Problem 6 - Calculus Early Transcendentals

Question

Let \(S=\\{(u, v): 0 \leq u \leq 1\) \(0 \leq v \leq 1\\}\) be a unit square in the uv-plane. Find the image of \(S\) in the xy-plane under the following transformations. $$T: x=-u, y=-v$$

Step-by-Step Solution

Answer

The image of the unit square S in the xy-plane under the given transformation T is a unit square with opposite corners at (0,0) and (-1,-1).

Step 1: Identify the corners of the unit square S

The unit square S has the following corners: \((0,0)\), \((1,0)\), \((0,1)\), and \((1,1)\).

Step 2: Apply the transformation T to each corner

Now, we need to apply the given transformation T to each of the corners: 1. For corner \((0,0)\): $$x = -u = -0 = 0$$ $$y = -v = -0 = 0$$ So, the new coordinates are \((0,0)\). 2. For corner \((1,0)\): $$x = -u = -1$$ $$y = -v = -0 = 0$$ So, the new coordinates are \((-1,0)\). 3. For corner \((0,1)\): $$x = -u = -0 = 0$$ $$y = -v = -1$$ So, the new coordinates are \((0,-1)\). 4. For corner \((1,1)\): $$x = -u = -1$$ $$y = -v = -1$$ So, the new coordinates are \((-1,-1)\).

Step 3: Identify the new image in the xy-plane

After applying the transformation T to each corner of the unit square S, we get the following coordinates in the xy-plane: \((0,0)\), \((-1,0)\), \((0,-1)\), and \((-1,-1)\). These points form a unit square in the xy-plane with opposite corners at \((0,0)\) and \((-1,-1)\). So, the image of the unit square S in the xy-plane under the given transformation T is a unit square with opposite corners \((0,0)\) and \((-1,-1)\).

Key Concepts

Unit Square

A unit square is a fundamental concept in geometry and is defined as a square with side lengths equal to one unit. It is usually denoted in a coordinate system, such as the uv-plane or xy-plane, and is often used to simplify problems involving transformations.

In our context, the unit square in the uv-plane is defined by points bounded within:

• The horizontal axis from 0 to 1 (0 ≤ u ≤ 1)
• The vertical axis from 0 to 1 (0 ≤ v ≤ 1)

The corners of this square are represented with the coordinates (0,0), (1,0), (0,1), and (1,1). These points mark the vertices of the square, giving it a fixed space in the uv-plane. Understanding this structure is crucial as it provides a starting reference when applying transformations to map this square to other planes.

xy-plane

The xy-plane is another traditional coordinate plane that is commonly used in mathematical contexts. It is a two-dimensional flat surface where any point can be represented by a unique pair of numbers:

• The x-coordinate
• The y-coordinate

These numbers determine the point's horizontal and vertical positions, respectively.

In problems related to coordinate transformation, the xy-plane serves as the destination plane. Here, shapes and points from the uv-plane are transformed to, using mathematical transformations. This plane is a part of the Cartesian coordinate system, a format that is widely used to handle multi-dimensional data. Understanding the structure and representation of points in the xy-plane is fundamental when mapping or transforming coordinates from another plane, such as our unit square from the uv-plane.

uv-plane

The uv-plane uses the letters 'u' and 'v' for its coordinates, which can sometimes represent a different space or a different set of characteristics compared to the typical xy-plane.

In many transformation problems, starting with the uv-plane allows us to clearly define original shapes or positions before applying changes or shifts to a different plane, such as the xy-plane. One can think of the uv-plane as a placeholder or a reference sheet where initial data or geometric figures are set up for transformation. It plays a critical role in scenarios where mapping transformations are used to transpose points into another space, making it essential to understand in such exercises. By studying transformations that involve the uv-plane, you can deepen your understanding of coordinate systems and how they interrelate through various transformations.

Transformation

Transformation is the process in which coordinates from one plane are mathematically mapped to another plane, following a set of rules or formulas.

In our exercise, the transformation involves a mapping defined by:

• For the x-coordinate: \( x = -u \)
• For the y-coordinate: \( y = -v \)

This specific transformation inverts each coordinate, flipping the unit square from the uv-plane into the xy-plane.

Through this operation, each point on the unit square finds new locations based on the transformation rules, effectively forming a new image in the xy-plane.Transformations are powerful mathematical tools used for various applications, such as computer graphics, physics, and engineering problems. Understanding how transformations work is essential for applying changes to coordinates, which is a common necessity in advanced mathematics.