Question

In Exercises 27–32, functions x = x(u, v) and y = y(u, v) are given that determine transformations from an XY-coordinate system to a UV-coordinate system in R2. Use these functions to determine a region in the XY-plane that has the image specified for the given values of u and v, and find the Jacobian of the transformation.

$x=usecvandy=u\tan vfor0\le u\le 2and0\le v\le \frac{\mathrm{\pi }}{4}$

Short answer

The Jacobian is equal to$J=usecv$.

Step-by-step solution

Given information

The functions are,

$x=usecvandy=u\tan vfor0\le u\le 2and0\le v\le \frac{\mathrm{\pi }}{4}$

Find the Jacobian

The Jacobian is computed as,

$$\begin{gathered} \frac{\partial (x,y)}{\partial (u,v)}=det\left[\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial y}{\partial u}\\ \frac{\partial x}{\partial v}& \frac{\partial y}{\partial v}\end{array}\right] \\ \frac{\partial (x,y)}{\partial (u,v)}=det\left[\begin{array}{cc}secv& \tan v\\ usecv\tan v& use{c}^{2}v\end{array}\right] \\ \frac{\partial (x,y)}{\partial (u,v)}=use{c}^{3}v-usecv{\tan }^{2}v \\ \frac{\partial (x,y)}{\partial (u,v)}=use{c}^{3}v-usecv\left(se{c}^{2}v-1\right) \\ \frac{\partial (x,y)}{\partial (u,v)}=use{c}^{3}v-use{c}^{3}v+usecv \\ \frac{\partial (x,y)}{\partial (u,v)}=usecv \end{gathered}$$