Q. 30 - 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.  

$x = u^2 + v^2 and y = u^2 - v^2 for 0 \le u \le 4 and 0 \le v \le 4$

Step-by-Step Solution

Answer

The Jacobian is equal to $J=-8uv$.

Step 1: Given information

The functions are,

$x = u^2 + v^2 and y = u^2 - v^2 for 0 \le u \le 4 and 0 \le v \le 4$

Step 2: Find the Jacobian

The Jacobian is computed as,

$$\begin{gathered} \frac{\partial (x, y)}{\partial (u,v)}=det\left[\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial y}{\partial u}\\ \frac{\partial x}{\partial v}& \frac{\partial y}{\partial v}\end{array}\right] \\ \frac{\partial (x, y)}{\partial (u,v)}=det\left[\begin{array}{cc}2u& 2u\\ 2v& -2v\end{array}\right] \\ \frac{\partial (x, y)}{\partial (u,v)}=-4uv-4uv \\ \frac{\partial (x, y)}{\partial (u,v)}=-8uv \end{gathered}$$