Question

A domain \(D\) in space is given. Parametrize each of the bounding surfaces of \(D\). \(D\) is the domain bounded by the planes \(z=\frac{1}{2}(3-x), x=1\) \(y=0, y=2\) and \(z=0\).

Short answer

The domain \(D\) is parametrized by its bounding top, bottom, and side surfaces using their respective plane equations and variable ranges.

Step-by-step solution

Understand the Domain

First, identify the bounding conditions for the domain \(D\). The domain is bounded by planes given by the equations: \(z = \frac{1}{2}(3-x)\), \(x = 1\), \(y = 0\), \(y = 2\), and \(z = 0\). This forms a three-dimensional space with limits in the \(xy\) plane and in depth along \(z\).

Parametrize the Top Surface

The top surface is described by the plane \(z = \frac{1}{2}(3-x)\). We can parametrize this surface with variables \(x\) and \(y\) varying over their respective ranges. Let \(x\) and \(y\) vary as free parameters. Then, the parametrization is \(r_{top}(x, y) = (x, y, \frac{1}{2}(3-x))\) for \(0 \leq x \leq 1\) and \(0 \leq y \leq 2\).

Parametrize the Bottom Surface

The bottom surface is simply the \(xy\)-plane where \(z=0\). Using parameters \(x\) and \(y\) again, the parametrization is \(r_{bottom}(x, y) = (x, y, 0)\) for \(0 \leq x \leq 1\) and \(0 \leq y \leq 2\).

Parametrize the Plane x = 1

For the plane \(x = 1\), set \(x = 1\) and let \(y\) and \(z\) vary. Thus, the parametrization is \(r_{x=1}(y, z) = (1, y, z)\) where \(0 \leq y \leq 2\) and \(0 \leq z \leq \frac{1}{2}(3-1) = 1\).

Parametrize the Plane y = 0

For the plane \(y = 0\), set \(y = 0\) and allow \(x\) and \(z\) to vary. The parametrization is \(r_{y=0}(x, z) = (x, 0, z)\) with \(0 \leq x \leq 1\) and \(0 \leq z \leq \frac{1}{2}(3-x)\).

Parametrize the Plane y = 2

For the plane \(y = 2\), set \(y = 2\) and let \(x\) and \(z\) vary. So, the parametrization is \(r_{y=2}(x, z) = (x, 2, z)\) for \(0 \leq x \leq 1\) and \(0 \leq z \leq \frac{1}{2}(3-x)\).

Review and Consolidate

We've now parametrized all bounding surfaces of domain \(D\). Each surface has been described using two free parameters, and their corresponding ranges have been identified, ensuring they match the geometric configuration of the problem.

Key concepts

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions involving more than one variable. This branch of mathematics is essential for understanding and solving problems in three-dimensional spaces, where phenomena depend on multiple factors.
Calculating derivatives and integrals of functions with several variables, multivariable calculus allows you to analyze how variables change simultaneously. It also introduces new concepts like partial derivatives and multiple integrals.
When dealing with surfaces and volumes, parameters in multivariable calculus help define the limits of integration. This is crucial for determining the behavior and constraints of objects in space.

• Partial derivatives: Show how a function changes with respect to one variable while keeping others constant.
• Multiple integrals: Used to compute volumes and areas in higher dimensions.
• Gradient and differential operators: Help understand directional changes along surfaces.
• Applications: From physics to engineering, these concepts allow for the solution of more complex problems.

Three-Dimensional Geometry

Three-dimensional geometry involves the study of objects in a three-dimensional space defined by axes: usually labeled as x, y, and z. This branch of mathematics explores spatial relationships and properties between figures. It is fundamental in visualizing and solving problems involving surfaces like the domain described.
It examines shapes, planes, lines, and points within this space, offering a broader view of simple shapes extended into three dimensions.
In this context, understanding how different planes intersect and define shapes is paramount to solving exercises based in 3D geometry. This includes:

• Planes: Flat surfaces that extend infinitely in two directions.
• Lines: The shortest distance between two points in space, lying on planes.
• Shapes and Solids: Objects defined by multiple surfaces, like polyhedra.
• Coordinates: Points in space defined by (x, y, z) allowing for easy location of entities.

Bounding Planes

Bounding planes are used to enclose a region or space in a three-dimensional setting. These planes act as the limits or boundaries for a space, such as the domain described in the problem.
An understanding of bounding planes is essential because it helps to visualize and define the problem's space accurately. These conceptual borders transform the abstract notion of a volume into a mathematical equation using coordinates and geometry principles.
In our domain example, specific planes z, x, and y bounds constrain the region:

• Top and bottom planes define the vertical extent (involving z values).
• Side planes set horizontal boundaries (involving x and y values).

Key to mastering these concepts is understanding how these planes interact to form enclosed spaces, crucial for integrations and calculations in calculus.

Parametric Equations

Parametric equations allow the representation of surfaces or curves in three-dimensional space. Instead of expressing a surface as a function of independent variables, you express all coordinates in terms of one or more parameters.
This approach is beneficial for defining complex surfaces, where traditional functions might fall short. Each parameter often represents a dimension, with the combination of parameters mapping out positions on the surface.
For the domain given, parameters x, y, and z are represented with equations relying on free variation of two parameters to describe distinct planes:

• Flexible framework: Easily adapt to describe varied shapes.
• Mapping: Link each point on a surface to parameters.
• Simplicity: Simplify calculations by using shared parameters.
• Applications: In engineering and graphing, especially for non-linear shapes.

The ability to precisely define the edges and surfaces of a region using parametric equations is a crucial skill in both geometry and calculus.