r\sin\theta\sin\phi & r\sin\theta\cos\phi \\ \end{vmatrix}$$ and that is $r^2\sin\theta\cos\phi$ which is dS(?)
I need to calculate the dS and after that I can calculate the area and after that the temperatures, but how?
How do I proceed from here?
First your formula, wich says temperature changes going east or west, but not going north or south seams very wrong, check it!
To your calculation, the spere lives in 3d, but you forgot the third coordinate $$z=r*cos(\theta)$$ so your dS is wrong the surface of a sphere you should know as $$A=4*\pi*r^2, dS=r^2sin(\theta)*d\phi*d\theta$$