Problem 15 - Thomas Calculus

Question

In Exercises \(13-18\) , describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. $$ (a)x^{2}+y^{2}+z^{2} \leq 1 \quad \text { b. } x^{2}+y^{2}+z^{2}>1 $$

Step-by-Step Solution

Answer

(a) Inside or on the sphere; (b) Outside the sphere.

Step 1: Identify the Inequality for (a)

The inequality given is \( x^2 + y^2 + z^2 \leq 1 \). This represents all points \((x, y, z)\) in space that satisfy this condition.

Step 2: Interpret the Set for (a)

The inequality \( x^2 + y^2 + z^2 \leq 1 \) describes the region within or on the surface of a sphere centered at the origin with radius 1.

Step 3: Identify the Inequality for (b)

The inequality given is \( x^2 + y^2 + z^2 > 1 \). This represents all points \((x, y, z)\) in space that satisfy this condition.

Step 4: Interpret the Set for (b)

The inequality \( x^2 + y^2 + z^2 > 1 \) describes the region outside of a sphere centered at the origin with radius 1, excluding the surface itself.

Key Concepts

Inequalities

Inequalities in 3D coordinate geometry play a crucial role in defining regions of space. An inequality, like the ones given in this exercise, helps determine whether a point

• satisfies a specific condition,
• belongs within a certain region,
• or lies outside it.

For example, the inequality \( x^2 + y^2 + z^2 \leq 1 \) includes not only points inside a sphere but also those on its surface. It indicates a boundary—or a limit—that points must meet or stay within. Similarly, \( x^2 + y^2 + z^2 > 1 \) signifies a set of points lying outside this sphere, meaning the sum of the squares of their respective coordinates must exceed the value of 1. This exclusion of the surface is key because these inequalities precisely describe bounded or unbounded spaces in the three-dimensional world.

Spheres

In this problem, the concepts of spheres aid in visualizing the constraints imposed by the inequalities. The equation \( x^2 + y^2 + z^2 = 1 \) defines a sphere centered at the origin

• with a radius of 1.
• The sphere is the collection of all points
  • (x, y, z)
  that maintain this specific distance from the center.

The inequality \( x^2 + y^2 + z^2 \leq 1 \) then takes this sphere and fills it with all its interior points, creating what is known as a solid sphere. This solid sphere includes everything from the very center to the edge of the sphere's surface. Conversely, \( x^2 + y^2 + z^2 > 1 \) describes all points in space outside the sphere, marking a three-dimensional region that exists beyond the surface of the defined sphere.

Coordinate systems

A Cartesian coordinate system, which this exercise uses, is fundamental for representing points in space. This coordinate system

• defines each point by a set of numerical values,
  • or coordinates \((x, y, z)\),
• relative to three perpendicular axes
• (x-axis, y-axis, and z-axis).

By plotting these coordinates, one can visualize geometric shapes and regions such as the equations and inequalities represent. In this exercise, the origin—designated as the point where all three axes intersect—becomes the center of the sphere, illustrating how each point's distance in space is evaluated based on its coordinates. Understanding coordinate systems allows us to parse complex spatial relationships easily, as positions and distances can be consistently calculated and interpreted.