Mathematical Methods for Physics and Engineering - Matematica.NET

6Multiple integralsFor functions of several variables, just as we may consider derivatives with respectto two or more of them, so may the integral of the function with respect to morethan one variable be formed. The formal definitions of such multiple integrals areextensions of that for a single variable, discussed in chapter 2. We first discussdouble and triple integrals and illustrate some of their applications. We thenconsider changing the variables in multiple integrals and discuss some generalproperties of Jacobians.6.1 Double integralsFor an integral involving two variables – a double integral – we have a function,f(x, y) say, to be integrated with respect to x and y between certain limits. Theselimits can usually be represented by a closed curve C bounding a region R in thexy-plane. Following the discussion of single integrals given in chapter 2, let usdivide the region R into N subregions ∆R p of area ∆A p , p =1, 2,...,N, and let(x p ,y p ) be any point in subregion ∆R p . Now consider the sumS =N∑f(x p ,y p )∆A p ,p=1and let N →∞as each of the areas ∆A p → 0. If the sum S tends to a uniquelimit, I, then this is called the double integral of f(x, y) over the region R and iswritten∫I = f(x, y) dA, (6.1)Rwhere dA stands for the element of area in the xy-plane. By choosing thesubregions to be small rectangles each of area ∆A =∆x∆y, and letting both ∆x187