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Chapter 13: Problem 18

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

Short Answer

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

Step by step solution

01

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: - \(\frac{\partial y}{\partial u} = -2\) - \(\frac{\partial y}{\partial v} = 0\)
02

Construct the Jacobian matrix

Now, arrange the partial derivatives found in the previous step to form the Jacobian matrix: $$ \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} = \begin{bmatrix} 0 & 4 \\ -2 & 0 \end{bmatrix} $$
03

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.

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