I want to calculate the surface area of a sphere with radius $R$.
As I was calculating the norm of the cross product of partial derivatives of the surface parametrisation, I stumbled to a problem.
I use the spherical coordinates as the parametrisation for the sphere.
Let $x(\phi,\theta) = R\cdot \sin(\phi)\cos(\theta)$
$y(\phi,\theta) = R\cdot \sin(\phi)\sin(\theta)$
$z(\phi,\theta) = R\cdot \cos(\phi)$
$|| r_\phi \times r_\theta || = || R\cdot \sin(\phi) (x,y,z) || = |R\sin(\phi)| \cdot||(x,y,z)||$
$\sin(\phi)$ is a function of $\phi$, so why are we able to treat it as a constant and factor it from the norm?