Change of Variables

For more general transformations $x = x(u,v)$ and $y = y(u,v)$ we can use the jacobian function in the linearalgebra package, which is loaded by default. Let $f(x,y)=x+y$, and we will calculate

$$\int\int_R \left( x+y\right) \mathrm{d}A.$$

Let's make the transformation $x=u^3-v^4$ and $y = 5uv$.

We need the Jacobian:

We were lucky in this example because the Jacobian is positive as long as $u$ is positive (or just not terribly negative). If this had not happened then we would need to be careful about the sign of $J$, which in principle could be a problem because Maxima does not integrate absolute value. Nevertheless, we finish up with the integration. It takes the form (we will make up random limits of integration, but in a given problem we would need to determine these)

$$\int_3^4\int_1^2 f\left( x(u,v),y(u,v) \right) \left\ver... ...al(x,y)}{\partial(u,v)} \right\vert \mathrm{d}u \mathrm{d}v.$$