Problem 9 - 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. $$T: x=u^{2}-v^{2}, y=2 u v$$

Step-by-Step Solution

Answer

The image of the unit square S in the xy-plane under transformation T is a geometric figure with vertices (0, 0), (1, 0), (-1, 0), and (0, 2).

Step 1: Identify corner coordinates of the unit square

First, we need to identify the corner coordinates of the given unit square \(S\). Since it is a unit square in the uv-plane, the corners will have coordinates \((0,0)\), \((1,0)\), \((0,1)\), and \((1,1)\).

Step 2: Apply the transformation to each corner

Now we need to apply the given transformation \(T\) to the corner coordinates of \(S\). Transformation \(T\) is given by two equations: $$ x = u^2 - v^2 \\ y = 2uv $$ We will substitute the corner coordinates \((u, v)\) into these equations to find the corresponding \((x, y)\) coordinates in the xy-plane. Corner 1: \((u, v) = (0, 0)\) $$ x = (0)^2 - (0)^2 = 0 \\ y = 2(0)(0) = 0 $$ Transformed Corner 1: \((x, y) = (0, 0)\) Corner 2: \((u, v) = (1, 0)\) $$ x = (1)^2 - (0)^2 = 1 \\ y = 2(1)(0) = 0 $$ Transformed Corner 2: \((x, y) = (1, 0)\) Corner 3: \((u, v) = (0, 1)\) $$ x = (0)^2 - (1)^2 = -1 \\ y = 2(0)(1) = 0 $$ Transformed Corner 3: \((x, y) = (-1, 0)\) Corner 4: \((u, v) = (1, 1)\) $$ x = (1)^2 - (1)^2 = 0 \\ y = 2(1)(1) = 2 $$ Transformed Corner 4: \((x, y) = (0, 2)\)

Step 3: Determine the image of S in the xy-plane

Now that we have computed the coordinates of the transformed corners, we can determine the overall image of the square S in the xy-plane. We have the following corner coordinates in the xy-plane: Transformed Corner 1: \((0, 0)\) Transformed Corner 2: \((1, 0)\) Transformed Corner 3: \((-1, 0)\) Transformed Corner 4: \((0, 2)\) The image of S in the xy-plane under transformation T is a geometric figure formed by the four points: \((0, 0)\), \((1, 0)\), \((-1, 0)\), and \((0, 2)\).

Key Concepts

Unit Square

The unit square is a simple, yet fundamental concept in geometry. It is defined as a square with sides of length one. In the uv-plane, the unit square consists of all points

• where both coordinates, \(u\) and \(v\), range from 0 to 1.
• Formally expressed, this is the set \( S = \{(u, v): 0 \leq u \leq 1,\, 0 \leq v \leq 1\} \).
• The corners of a unit square in the uv-plane are (0,0), (1,0), (0,1), and (1,1).

Since it's such a regular shape, transformations applied to a unit square can help illustrate basic concepts of geometry. These corners are significant because transformations are often applied to them to understand how the entire shape changes.

Coordinate Transformation

Coordinate transformation involves changing the coordinates of points from one system to another. In our exercise, we are transforming points from the uv-plane to the xy-plane. The given transformation equations are:

• \( x = u^2 - v^2 \)
• \( y = 2uv \)

These equations map each point \((u, v)\) to a new point \((x, y)\). Such transformations can dramatically alter the original shape. By using these equations, the pattern and layout of points from the initial system change, allowing us to analyze different geometric properties.

xy-plane

The xy-plane is a two-dimensional coordinate plane defined by x and y axes. Transformations such as the one in this exercise enable us to visualize points from a different system, like the uv-plane, in the xy-plane.

• This is especially useful for understanding changes in shapes due to operations such as translation, scaling, or rotation.
• In this specific instance, the corners of the unit square from the uv-plane take new positions in the xy-plane once transformed.

Understanding the xy-plane is crucial in graphing and visualizing mathematical relationships, showing how figures are manipulated and perceived through transformations.

Transformed Coordinates

Transformed coordinates result from applying transformation equations to original points. The transformation \(T\) in this exercise takes points \((u, v)\) from the unit square and moves them to new points \((x, y)\) in the xy-plane.The transformed corners are:

• From \((0,0)\) to \((0,0)\)
• From \((1,0)\) to \((1,0)\)
• From \((0,1)\) to \((-1,0)\)
• From \((1,1)\) to \((0,2)\)

These coordinates form a new shape in the xy-plane.
By identifying the transformed coordinates, we see how the original unit square is stretched and shifted. Each transformation expresses certain properties and symmetries, offering deeper insight into geometric manipulations.