Problem 11 - APEX Calculus

Question

In Exercises 11-14, describe the region in space defined by the inequalities. \(x^{2}+y^{2}+z^{2}<1\)

Step-by-Step Solution

Answer

The region is an open ball (interior of a sphere) of radius 1 centered at the origin.

Step 1: Analyzing the Inequality

The inequality \(x^2 + y^2 + z^2 < 1\) represents a geometric shape. We begin by comparing it to the equation \(x^2 + y^2 + z^2 = 1\), which defines a sphere centered at the origin (0, 0, 0) with a radius of 1.

Step 2: Understanding the Sphere's Properties

The inequality uses the spherical equation \(x^2 + y^2 + z^2 = 1\) to describe a complete sphere. Therefore, for the inequality \(x^2 + y^2 + z^2 < 1\), it defines the interior of this sphere. This means it includes all the points inside the sphere but excludes those on the surface.

Step 3: Defining the Region in Space

The inequality \(x^2 + y^2 + z^2 < 1\) describes a region in space that contains all points within a sphere of radius 1, centered at the origin. It is not just the surface; it includes everything inside up to just before the boundary. Hence, the region is an open ball with radius 1 centered at the origin.

Key Concepts

Spheres

A sphere is a perfectly symmetrical geometric shape. It is defined as the set of all points in three-dimensional space that are at a fixed distance, known as the radius, from a given point called the center. Imagine the globe we live on; it's an example of a three-dimensional sphere. In mathematical terms, if the sphere's center is at the origin

• the center of the coordinate system (0, 0, 0),
• the equation would be \[ x^2 + y^2 + z^2 = r^2 \]
• where \(r\) is the radius.

For instance, an equation such as \[ x^2 + y^2 + z^2 = 1 \] specifically describes a sphere centered at the origin with a radius of 1. In essence, it includes all the space on the boundary at exactly a 1-unit distance from the center in all directions. Essentially, each point on this sphere is equidistant from the center, forming a closed surface that is uniform in every dimension.

Inequalities

When dealing with mathematical expressions, inequalities are statements that one quantity is less than or greater than another. They help in defining ranges and boundaries within mathematical spaces. Let's say you have an inequality like \[ x^2 + y^2 + z^2 < 1 \]. This inequality means that we are looking at the set of all points that are strictly inside a unit sphere, not touching its surface. In other words, all combinations of \(x, y, z\) such that

• their squared sum is less than 1
• fit inside this region.

Unlike equations, which denote exact positions or surfaces, inequalities signal entire regions or volumes. They are essential in defining areas like 'inside' or 'outside' a given shape. In this context, inequalities create what's known as an 'open' region because the boundary points of the sphere are not included in the set.

Open Ball

An open ball is a concept that describes a region in space, particularly within the field of metric spaces, without including the boundary. Think of it as a sphere that lacks its outer shell. In three-dimensional geometry, if you have a center (say at the origin) and a radius, the space inside this boundary is termed an 'open ball'. Consider the inequality \[ x^2 + y^2 + z^2 < 1 \], which defines an open ball around the origin with a radius of 1. It encompasses

• every point inside the spherical space,
• omitting those exactly on its boundary.

An open ball is crucial because it helps delineate spaces that are entirely contained within a spherical boundary. These allow mathematicians and scientists to specify and study the parts of space that are accessible or affected within these limits, ensuring boundary points do not influence results or considerations in certain analyses.