Mathematical Methods for Physics and Engineering - Matematica.NET

11Line, surface and volume integralsIn the previous chapter we encountered continuously varying scalar and vectorfields and discussed the action of various differential operators on them. Inaddition to these differential operations, the need often arises to consider theintegration of field quantities along lines, over surfaces and throughout volumes.In general the integrand may be scalar or vector in nature, but the evaluationof such integrals involves their reduction to one or more scalar integrals, whichare then evaluated. In the case of surface and volume integrals this requires theevaluation of double and triple integrals (see chapter 6).11.1 Line integralsIn this section we discuss line or path integrals, in which some quantity relatedto the field is integrated between two given points in space, A and B, along aprescribed curve C that joins them. In general, we may encounter line integralsof the forms ∫ ∫∫φdr, a · dr, a × dr, (11.1)CCwhere φ is a scalar field and a is a vector field. The three integrals themselves arerespectively vector, scalar and vector in nature. As we will see below, in physicalapplications line integrals of the second type are by far the most common.The formal definition of a line integral closely follows that of ordinary integralsand can be considered as the limit of a sum. We may divide the path C joiningthe points A and B into N small line elements ∆r p , p =1,...,N.If(x p ,y p ,z p )isany point on the line element ∆r p then the second type of line integral in (11.1),for example, is defined as∫∑Na · dr = lim a(x p ,y p ,z p ) · ∆r p ,CN→∞p=1where it is assumed that all |∆r p |→0asN →∞.377C