https://file.aiquickdraw.com/mathbot/file/1729740903548hm2joxt1.jpg

Let's compute the Jacobian for the given transformation.

The transformation is defined as:
\[
x = 8u + 5v, \quad y = u - 2v
\]

The Jacobian matrix \( J(u, v) \) is the matrix of partial derivatives of \( (x, y) \) with respect to \( (u, v) \), which is:

\[
J(u, v) = \begin{bmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{bmatrix}
\]

### Step 1: Compute the partial derivatives
- \( \frac{\partial x}{\partial u} = \frac{\partial}{\partial u}(8u + 5v) = 8 \)
- \( \frac{\partial x}{\partial v} = \frac{\partial}{\partial v}(8u + 5v) = 5 \)
- \( \frac{\partial y}{\partial u} = \frac{\partial}{\partial u}(u - 2v) = 1 \)
- \( \frac{\partial y}{\partial v} = \frac{\partial}{\partial v}(u - 2v) = -2 \)

### Step 2: Form the Jacobian matrix
\[
J(u, v) = \begin{bmatrix}
8 & 5 \\
1 & -2
\end{bmatrix}
\]

### Step 3: Compute the determinant of the Jacobian
The determinant of the Jacobian matrix is:

\[
\det(J(u, v)) = (8)(-2) - (5)(1) = -16 - 5 = -21
\]

Thus, the Jacobian determinant is:
\[
\det(J(u, v)) = -21
\]

This confirms that the Jacobian is not always positive, as noted in the problem.

Would you like more details or have any other questions? 

Here are 5 related questions to consider:
1. What is the significance of the Jacobian determinant in transformations?
2. How does the sign of the Jacobian determinant affect the orientation of the transformation?
3. Could the Jacobian determinant be zero? What would that imply about the transformation?
4. How would you interpret a negative Jacobian determinant geometrically?
5. How can you apply the Jacobian matrix in higher dimensions, such as in 3D transformations?

**Tip:** The Jacobian determinant helps measure how a transformation affects area (in 2D) or volume (in higher dimensions).

Compute the Jacobian for the transformation x = 8u + 5v, y = u - 2v.

Step-by-step computation of the Jacobian for the transformation x = 8u + 5v, y = u - 2v. This involves calculating partial derivatives and forming a Jacobian matrix, suitable for high school and undergraduate students.

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