Section 15.9: Example 2: Change of variables: x = g(u,v) and y = h(u,v)

g(u,v) := u^2-v^2;

h(u,v) := 2*u*v;

jacobian([g(u,v),h(u,v)],[u,v]);

J: determinant(jacobian([g(u,v),h(u,v)],[u,v]));

Region: R = {(u^2-v^2,2*u*v): 0 < u < 1, 0 < v < 1}

f(x,y) := y;

'integrate('integrate(f(g(u,v),h(u,v))*J,u,0,1),v,0,1);

integrate(integrate(f(g(u,v),h(u,v))*J,u,0,1),v,0,1);

Region: R = {(x,y): 0 < y < 2, y^2/4-1 < x < 1-y^2/4}

integrate(integrate(y,x,y^2/4-1,1-y^2/4),y,0,2);

Example 3: Region: R = {((u+v)/2,(u-v)/2): 1 < v < 2, -v < u < v}

g(u,v) := (u+v)/2;

h(u,v) := (u-v)/2;

jacobian([g(u,v),h(u,v)],[u,v]);

determinant(jacobian([g(u,v),h(u,v)],[u,v]));

J: abs(determinant(jacobian([g(u,v),h(u,v)],[u,v])));

f(x,y) := exp((x+y)/(x-y));

'integrate('integrate(f(g(u,v),h(u,v))*J,u,-v,v),v,1,2);

'integrate('integrate(ratsimp(f(g(u,v),h(u,v))*J),u,-v,v),v,1,2);

integrate(integrate(f(g(u,v),h(u,v))*J,u,-v,v),v,1,2);

expand(ratsimp(%));