Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

6


Multiple integrals


For functions of several variables, just as we may consider derivatives with respect


to two or more of them, so may the integral of the function with respect to more


than one variable be formed. The formal definitions of such multiple integrals are


extensions of that for a single variable, discussed in chapter 2. We first discuss


double and triple integrals and illustrate some of their applications. We then


consider changing the variables in multiple integrals and discuss some general


properties of Jacobians.


6.1 Double integrals

For an integral involving two variables – a double integral – we have a function,


f(x, y) say, to be integrated with respect toxandybetween certain limits. These


limits can usually be represented by a closed curveCbounding a regionRin the


xy-plane. Following the discussion of single integrals given in chapter 2, let us


divide the regionRintoNsubregions ∆Rpof area ∆Ap,p=1, 2 ,...,N, and let


(xp,yp) be any point in subregion ∆Rp. Now consider the sum


S=

∑N

p=1

f(xp,yp)∆Ap,

and letN→∞as each of the areas ∆Ap→0. If the sumStends to a unique


limit,I, then this is called thedouble integral off(x, y)over the regionRand is


written


I=


R

f(x, y)dA, (6.1)

wheredAstands for the element of area in thexy-plane. By choosing the


subregions to be small rectangles each of area ∆A=∆x∆y, and letting both ∆x

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