Section 14.8 Homework

From Calculus 10e by Larson and Edwards, p. 1032. Exercises 3, 11

Exercise 14.8.3 Finding a Jacobian

Find the Jacobian $\partial(x,y)/\partial(u,v)$ for the indicated variable change.

$x=u-v^{2}, \: y=u+v$

Solution

$$\frac{\partial(x,y)}{\partial(u,v)} $$	$$= \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v }\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{vmatrix}$$
	$$=\begin{vmatrix} 1 & -2v \\ 1 & 1 \end{vmatrix} = 1+2v$$

Figure 1

Figure 2

Sketch the image $S$ in the $uv$-plane for the region $R$ in the $xy$-plane using the following transformations, as shown in Figure 1.

$$x= \frac{1}{2}(u+v) $$

$$y= \frac{1}{2}(u-v) $$

Solution Let $u=x+y$ and $v=y-x$. The region $S$ is shown in Figure 2.