Mathematical Methods for Physics and Engineering - Matematica.NET

11.5 SURFACE INTEGRALS◮Find the vector area of the surface ∮ of the hemisphere x 2 + y 2 + z 2evaluating the line integral S = 1 r × dr around its perimeter.2 C= a 2 , z ≥ 0, byThe perimeter C of the hemisphere is the circle x 2 + y 2 = a 2 , on which we haver = a cos φ i + a sin φ j, dr = −a sin φdφi + a cos φdφj.Therefore the cross product r × dr is given byi j kr × dr =a cos φ asin φ 0∣ −a sin φdφ acos φdφ 0 ∣ = a2 (cos 2 φ +sin 2 φ) dφ k = a 2 dφ k,and the vector area becomes∫ 2πS = 1 2 a2 k dφ = πa 2 k. ◭011.5.3 Physical examples of surface integralsThere are many examples of surface integrals in the physical sciences. Surfaceintegrals of the form (11.8) occur in computing the total electric charge on asurface or the mass of a shell, ∫ Sρ(r) dS, given the charge or mass density ρ(r).For surface integrals involving vectors, the second form in (11.9) is the mostcommon. For a vector field a, the surface integral ∫ Sa · dS is called the fluxof a through S. Examples of physically important flux integrals are numerous.For example, let us consider a surface S in a fluid with density ρ(r) that has avelocity field v(r). The mass of fluid crossing an element of surface area dS intime dt is dM = ρv · dS dt. Therefore the net total mass flux of fluid crossing Sis M = ∫ Sρ(r)v(r) · dS. As a another example, the electromagnetic flux of energyout of a given volume V bounded by a surface S is ∮ S(E × H) · dS.The solid angle, to be defined below, subtended at a point O by a surface (closedor otherwise) can also be represented by an integral of this form, although it isnot strictly a flux integral (unless we imagine isotropic rays radiating from O).The integral∫ ∫r · dS ˆr · dSΩ=S r 3 =S r 2 , (11.11)gives the solid angle Ω subtended at O by a surface S if r is the position vectormeasured from O of an element of the surface. A little thought will show that(11.11) takes account of all three relevant factors: the size of the element ofsurface, its inclination to the line joining the element to O and the distance fromO. Such a general expression is often useful for computing solid angles when thethree-dimensional geometry is complicated. Note that (11.11) remains valid whenthe surface S is not convex and when a single ray from O in certain directionswould cut S in more than one place (but we exclude multiply connected regions).395