Question

A surface \(\mathcal{S}\) in space is described that cannot be defined in terms of a function \(z=f(x, y)\). Give a parametrization of \(\mathcal{S}\). \(\mathcal{S}\) is the rectangle in space with corners at \((0,0,0),(0,2,0),\) (0,2,1) and (0,0,1).

Short answer

The parametrization of \(\mathcal{S}\) is \(r(v, w) = (0, v, w)\) with \(0 \le v \le 2\) and \(0 \le w \le 1\).

Step-by-step solution

Identify the Properties of the Surface

The described surface \(\mathcal{S}\) is a rectangle in space. The given corners are \((0,0,0)\), \((0,2,0)\), \((0,2,1)\), and \((0,0,1)\). This rectangle lies on a plane parallel to the yz-plane at \(x=0\). The lengths of sides suggest it spans from 0 to 2 along the y-axis and from 0 to 1 along the z-axis.

Determine Parametric Equations

To parameterize the rectangle, we need two parameters. Let's use \(v\) for the y-direction and \(w\) for the z-direction. The parametric equations will be \(x = 0\), \(y = v\), and \(z = w\). The parameters \(v\) and \(w\) will range as: \(0 \le v \le 2\) and \(0 \le w \le 1\).

Define the Parametric Representation

The parametric representation of the surface \(\mathcal{S}\) is given by \(r(v, w) = (0, v, w)\), where \(0 \le v \le 2\) and \(0 \le w \le 1\). This equation traces each point on the rectangle as \(v\) varies along the y-axis and \(w\) varies along the z-axis.

Key concepts

Parametric Equations

Parametric equations are a fundamental tool in mathematics, especially in the context of describing geometric surfaces. Unlike regular equations that express variables in terms of each other, parametric equations use one or more auxiliary parameters to capture more complex movements in space. These parameters provide a way to capture the dynamics of curves or surfaces that may not fit into traditional function formats, like a plane in three-dimensional space.
By defining two or more equations with these parameters, we can describe a path that traces a curve or a surface. This is particularly useful for surfaces that cannot be defined by a single expression like functions of the form \(z=f(x, y)\). In our rectangle example, the parameters \(v\) and \(w\) describe the ranges along the y and z axes respectively, capturing the breadth and height of the rectangle. It showcases how parametric equations offer great flexibility, allowing us to break free from the restrictions of traditional function definitions.

Rectangle in Space

A rectangle in space is a four-sided flat shape with distinct corners, just as it appears on a sheet of paper, but existing within a three-dimensional environment. In this scenario, we've been tasked with parametrizing a rectangle defined by the corners \((0,0,0)\), \((0,2,0)\), \((0,2,1)\), and \((0,0,1)\).
What makes this rectangle unique is its positioning. It lies parallel to the yz-plane, and extends along the y-axis from 0 to 2, while along the z-axis from 0 to 1. The importance of understanding these spatial features cannot be overstated. Recognizing how these corners form a rectangle in space enables the correct application of parametric equations to describe the geometry in its entirety.

• The parameter \(v\) captures the movement along the y-axis.
• The parameter \(w\) defines motion along the z-axis.

YZ-plane

The yz-plane is one of the coordinate planes in three-dimensional space, and it plays a crucial role in this exercise. The yz-plane is essentially a plane where the x-coordinate is zero. This means any point or surface lying entirely in this plane will have an equation where \(x = 0\).
Understanding the yz-plane helps in visualizing spatial structures. In our problem, the rectangle is parallel to and lies completely on this plane. This understanding guided the creation of our parametric equations where \(x = 0\), emphasizing the entirety of the rectangle exists within the yz-plane.
Visualizing spatial relationships, such as how a shape interacts with a particular plane, enhances comprehension of three-dimensional parametric equations. It's the key to unlocking how different objects and surfaces exist and interact within 3D spaces, providing essential groundwork for further studies in mathematics and sciences.