Problem 10 - 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. $$T: x=2 u v, y=u^{2}-v^{2}$$

Step-by-Step Solution

Answer

Answer: The image of the unit square S in the xy-plane under the given transformation is a right-angled trapezoid.

Step 1: Find the vertices of the square S in the uv-plane

To find the vertices of the unit square S in the uv-plane, we will look at the possible combinations of u and v based on the given constraints: \(0 \leq u \leq 1\) and \(0 \leq v \leq 1\). The four vertices of the unit square S are: - Vertex A: \((u=0, v=0)\) - Vertex B: \((u=1, v=0)\) - Vertex C: \((u=1, v=1)\) - Vertex D: \((u=0, v=1)\)

Step 2: Apply the transformation T to the vertices of S

We will now apply the given transformation $$T: x=2 u v, y=u^{2}-v^{2}$$ to the vertices A, B, C, and D to find their images in the xy-plane. - Image of Vertex A under T: \((x=2(0)(0), y=(0)^{2}-(0)^{2}) = (0, 0)\) - Image of Vertex B under T: \((x=2(1)(0), y=(1)^{2}-(0)^{2}) = (0, 1)\) - Image of Vertex C under T: \((x=2(1)(1), y=(1)^{2}-(1)^{2}) = (2, 0)\) - Image of Vertex D under T: \((x=2(0)(1), y=(0)^{2}-(1)^{2}) = (0, -1)\)

Step 3: Find the shape of the image of S in the xy-plane

Now that we have found the images of the vertices of S under the transformation T, we will connect the points to determine the shape of the image of S in the xy-plane. - The image of Vertex A connects to the image of Vertex B: Connect point (0, 0) to (0, 1). - The image of Vertex B connects to the image of Vertex C: Connect point (0, 1) to (2, 0). - The image of Vertex C connects to the image of Vertex D: Connect point (2, 0) to (0, -1). - The image of Vertex D connects back to the image of Vertex A: Connect point (0, -1) to (0, 0). Upon connecting these points, we can see that the image of S in the xy-plane is a right-angled trapezoid with vertices at (0, 0), (0, 1), (2, 0), and (0, -1).

Key Concepts

Unit Square

A unit square is a square with sides of length one. It’s a fundamental shape often used in geometry to understand transformations. In the context of this exercise:

• The unit square is defined in the uv-plane with corners determined by the values of \(u\) and \(v\).
• Since both \(u\) and \(v\) can range from 0 to 1, the unit square will have sides parallel to the axis.
• The corners of the square are calculated by combining the minimum and maximum values of \(u\) and \(v\).

Understanding how these vertices transform helps reveal the behavior of functions and their geometric interpretations.

Vertices

Vertices are the corner points of geometric shapes. For the unit square in this problem:

• Vertex \(A\) is \( (0, 0) \)
• Vertex \(B\) is \( (1, 0) \)
• Vertex \(C\) is \( (1, 1) \)
• Vertex \(D\) is \( (0, 1) \)

Mapping these vertices through the transformation function \(T\) helps us determine how the entire shape changes:

• The transformation maps the original points to new locations in the xy-plane.
• This mapping visualizes how each corner moves, giving insight into the resulting shape.

Recognizing the roles of these vertices aids in understanding complex transformations.

Trapezoid

A trapezoid is a four-sided figure with at least one pair of parallel sides. In this exercise, the transformed unit square becomes a right-angled trapezoid:

• The vertices after transformation are \( (0, 0), (0, 1), (2, 0), \) and \( (0, -1) \).
• Connecting these points, we see one pair of parallel sides, forming the trapezoid.
• This shape is useful for visual interpretation of the transformation’s impact on the unit square.

The transformation turns the familiar square into a less symmetric shape, demonstrating how linear changes in inputs can lead to significant visual differences.

xy-plane

The xy-plane is the standard two-dimensional plane used in coordinate geometry:

• In this exercise, the xy-plane is the space where the transformed unit square is represented.
• Axes \(x\) and \(y\) are used to plot points and visualize transformations.
• The transformation \(T\) moves points from the uv-plane to new locations on this plane, showing the effect of the transformation.

The ability to switch between planes and understand how points change enhances one's grasp of geometric transformations and their applications. It's an essential skill for working with more advanced mathematical concepts.