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Chapter 13: Problem 10

Find the image of the rectangle with the given corners and find the Jacobian of the transformation. \(x=u, y=u^{2}-v^{2} ;(0,0),(3,0),(3,1),(0,1)\)

Short Answer

Expert verified
Images: (0,0), (3,9), (3,8), (0,-1). Jacobian determinant: \(-2v\).

Step by step solution

01

Understanding the Transformation

We have the transformation given by the equations \( x = u \) and \( y = u^2 - v^2 \). This means that for a given point \((u, v)\) in the parameter space, the point \((x, y)\) can be calculated using these equations.
02

Determine Image of Corner (0,0)

For corner \((u, v) = (0, 0)\), we substitute into the transformation equations: \( x = 0 \) and \( y = 0^2 - 0^2 = 0 \). Thus, the image is \((0, 0)\).
03

Determine Image of Corner (3,0)

For corner \((u, v) = (3, 0)\), we substitute into the transformation equations: \( x = 3 \) and \( y = 3^2 - 0^2 = 9 \). Thus, the image is \((3, 9)\).
04

Determine Image of Corner (3,1)

For corner \((u, v) = (3, 1)\), we substitute into the transformation equations: \( x = 3 \) and \( y = 3^2 - 1^2 = 8 \). Thus, the image is \((3, 8)\).
05

Determine Image of Corner (0,1)

For corner \((u, v) = (0, 1)\), we substitute into the transformation equations: \( x = 0 \) and \( y = 0^2 - 1^2 = -1 \). Thus, the image is \((0, -1)\).
06

Find the Jacobian

To find the Jacobian of the transformation, calculate the matrix of partial derivatives:\[J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 2u & -2v \end{bmatrix}\]
07

Calculate the Determinant of the Jacobian

The determinant of the Jacobian matrix is:\[\text{det}(J) = \left(1 \cdot (-2v) - (0 \cdot 2u)\right) = -2v\]Thus, the Jacobian determinant is \(-2v\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Jacobian Matrix
The Jacobian matrix is a fundamental structure in calculus, specifically in the context of transformations. It helps us understand how a transformation affects areas or volumes locally around a point. In our exercise, the transformation is given by the equations:
  • \( x = u \)
  • \( y = u^2 - v^2 \)
The Jacobian matrix is composed of partial derivatives of these equations with respect to the parameters \(u\) and \(v\). It includes:
  • \( \frac{\partial x}{\partial u} = 1 \), \( \frac{\partial x}{\partial v} = 0 \)
  • \( \frac{\partial y}{\partial u} = 2u \), \( \frac{\partial y}{\partial v} = -2v \)
This builds the matrix:\[J = \begin{bmatrix} 1 & 0 \ 2u & -2v \end{bmatrix}\]Understanding this matrix lays the groundwork for further calculations in transformations, such as determining the effect on area.
Determinant Calculation
The determinant of the Jacobian matrix provides insight into how the transformation scales area locally. Specifically, it tells us how the area around a point in the parameter space is stretched or shrunk when mapped to the image space.
For the given transformation, the determinant is calculated from the Jacobian matrix:
  • Jacobian matrix: \( J = \begin{bmatrix} 1 & 0 \ 2u & -2v \end{bmatrix} \)
  • Determinant formula: \( \text{det}(J) = (1)(-2v) - (0)(2u) = -2v \)
Thus, the determinant of the Jacobian is calculated as \(-2v\). This indicates that the transformation affects the area by a factor of \(-2v\), depending on the variable \(v\).
Image of Transformation
Finding the image of a transformation involves mapping specified points in one space (like a plane or rectangle) to another using transformation equations. In this exercise, we've provided four corners of a rectangle defined by parameter coordinates:\( (0,0), (3,0), (3,1), (0,1) \). To find the image in the \(x, y\) space:
  • Corner \((0, 0)\) is mapped to \((0, 0)\).
  • Corner \((3, 0)\) is mapped to \((3, 9)\).
  • Corner \((3, 1)\) is mapped to \((3, 8)\).
  • Corner \((0, 1)\) is mapped to \((0, -1)\).
These mappings show how the original rectangle's corners get transformed in the \(x, y\) plane using the given equations. This gives us a clear picture of the geometric implications of the transformation.

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