Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The image is a quadrilateral with vertices at (0,0), (3,9), (3,8), (0,-1) and the Jacobian determinant is \(-2v\).

Step 1: Understand the Transformation

We begin with the given transformation equations: \( x = u \) and \( y = u^2 - v^2 \). Our goal is to apply these equations to transform the corners of the rectangle from the \((u, v)\) coordinates to the \((x, y)\) coordinates.

Step 2: Transform the Corner (0,0)

For the corner \((0,0)\), substitute \( u = 0 \) and \( v = 0 \) into the transformation equations. We get \( x = 0 \) and \( y = 0^2 - 0^2 = 0 \). Thus, \((x, y) = (0, 0)\).

Step 3: Transform the Corner (3,0)

For the corner \((3,0)\), substitute \( u = 3 \) and \( v = 0 \) into the transformation equations. We get \( x = 3 \) and \( y = 3^2 - 0^2 = 9 \). So, \((x, y) = (3, 9)\).

Step 4: Transform the Corner (3,1)

For the corner \((3,1)\), substitute \( u = 3 \) and \( v = 1 \) into the transformation equations. This gives us \( x = 3 \) and \( y = 3^2 - 1^2 = 8 \). Thus, \((x, y) = (3, 8)\).

Step 5: Transform the Corner (0,1)

For the corner \((0,1)\), substitute \( u = 0 \) and \( v = 1 \) into the transformation equations. We get \( x = 0 \) and \( y = 0^2 - 1^2 = -1 \). So, \((x, y) = (0, -1)\).

Step 6: Identify the Image of the Rectangle

The image of the rectangle in the \((u,v)\) space through the transformation is a quadrilateral with transformed vertices \((0,0), (3,9), (3,8), (0,-1)\).

Step 7: Calculate the Jacobian of the Transformation

The Jacobian matrix \( J \) of the transformation is given by the partial derivatives:\[J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix}\]Where:- \( \frac{\partial x}{\partial u} = 1 \), \( \frac{\partial x}{\partial v} = 0 \)- \( \frac{\partial y}{\partial u} = 2u \), \( \frac{\partial y}{\partial v} = -2v \)Thus:\[J = \begin{bmatrix} 1 & 0 \ 2u & -2v \end{bmatrix}\]

Step 8: Determine the Determinant of the Jacobian

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

Key Concepts

Jacobian

In mathematics, the Jacobian is a central concept used to understand transformations of coordinates. It is essentially a matrix of all first-order partial derivatives of a vector function. When trying to understand how a transformation affects area, volume, or shape in multidimensional space, the Jacobian provides the scale of this transformation.

### How to Find the JacobianTo calculate the Jacobian for a transformation, organize variables in a function and calculate the partial derivatives:

• Given a function: \( x = u \) and \( y = u^2 - v^2 \),
• The Jacobian matrix, \( J \), is:
  \[ J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} \]

### Interpreting the DeterminantOnce the Jacobian matrix is constructed, its determinant helps to determine how much a small area around a point is scaled when transforming from one system of coordinates to another. A positive determinant indicates a scale up, while a negative suggests an inversion.

In our case, the determinant is \(-2v\), indicating that the transformation on the \((u, v)\) plane experiences compression/expansion dependent on \(v\) and an inversion depending on the sign of \(-2v\).

Coordinate Transformation

Coordinate transformation is a fundamental concept to help change perspective, location, or focus in mathematical problems. It involves altering the coordinate system to express a function or a problem differently.

### Example of Coordinate TransformationIn our original problem, the transformation from \( (u,v) \) coordinates to \( (x,y) \) coordinates via the functions

• \( x = u \)
• \( y = u^2 - v^2 \)

helps reshape a rectangle into a new geometric figure.

### Practical ImportanceSuch transformations are more than just academic exercises:

• They can simplify complex geometric problems.
• Coordinate transformations are pivotal in fields like computer graphics, physics, and engineering.

Through these transformations, calculations become more manageable, making abstract concepts more accessible.

Rectangular to Quadrilateral transformation

Understanding the transformation from a rectangle to a quadrilateral provides deep insights into geometric manipulations using math. Initially, we have a rectangular shape defined in \( (u,v) \) space by corners \( (0,0), (3,0), (3,1), (0,1) \). Using the transformation equations, each corner is mapped to a new point in \( (x,y) \) coordinates.

### The Transformation ProcessThrough the exercises, the rectangle gets its vertices remapped, creating a distorted yet fascinating quadrilateral:

• From \( (0,0) \) to \( (0,0) \)
• From \( (3,0) \) to \( (3,9) \)
• From \( (3,1) \) to \( (3,8) \)
• From \( (0,1) \) to \( (0,-1) \)

### Visualization in MathematicsThis transformation illustrates how algebraic manipulations map geometric figures into different shapes, crucial in visualization, simulations, and various analyses. By changing only our view or expression of coordinates, what was once a simple rectangle translates into a more complex geometric form.