Problem 2 - Calculus Early Transcendentals

Question

Explain how to compute the Jacobian of the transformation \(T: x=g(u, v), y=h(u, v)\)

Step-by-Step Solution

Answer

Answer: The Jacobian matrix represents how the transformation T influences the area of a shape when transforming it from the (u,v) space to the (x,y) space. To compute the Jacobian matrix, first, calculate the partial derivatives of the component functions with respect to u and v. Then, arrange these derivatives into a 2x2 matrix. The determinant of the Jacobian matrix, called the Jacobian determinant, represents the scaling factor of the area when passing through the transformation and indicates if the local orientation is preserved or reversed.

Step 1: Find the partial derivatives of g(u,v) and h(u,v) with respect to u and v

First, we need to compute the following partial derivatives: 1. \(\frac{\partial x}{\partial u}\): the partial derivative of \(x=g(u, v)\) with respect to \(u\). 2. \(\frac{\partial x}{\partial v}\): the partial derivative of \(x=g(u, v)\) with respect to \(v\). 3. \(\frac{\partial y}{\partial u}\): the partial derivative of \(y=h(u, v)\) with respect to \(u\). 4. \(\frac{\partial y}{\remaining.paths.folder_id.remaining.executions_count.v}\): the partial derivative of \(y=h(u, v)\) with respect to \(v\).

Step 2: Create the Jacobian matrix

Once all the partial derivatives in step 1 are calculated, we can write down the Jacobian matrix of the transformation \(T\). The Jacobian matrix is given by the following 2x2 matrix: $$ \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\remaining.paths.folder_id.remaining.executions_count.v}\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\remaining.paths.folder_id.remaining.executions_count.v} \end{bmatrix} $$

Step 3: Interpret the result

The Jacobian matrix obtained above shows us how the transformation \(T\) influences the area of a shape when transforming it from the (u,v) space to the (x,y) space. The determinant of the Jacobian matrix, called the Jacobian determinant, represents the scaling factor of the area when passing through the transformation. The determinant can be computed as follows: $$ \text{Jacobian determinant} = \frac{\partial x}{\partial u} \frac{\partial y}{\remaining.paths.folder_id.remaining.executions_count.v} - \frac{\partial x}{\remaining.paths.folder_id.remaining.executions_count.v} \frac{\partial y}{\partial u} $$ If the Jacobian determinant is positive, it means that the local orientation is preserved. If the Jacobian determinant is negative, it means that the local orientation is reversed. The absolute value of the Jacobian determinant indicates the scaling factor.