Mathematical Methods for Physics and Engineering - Matematica.NET

LINE, SURFACE AND VOLUME INTEGRALSIn particular, when the surface is closed Ω = 0 if O is outside S and Ω = 4π if Ois an interior point.Surface integrals resulting in vectors occur less frequently. An example isafforded, however, by the total resultant force experienced by a body immersed ina stationary fluid in which the hydrostatic pressure is given by p(r). The pressureis everywhere inwardly directed and the resultant force is F = − ∮ SpdS, takenover the whole surface.11.6 Volume integralsVolume integrals are defined in an obvious way and are generally simpler thanline or surface integrals since the element of volume dV is a scalar quantity. Wemay encounter volume integrals of the forms∫∫φdV, a dV . (11.12)VClearly, the first form results in a scalar, whereas the second form yields a vector.Two closely related physical examples, one of each kind, are provided by the totalmass of a fluid contained in a volume V , given by ∫ Vρ(r) dV , and the total linearmomentum of that same fluid, given by ∫ Vρ(r)v(r) dV where v(r) is the velocityfield in the fluid. As a slightly more complicated example of a volume integral wemay consider the following.◮Find an expression for the angular momentum of a solid body rotating with angularvelocity ω about an axis through the origin.Consider a small volume element dV situated at position r; its linear momentum is ρdVṙ,where ρ = ρ(r) is the density distribution, and its angular momentum about O is r × ρṙ dV .Thus for the whole body the angular momentum L is∫L = (r × ṙ)ρdV.Putting ṙ = ω × r yields∫∫L = [r × (ω × r)] ρdV =VVVV∫ωr 2 ρdV − (r · ω)rρdV. ◭VThe evaluation of the first type of volume integral in (11.12) has already beenconsidered in our discussion of multiple integrals in chapter 6. The evaluation ofthe second type of volume integral follows directly since we can write∫ ∫ ∫∫a dV = i a x dV + j a y dV + k a z dV , (11.13)VVwhere a x ,a y ,a z are the Cartesian components of a. Of course, we could havewritten a in terms of the basis vectors of some other coordinate system (e.g.spherical polars) but, since such basis vectors are not, in general, constant, they396VV