a. Solve the system u = 3x+4y, v=x + 4y for x and y in terms of u and v. Then find the value of the Jacobian a(x,y) a(u,v) b. Find the image under the transformation u = 3x+4y, v=x+4y of the triangular region the xy-plane bounded by the x-axis, the y-axis, and the line x + y = 2. Sketch the transformed image in the uv-plane.
a. Solve the system u = 3x+4y, v=x + 4y for x and y in terms of u and v. Then find the value of the Jacobian a(x,y) a(u,v) b. Find the image under the transformation u = 3x+4y, v=x+4y of the triangular region the xy-plane bounded by the x-axis, the y-axis, and the line x + y = 2. Sketch the transformed image in the uv-plane.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 39RE
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![### Problem Statement
#### Part (a)
Solve the system \( u = 3x + 4y \), \( v = x + 4y \) for \( x \) and \( y \) in terms of \( u \) and \( v \). Then find the value of the Jacobian \(\frac{\partial(x,y)}{\partial(u,v)}\).
#### Solution:
Given the equations:
\[ u = 3x + 4y \]
\[ v = x + 4y \]
First, solve for \( x \) and \( y \) in terms of \( u \) and \( v \).
1. Subtract the second equation from the first:
\[
u - v = (3x + 4y) - (x + 4y)
\]
\[
u - v = 2x
\]
\[
x = \frac{u - v}{2}
\]
2. Substitute \( x = \frac{u - v}{2} \) into the second equation:
\[
v = \frac{u - v}{2} + 4y
\]
\[
2v = u - v + 8y
\]
\[
2v + v = u + 8y
\]
\[
3v = u + 8y
\]
\[
8y = 3v - u
\]
\[
y = \frac{3v - u}{8}
\]
So the solutions are:
\[ x = \frac{u - v}{2} \]
\[ y = \frac{3v - u}{8} \]
Next, find the Jacobian \(\frac{\partial(x,y)}{\partial(u,v)}\).
The Jacobian determinant \(J_{xy/uv}\) is given by:
\[ J = \begin{vmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{vmatrix} \]
Calculate the partial derivatives:
\[ \frac{\partial x}{\partial u} = \frac{1}{2}, \quad \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1840ebe-b0df-4fe8-9210-d8e3dcfa32cc%2F86a800c1-5679-42a4-b385-a1024cafa0f5%2Fdcnbivu_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
#### Part (a)
Solve the system \( u = 3x + 4y \), \( v = x + 4y \) for \( x \) and \( y \) in terms of \( u \) and \( v \). Then find the value of the Jacobian \(\frac{\partial(x,y)}{\partial(u,v)}\).
#### Solution:
Given the equations:
\[ u = 3x + 4y \]
\[ v = x + 4y \]
First, solve for \( x \) and \( y \) in terms of \( u \) and \( v \).
1. Subtract the second equation from the first:
\[
u - v = (3x + 4y) - (x + 4y)
\]
\[
u - v = 2x
\]
\[
x = \frac{u - v}{2}
\]
2. Substitute \( x = \frac{u - v}{2} \) into the second equation:
\[
v = \frac{u - v}{2} + 4y
\]
\[
2v = u - v + 8y
\]
\[
2v + v = u + 8y
\]
\[
3v = u + 8y
\]
\[
8y = 3v - u
\]
\[
y = \frac{3v - u}{8}
\]
So the solutions are:
\[ x = \frac{u - v}{2} \]
\[ y = \frac{3v - u}{8} \]
Next, find the Jacobian \(\frac{\partial(x,y)}{\partial(u,v)}\).
The Jacobian determinant \(J_{xy/uv}\) is given by:
\[ J = \begin{vmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{vmatrix} \]
Calculate the partial derivatives:
\[ \frac{\partial x}{\partial u} = \frac{1}{2}, \quad \
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