To prove the divergence theorem for a general vector field F =
á M,N,P
ñ , we begin by noticing that F = F1+F2+F3, where
F1 =
á M,0,0
ñ , F2 =
á 0,N,0
ñ , andF3 =
á0,0,P
ñ
Suppose now that W is a solid that can be described in three
different ways- (1) as a solid bounded between two surfaces x = p(y,z) and x = q( y,z) , (2) as a solid bound between two
surfaces y = f( x,z) and y = g( x,z) , and (3) as a
solid bound between two surfaces z = a( x,y) and z = b(x,y).