Advanced Calculus fi..

Green's theorem now givesChapter 5 Vector Integral Calculus 333IThe line integral JQ dy can at once be written as a line integral in the uv-plane: 1The differentials here should all be thought of as expressed in terms of a parametert: dy = (dy/dt)dt, du = (du/dt)dt, dv = (dv/dt)dt. As t goes from h to k, C,,is to be traced just once in the positive direction, so that C,, is traced just once inthe positive or negative direction. The f sign corresponds to these two cases.The line integral on the right is of formwhere PI and Ql are defined in D,, and have continuous derivatives there. HenceGreen's theorem is applicable:since aQ/ax = F(x, y). One thus concludesJJ ~ ( x Y) , dx dy = f ~ [(u, f v), g(u, U)I---- a(x9 du dv,. a(u9 v)R,,as was to be shown.JJR,,,.t