Mathematical Methods for Physics and Engineering - Matematica.NET

11.8 DIVERGENCE THEOREM AND RELATED THEOREMS11.8 Divergence theorem and related theoremsThe divergence theorem relates the total flux of a vector field out of a closedsurface S to the integral of the divergence of the vector field over the enclosedvolume V ; it follows almost immediately from our geometrical definition ofdivergence (11.15).Imagine a volume V , in which a vector field a is continuous and differentiable,to be divided up into a large number of small volumes V i . Using (11.15), we havefor each small volume∮(∇ · a)V i ≈ a · dS,S iwhere S i is the surface of the small volume V i . Summing over i we find thatcontributions from surface elements interior to S cancel since each surface elementappears in two terms with opposite signs, the outward normals in the two termsbeing equal and opposite. Only contributions from surface elements that are alsoparts of S survive. If each V i is allowed to tend to zero then we obtain thedivergence theorem,∫∮∇ · a dV = a · dS. (11.18)VSWe note that the divergence theorem holds for both simply and multiply connectedsurfaces, provided that they are closed and enclose some non-zero volumeV . The divergence theorem may also be extended to tensor fields (see chapter 26).The theorem finds most use as a tool in formal manipulations, but sometimesit is of value in transforming surface integrals of the form ∫ Sa · dS into volumeintegrals or vice versa. For example, setting a = r we immediately obtain∫∫∮∇ · r dV = 3 dV =3V = r · dS,VVwhich gives the expression for the volume of a region found in subsection 11.6.1.The use of the divergence theorem is further illustrated in the following example.◮Evaluate the surface integral I = ∫ S a · dS, wherea =(y − x) i + x2 z j +(z + x 2 ) k and Sis the open surface of the hemisphere x 2 + y 2 + z 2 = a 2 , z ≥ 0.We could evaluate this surface integral directly, but the algebra is somewhat lengthy. Wewill therefore evaluate it by use of the divergence theorem. Since the latter only holdsfor closed surfaces enclosing a non-zero volume V , let us first consider the closed surfaceS ′ = S + S 1 ,whereS 1 is the circular area in the xy-plane given by x 2 + y 2 ≤ a 2 , z =0;S ′then encloses a hemispherical volume V . By the divergence theorem we have∫∮ ∫ ∫∇ · a dV = a · dS = a · dS + a · dS.VS ′ SS 1Now ∇ · a = −1+0+1=0,sowecanwrite∫∫a · dS = − a · dS.SS 1401S