Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Finding the Jacobian Matrix
To determine the Jacobian for the given transformation, we need to start with the equations provided:
\[ x = u^2 + 8uv \]
\[ y = 6uv^2 \]
### Jacobian Matrix
The Jacobian matrix \( J \) is a matrix of all first-order partial derivatives of a vector-valued function. For this transformation, the Jacobian matrix is given by:
\[ J = \begin{bmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{bmatrix} \]
### Steps to Find the Partial Derivatives
1. **Calculate \( \frac{\partial x}{\partial u} \):**
\[ \frac{\partial x}{\partial u} = \frac{\partial (u^2 + 8uv)}{\partial u} \]
\[ = 2u + 8v \]
2. **Calculate \( \frac{\partial x}{\partial v} \):**
\[ \frac{\partial x}{\partial v} = \frac{\partial (u^2 + 8uv)}{\partial v} \]
\[ = 8u \]
3. **Calculate \( \frac{\partial y}{\partial u} \):**
\[ \frac{\partial y}{\partial u} = \frac{\partial (6uv^2)}{\partial u} \]
\[ = 6v^2 \]
4. **Calculate \( \frac{\partial y}{\partial v} \):**
\[ \frac{\partial y}{\partial v} = \frac{\partial (6uv^2)}{\partial v} \]
\[ = 12uv \]
### Jacobian Matrix
By substituting these partial derivatives into the Jacobian matrix, we get:
\[ J = \begin{bmatrix}
2u + 8v & 8u \\
6v^2 & 12uv
\end{bmatrix} \]
### Final Answer
Thus, the Jacobian matrix \( J \) for the given transformation is:
\[ \boxed{\begin{bmatrix} 2u + 8v & 8u \\ 6](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74b6836e-7075-4604-8b7e-b1441272473c%2F2867ff53-62b7-4e68-b7aa-242325b2025f%2Fb5jwd97_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Jacobian Matrix
To determine the Jacobian for the given transformation, we need to start with the equations provided:
\[ x = u^2 + 8uv \]
\[ y = 6uv^2 \]
### Jacobian Matrix
The Jacobian matrix \( J \) is a matrix of all first-order partial derivatives of a vector-valued function. For this transformation, the Jacobian matrix is given by:
\[ J = \begin{bmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{bmatrix} \]
### Steps to Find the Partial Derivatives
1. **Calculate \( \frac{\partial x}{\partial u} \):**
\[ \frac{\partial x}{\partial u} = \frac{\partial (u^2 + 8uv)}{\partial u} \]
\[ = 2u + 8v \]
2. **Calculate \( \frac{\partial x}{\partial v} \):**
\[ \frac{\partial x}{\partial v} = \frac{\partial (u^2 + 8uv)}{\partial v} \]
\[ = 8u \]
3. **Calculate \( \frac{\partial y}{\partial u} \):**
\[ \frac{\partial y}{\partial u} = \frac{\partial (6uv^2)}{\partial u} \]
\[ = 6v^2 \]
4. **Calculate \( \frac{\partial y}{\partial v} \):**
\[ \frac{\partial y}{\partial v} = \frac{\partial (6uv^2)}{\partial v} \]
\[ = 12uv \]
### Jacobian Matrix
By substituting these partial derivatives into the Jacobian matrix, we get:
\[ J = \begin{bmatrix}
2u + 8v & 8u \\
6v^2 & 12uv
\end{bmatrix} \]
### Final Answer
Thus, the Jacobian matrix \( J \) for the given transformation is:
\[ \boxed{\begin{bmatrix} 2u + 8v & 8u \\ 6
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