Find the Jacobian of the transformation x = u - 3v, y = u² − 6v
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 given by the equations:
\[ x = u - 3v, \quad y = u^2 - 6v. \]
**Explanation**
In this problem, you are asked to determine the Jacobian matrix for the transformation between the variables \((u, v)\) and \((x, y)\). To do this, you will calculate the partial derivatives of \(x\) and \(y\) with respect to \(u\) and \(v\). The Jacobian matrix \(\mathbf{J}\) is:
\[
\mathbf{J} = \begin{bmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{bmatrix}
\]
Where:
- \(\frac{\partial x}{\partial u}\) and \(\frac{\partial x}{\partial v}\) are the partial derivatives of \(x\) with respect to \(u\) and \(v\), respectively.
- \(\frac{\partial y}{\partial u}\) and \(\frac{\partial y}{\partial v}\) are the partial derivatives of \(y\) with respect to \(u\) and \(v\), respectively.
Calculate these derivatives to construct the Jacobian matrix, which provides insight into how the transformation affects areas and volumes in the coordinate space defined by \((u, v)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b463452-b960-4b00-bce0-3a69d9f467e2%2Fc15ed6f9-2cb5-49fe-a1c8-9b935bff83a8%2F5u1oip9_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Find the Jacobian of the transformation given by the equations:
\[ x = u - 3v, \quad y = u^2 - 6v. \]
**Explanation**
In this problem, you are asked to determine the Jacobian matrix for the transformation between the variables \((u, v)\) and \((x, y)\). To do this, you will calculate the partial derivatives of \(x\) and \(y\) with respect to \(u\) and \(v\). The Jacobian matrix \(\mathbf{J}\) is:
\[
\mathbf{J} = \begin{bmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{bmatrix}
\]
Where:
- \(\frac{\partial x}{\partial u}\) and \(\frac{\partial x}{\partial v}\) are the partial derivatives of \(x\) with respect to \(u\) and \(v\), respectively.
- \(\frac{\partial y}{\partial u}\) and \(\frac{\partial y}{\partial v}\) are the partial derivatives of \(y\) with respect to \(u\) and \(v\), respectively.
Calculate these derivatives to construct the Jacobian matrix, which provides insight into how the transformation affects areas and volumes in the coordinate space defined by \((u, v)\).
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