Thedivergenceof functionF=Pi+Qj+Rkis defined asdivF=r·F=@P@x+@Q@y+@R@z.Notice that the curl of a function is a vector and the divergence of a function is a scalar (just anumber).Surface IntegralsIfFis a continuous vector field defined on an oriented surfaceSwith unit normal vectorn, thenthe surface integral ofFoverSisZZSF·dS=ZZSF·ndS.IfSis given by a vector functionr(u, v), then the above integral can be written asZZSF·dS=ZZDF·(ru⇥rv)dAwhereDis the parameter domain (the projection ofSonto a plane).If we are givenSis the functionz=g(x, y), we can define the parametrization ofSto ber(x, y) = (x, y, g(x, y)).Then the surface integral ofFoverSis given byZZSF·dS=ZZDF·(rx⇥ry)dAwhereDis the projection ofz=g(x, y) onto thexyplane.Stokes’ TheoremTheorem(Stokes’ Theorem).LetSbe an oriented piecewise-smooth surface that is bounded bya simple, closed, piecewise-smooth boundary curveCwith positive orientation. LetFbe a vectorfield whose components have continuous partial derivatives on an open region inR3that containsS. ThenZCF·dr=ZZScurlF·dS.In other words, the line integral around the boundary curve of a surfaceSof the tangential com-ponent ofFis equal to the surface integral of the normal component of the curl ofF.Note: This looks very similar to Green’s Theorem! In fact, Green’s Theorem is a special case ofStokes’ Theorem; it is whenFis restricted to thexyplane. Stokes’ Theorem can then be thoughtof as the higher-dimensional version of Green’s Theorem.