Section 15.9 (6th ed): Exercise 2: Find the Jacobian

[x,y]: [u*v, u/v];

J: determinant(jacobian([x,y],[u,v]));

Exercise 12: R is bounded by y=x+4, y=x-4, y=-3x+8, and y=-3x.

[x,y]: [(u+v)/4, (v-3*u)/4];

Find R = {((u+v)/4,(v-3u)/4): a<u<b, c<v<d}

solve(y=x+4,u); solve(y=x-4,u); solve(y=-3*x+8,v); solve(y=-3*x,v);

f: 4*x+8*y, expand;

J: determinant(jacobian([x,y],[u,v]));

integrate(integrate(f*J,u,-4,4),v,0,8);

Exercise 20: R is bounded by x-y=0, x-y=2, x+y=0, and x+y=3.

[x,y]: [(u+v)/2, (u-v)/2];

Find R = {((u+v)/2,(u-v)/2): a<u<b, c<v<d}

solve(x-y=0,v); solve(x-y=2,v); solve(x+y=0,u); solve(x+y=3,u);

f: (x+y)*exp(x^2-y^2), expand;

J: determinant(jacobian([x,y],[u,v]));

Jacobian has to be positive in change of variables.

J: abs(J);

integrate(integrate(f*J,v,0,2),u,0,3), expand;