Problem 7 - 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. $$ x=u+2 v, y=u-2 v ;(0,0),(2,0),(2,1),(0,1) $$

Step-by-Step Solution

Answer

Image rectangle: [(0,0), (2,2), (4,0), (2,-2)]; Determinant of Jacobian: -4.

Step 1: Compute the Jacobian

\(J = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix} = \begin{vmatrix} 1 & 2 \\ 1 & -2 \end{vmatrix} = -2 - 2 = -4\)

Step 2: Find the image

Map the corners: \((0,0)\to(0,0)\), \((2,0)\to(2,2)\), \((2,1)\to(4,0)\), \((0,1)\to(2,-2)\). The image is a parallelogram (rhombus) with these vertices.

Key Concepts

Transformation Geometry

In transformation geometry, we explore how shapes change or move in space using specific rules or equations. It's a branch of geometry that involves mapping points from one coordinate system to another, which we often refer to as transforming a shape. To illustrate this, consider the transformation with equations \(x = u + 2v\) and \(y = u - 2v\). These equations show how any point \((u, v)\) will relocate and morph into a new point \((x, y)\). Think of it like folding or stretching shapes on paper but using math to describe every move. Transformation geometry is crucial because:

• It helps us visualize algebraic equations.
• We can predict how objects behave under transformations.
• Engineers use it for design prototypes.

For our rectangle, the transformation suggests how each corner will shift when mapped to a new grid. This could change its size, shape, or orientation altogether.

Coordinate Transformation

When discussing coordinate transformation, we're delving into the idea of changing the way we describe point locations in space. Imagine two different map systems: one for your phone's GPS and another for a traditional paper map. Both can guide you to the same location but use different systems. The transformation equations \(x = u + 2v\) and \(y = u - 2v\) represent this shift from \((u, v)\) coordinates to \((x, y)\) coordinates. Such transformations are crucial for:

• Aligning different data sets for analysis.
• Converting between models in computer graphics and simulations.
• Understanding spatial relationships from different perspectives.

By applying the given transformations to the initial rectangle's corners, we observe not only how each point relocates but how the entire figure transforms in its new coordination framework.

Multivariable Calculus

Multivariable calculus extends the basics of calculus to functions of two or more variables. It allows us to examine concepts of change that occur in higher dimensions. For instance, the Jacobian matrix is a valuable tool in this realm. When dealing with transformations, particularly those modifying an entire shape or surface in space, the Jacobian matrix helps us study the behavior and rate of change of the transformation itself. Given the transformation equations \(x = u + 2v\) and \(y = u - 2v\), the Jacobian is derived by computing partial derivatives:

• \(\frac{\partial x}{\partial u} = 1, \frac{\partial x}{\partial v} = 2\)
• \(\frac{\partial y}{\partial u} = 1, \frac{\partial y}{\partial v} = -2\)

The Jacobian matrix, which aids in this process, is:\[J = \begin{bmatrix} 1 & 2 \ 1 & -2 \end{bmatrix}\] This matrix helps determine how areas and volumes change under the transformation, which is vital for calculus in multiple dimensions and understanding the impact transformations have on an object's size or orientation.