4.6.4 Area element and volume element on a sphere

When considering budgets over finite domains, integration over an element of the sphere is common. A useful bit of notation is the area of an infinitesimal element of the sphere

$$d\Omega = a^{2} \, \cos \phi \; d\phi \; d\lambda = a^{2} \, d(\sin\phi) \; d\lambda.$$ (4.69)

With this notation, the volume element on the sphere takes the form

$$d{\bf x} = d\Omega \, dz.$$ (4.70)