Problem 18 - Calculus Early Transcendentals

Question

Compute the Jacobian \(J(u, v)\) for the following transformations. $$T: x=4 v, y=-2 u$$

Step-by-Step Solution

Answer

Answer: The Jacobian determinant for the transformation T is 8.

Step 1: Compute the partial derivatives of x and y with respect to u and v

We have two functions: 1. \(x=4v\) 2. \(y=-2u\) We will compute the partial derivatives of each function with respect to u and v. For x: - \(\frac{\partial x}{\partial u} = 0\) - \(\frac{\partial x}{\partial v} = 4\) For y: - \(\frac{\partial y}{\partial u} = -2\) - \(\frac{\partial y}{\partial v} = 0\)

Step 2: Construct the Jacobian matrix

Now, arrange the partial derivatives found in the previous step to form the Jacobian matrix: $$ \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} = \begin{bmatrix} 0 & 4 \\ -2 & 0 \end{bmatrix} $$

Step 3: Compute the determinant of the Jacobian matrix

The determinant of a 2x2 matrix: $$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$ is calculated as \((ad - bc)\). In the case of our Jacobian matrix, this will be: $$J(u, v)=\det \begin{bmatrix} 0 & 4 \\ -2 & 0 \end{bmatrix}=(0)(0)-(4)(-2)=8 $$ The Jacobian \(J(u, v)\) for the given transformation T is 8.