derivation of surface area measure on sphere

The sphere of radius $r$ can be described parametrically by spherical coordinates (what else ;) ) as follows:

$$x=r\sin u\sin v$$

$$y=r\sin u\cos v$$

$$z=r\cos u$$

Then, using trigonometric identities to simplify expressions we find that

and hence, using more trigonometric identities, we find that

$$\sqrt{{\left(\frac{\partial (x,y)}{\partial (u,v)}\right)}^2+{\left(\frac{\partial (y,z)}{\partial (u,v)}\right)}^2+{\left(\frac{\partial (z,x)}{\partial (u,v)}\right)}^2}=$$

$$\sqrt{r^4{\cos }^{2}u{\sin }^{2}u+r^4{\sin }^{4}u{\sin }^{2}v+r^4{\sin }^{4}u{\cos }^{2}v}=r^2\sin u.$$

This means that, on a sphere

$$d^{2}A=r^2\sin ududv.$$

Note that in the case of a unit sphere, ($r=1$) this agrees with the formula presented in the second paragraph of subsection 2 of the main entry.

To return to the main entry http://planetmath.org/node/6660click here

Title	derivation of surface area measure on sphere
Canonical name	DerivationOfSurfaceAreaMeasureOnSphere
Date of creation	2013-03-22 14:57:55
Last modified on	2013-03-22 14:57:55
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	6
Author	rspuzio (6075)
Entry type	Derivation
Classification	msc 28A75