Q. 27 - Calculus

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. 

$x=u-v and y=u+v for 0\le u\le 2 and 0\le v\le 1$

Step-by-Step Solution

Answer

The Jacobian of transformation is equal to $2$.

Step 1: Given information

The functions are,

$x=u-v and y=u+v for 0\le u\le 2 and 0\le v\le 1$

Step 2: Find the Jacobian

The Jacobian is computed as,

$$\begin{gathered} \frac{\partial (x, y)}{\partial (u,v)}=det\left[\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial y}{\partial u}\\ \frac{\partial x}{\partial v}& \frac{\partial y}{\partial v}\end{array}\right] \\ \frac{\partial (x, y)}{\partial (u,v)}=det\left[\begin{array}{cc}1& 1\\ -1& 1\end{array}\right] \\ \frac{\partial (x, y)}{\partial (u,v)}=1-(-1) \\ \frac{\partial (x, y)}{\partial (u,v)}=1+1=2 \end{gathered}$$