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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![### Jacobian Determinant Calculation
#### Problem Statement
Find the Jacobian,
\[ \frac{\partial(x,y,z)}{\partial(s,t,u)} \]
where the transformation is given by the equations:
\[ x = s - 3t + u \]
\[ y = s + t - 5u \]
\[ z = s + 3t - 5u \]
#### Solution
To find the Jacobian determinant, we need to compute the partial derivatives of \(x\), \(y\), and \(z\) with respect to \(s\), \(t\), and \(u\) and arrange them in a matrix:
\[
J = \begin{vmatrix}
\frac{\partial x}{\partial s} & \frac{\partial x}{\partial t} & \frac{\partial x}{\partial u} \\
\frac{\partial y}{\partial s} & \frac{\partial y}{\partial t} & \frac{\partial y}{\partial u} \\
\frac{\partial z}{\partial s} & \frac{\partial z}{\partial t} & \frac{\partial z}{\partial u}
\end{vmatrix}
\]
The matrix of partial derivatives, known as the Jacobian matrix, will be evaluated as follows:
\[
\frac{\partial(x,y,z)}{\partial(s,t,u)} =
\begin{vmatrix}
\frac{\partial(s - 3t + u)}{\partial s} & \frac{\partial(s - 3t + u)}{\partial t} & \frac{\partial(s - 3t + u)}{\partial u} \\
\frac{\partial(s + t - 5u)}{\partial s} & \frac{\partial(s + t - 5u)}{\partial t} & \frac{\partial(s + t - 5u)}{\partial u} \\
\frac{\partial(s + 3t - 5u)}{\partial s} & \frac{\partial(s + 3t - 5u)}{\partial t} & \frac{\partial(s + 3t - 5u)}{\partial u}
\end{vmatrix}
\]
Therefore, the solved determinants will fill as follows:
\[
\frac{\partial(x,y,z)}{\partial(s,t,u)} =
\begin{vmatrix}
1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbbff2935-77bb-4550-bfd1-d595e6271f30%2F894dd5f8-c849-4ad8-a3bb-7a7f8583bca2%2Fgcwu34e_processed.png&w=3840&q=75)
Transcribed Image Text:### Jacobian Determinant Calculation
#### Problem Statement
Find the Jacobian,
\[ \frac{\partial(x,y,z)}{\partial(s,t,u)} \]
where the transformation is given by the equations:
\[ x = s - 3t + u \]
\[ y = s + t - 5u \]
\[ z = s + 3t - 5u \]
#### Solution
To find the Jacobian determinant, we need to compute the partial derivatives of \(x\), \(y\), and \(z\) with respect to \(s\), \(t\), and \(u\) and arrange them in a matrix:
\[
J = \begin{vmatrix}
\frac{\partial x}{\partial s} & \frac{\partial x}{\partial t} & \frac{\partial x}{\partial u} \\
\frac{\partial y}{\partial s} & \frac{\partial y}{\partial t} & \frac{\partial y}{\partial u} \\
\frac{\partial z}{\partial s} & \frac{\partial z}{\partial t} & \frac{\partial z}{\partial u}
\end{vmatrix}
\]
The matrix of partial derivatives, known as the Jacobian matrix, will be evaluated as follows:
\[
\frac{\partial(x,y,z)}{\partial(s,t,u)} =
\begin{vmatrix}
\frac{\partial(s - 3t + u)}{\partial s} & \frac{\partial(s - 3t + u)}{\partial t} & \frac{\partial(s - 3t + u)}{\partial u} \\
\frac{\partial(s + t - 5u)}{\partial s} & \frac{\partial(s + t - 5u)}{\partial t} & \frac{\partial(s + t - 5u)}{\partial u} \\
\frac{\partial(s + 3t - 5u)}{\partial s} & \frac{\partial(s + 3t - 5u)}{\partial t} & \frac{\partial(s + 3t - 5u)}{\partial u}
\end{vmatrix}
\]
Therefore, the solved determinants will fill as follows:
\[
\frac{\partial(x,y,z)}{\partial(s,t,u)} =
\begin{vmatrix}
1
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