Find the Jacobian of the transformation x = · 3u − 2v, y = u² + 3v.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
5.7.1
![**Problem Statement:**
Find the Jacobian of the transformation \( x = -3u - 2v, \quad y = u^2 + 3v \).
**Explanation:**
To find the Jacobian of the given transformation from \( (u, v) \) to \( (x, y) \), we need to compute the determinant of the Jacobian matrix. The Jacobian matrix is composed of the partial derivatives of \( x \) and \( y \) with respect to \( u \) and \( v \). It is structured as follows:
\[
J = \begin{bmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{bmatrix}
\]
Calculating the partial derivatives:
- \(\frac{\partial x}{\partial u} = -3\)
- \(\frac{\partial x}{\partial v} = -2\)
- \(\frac{\partial y}{\partial u} = 2u\)
- \(\frac{\partial y}{\partial v} = 3\)
Thus, the Jacobian matrix \( J \) becomes:
\[
J = \begin{bmatrix}
-3 & -2 \\
2u & 3
\end{bmatrix}
\]
The Jacobian determinant is then computed as follows:
\[
\text{det}(J) = (-3)(3) - (-2)(2u) = -9 + 4u = 4u - 9
\]
**Conclusion:**
The Jacobian determinant of the transformation is \( 4u - 9 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b463452-b960-4b00-bce0-3a69d9f467e2%2F35827bbf-58d2-43fd-a42a-b087d44cfad5%2Fv95n9qv_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the Jacobian of the transformation \( x = -3u - 2v, \quad y = u^2 + 3v \).
**Explanation:**
To find the Jacobian of the given transformation from \( (u, v) \) to \( (x, y) \), we need to compute the determinant of the Jacobian matrix. The Jacobian matrix is composed of the partial derivatives of \( x \) and \( y \) with respect to \( u \) and \( v \). It is structured as follows:
\[
J = \begin{bmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{bmatrix}
\]
Calculating the partial derivatives:
- \(\frac{\partial x}{\partial u} = -3\)
- \(\frac{\partial x}{\partial v} = -2\)
- \(\frac{\partial y}{\partial u} = 2u\)
- \(\frac{\partial y}{\partial v} = 3\)
Thus, the Jacobian matrix \( J \) becomes:
\[
J = \begin{bmatrix}
-3 & -2 \\
2u & 3
\end{bmatrix}
\]
The Jacobian determinant is then computed as follows:
\[
\text{det}(J) = (-3)(3) - (-2)(2u) = -9 + 4u = 4u - 9
\]
**Conclusion:**
The Jacobian determinant of the transformation is \( 4u - 9 \).
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