(a) For the transformation from Cartesian coordinates ( (x, y) ) to coordinates ( (u, v) ) where [ u=x+y, quad v=rac{y}{x} ] find ( x ) and ( y ) in terms of ( u ) and ( v ). Hence calculate the Jacobian ( partial(x, y) / partial(u, v) ). Sketch the region ( A ) in the ( (x, y) ) plane satisfying ( x leq y leq 2 x ) and ( 1-x leq y leq ) ( 2-x ), and the corresponding region ( A^{prime} ) in the ( (u, v) ) plane. By transforming to ( (u, v) ) coordinates, evaluate the integral [ I=iint_{A} rac{1}{x y} mathrm{~d} x mathrm{~d} y ] (b) By making ( x ) the last (outermost) variable of integration, evaluate the integral [ J=iiint_{V} x y z^{2} mathrm{~d} V ] for the wedge-shaped locus ( V ) of points that satisfy ( 0 leq x leq 2 ) with ( 1 leq y leq ) ( 1+(x / 2) ) and ( -1 leq z leq 1 ). Confirm your answer by repeating the calculation with the innermost partial integration with respect to ( x ).

(a) For the transformation from Cartesian coordinates (x,y) to coordinates (u,v) where u=x+y,v=xy, find x and y in terms of u and v. Hence calculate the Jacobian (x,y)/(u,v). Sketch the region A in the (x,y) plane satisfying xy2x and 1xy 2x, and the corresponding region A in the (u,v) plane. By transforming to (u,v) coordinates, evaluate the integral I=Axy1dxdy. (b) By making x the last (outermost) variable of integration, evaluate the integral J=Vxyz2dV for the wedge-shaped locus V of points that satisfy 0x2 with 1y 1+(x/2) and 1z1. Confirm your answer by repeating the calculation with the innermost partial integration with respect to x