Find the Jacobian of the transformation = 2u+7v, y u² - 5v. =
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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![**Problem Statement**
Find the Jacobian of the transformation \( x = 2u + 7v, \, y = u^2 - 5v \).
---
**Solution Explanation**
To find the Jacobian of the given transformation, we need to compute the determinant of the Jacobian matrix, which is constructed from the partial derivatives of \( x \) and \( y \) with respect to \( u \) and \( v \).
**Jacobian Matrix Calculation:**
The transformation is given by:
\[ x = 2u + 7v \]
\[ y = u^2 - 5v \]
1. Compute the partial derivatives:
- \(\frac{\partial x}{\partial u} = 2\)
- \(\frac{\partial x}{\partial v} = 7\)
- \(\frac{\partial y}{\partial u} = 2u\)
- \(\frac{\partial y}{\partial v} = -5\)
2. Form the Jacobian matrix:
\[
J = \begin{bmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{bmatrix}
=
\begin{bmatrix}
2 & 7 \\
2u & -5
\end{bmatrix}
\]
3. Compute the determinant of the Jacobian matrix:
\[
\det(J) = (2)(-5) - (7)(2u) = -10 - 14u = -10 - 14u
\]
The Jacobian determinant, which represents the factor by which the area is scaled under this transformation, is:
\[
-10 - 14u
\]
By finding the Jacobian, we can analyze how the transformation affects the differential elements and measure the local change in area induced by the transformation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd20dfe5a-a4c1-4793-9f05-a80ad59a67d4%2F94b81341-621f-445b-b756-d3ed33372272%2F45frwjp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Find the Jacobian of the transformation \( x = 2u + 7v, \, y = u^2 - 5v \).
---
**Solution Explanation**
To find the Jacobian of the given transformation, we need to compute the determinant of the Jacobian matrix, which is constructed from the partial derivatives of \( x \) and \( y \) with respect to \( u \) and \( v \).
**Jacobian Matrix Calculation:**
The transformation is given by:
\[ x = 2u + 7v \]
\[ y = u^2 - 5v \]
1. Compute the partial derivatives:
- \(\frac{\partial x}{\partial u} = 2\)
- \(\frac{\partial x}{\partial v} = 7\)
- \(\frac{\partial y}{\partial u} = 2u\)
- \(\frac{\partial y}{\partial v} = -5\)
2. Form the Jacobian matrix:
\[
J = \begin{bmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{bmatrix}
=
\begin{bmatrix}
2 & 7 \\
2u & -5
\end{bmatrix}
\]
3. Compute the determinant of the Jacobian matrix:
\[
\det(J) = (2)(-5) - (7)(2u) = -10 - 14u = -10 - 14u
\]
The Jacobian determinant, which represents the factor by which the area is scaled under this transformation, is:
\[
-10 - 14u
\]
By finding the Jacobian, we can analyze how the transformation affects the differential elements and measure the local change in area induced by the transformation.
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