Mathematical Methods for Physics and Engineering - Matematica.NET

MULTIPLE INTEGRALSyu =constantv =constantNKMLRCFigure 6.10 A region of integration R overlaid with a grid formed by thefamily of curves u =constantandv = constant. The parallelogram KLMNdefines the area element dA uv .xexpress a multiple integral in terms of a new set of variables. We now considerhow to do this.6.4.1 Change of variables in double integralsLet us begin by examining the change of variables in a double integral. Supposethat we require to change an integral∫∫I = f(x, y) dx dy,Rin terms of coordinates x and y, into one expressed in new coordinates u and v,given in terms of x and y by differentiable equations u = u(x, y) andv = v(x, y)with inverses x = x(u, v) andy = y(u, v). The region R in the xy-plane and thecurve C that bounds it will become a new region R ′ and a new boundary C ′ inthe uv-plane, and so we must change the limits of integration accordingly. Also,the function f(x, y) becomes a new function g(u, v) of the new coordinates.Now the part of the integral that requires most consideration is the area element.In the xy-plane the element is the rectangular area dA xy = dx dy generated byconstructing a grid of straight lines parallel to the x- andy- axes respectively.Our task is to determine the corresponding area element in the uv-coordinates. Ingeneral the corresponding element dA uv will not be the same shape as dA xy , butthis does not matter since all elements are infinitesimally small and the value ofthe integrand is considered constant over them. Since the sides of the area elementare infinitesimal, dA uv will in general have the shape of a parallelogram. We canfind the connection between dA xy and dA uv by considering the grid formed by thefamily of curves u =constantandv = constant, as shown in figure 6.10. Since v200