Problem 17 - 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. $$ \begin{array}{l}{\text { a. } x^{2}+y^{2}+z^{2}=1, \quad z \geq 0} \\ {\text { b. } x^{2}+y^{2}+z^{2} \leq 1, \quad z \geq 0}\end{array} $$

Step-by-Step Solution

Answer

Part a is the upper hemisphere surface; part b is the solid upper hemisphere.

Step 1: Understand the Equation

For part a, the equation \(x^2 + y^2 + z^2 = 1\) represents a sphere in 3D space with a radius of 1, centered at the origin \((0, 0, 0)\). The condition \(z \geq 0\) means we are only considering the upper hemisphere of this sphere where the z-coordinates are non-negative.

Step 2: Analyze Part a

Combining both the equation and the condition \(z \geq 0\), the set of points is the upper hemisphere of the unit sphere, which includes all points on the sphere's surface that rest in the half-space above the xy-plane.

Step 3: Understand the Inequality

For part b, the inequality \(x^2 + y^2 + z^2 \leq 1\) describes a solid sphere (including its interior) with radius 1 centered at the origin. The condition \(z \geq 0\) limits this region to the top half of the sphere, similar to the restriction in part a.

Step 4: Analyze Part b

The combination of the inequality and the condition \(z \geq 0\) results in a solid (filled) upper hemisphere. This includes all points with \(z \geq 0\) that lie inside or on the surface of the unit sphere.

Key Concepts

3D Coordinate System

When we consider the 3D coordinate system, we visualize it as a space where each point is defined by three coordinates: \(x\), \(y\), and \(z\). Unlike the 2D coordinate system, which is flat and consists of the axes \(x\) and \(y\), the 3D system introduces an additional dimension: height or depth, which is the \(z\)-axis.
In this system, the origin is at the point \((0, 0, 0)\), where all three axes intersect.

• The \(x\)-axis allows movement left or right.
• The \(y\)-axis allows movement forward or backward.
• The \(z\)-axis allows movement up or down.

This coordinate system is used in geometry to locate points in space and to describe the position of objects in a three-dimensional environment. Each axis is perpendicular to the other two, creating orthogonal lines that help clearly define every point's location. This understanding is crucial for visualizing and solving problems involving distances, volumes, and areas in space.

Unit Sphere

A unit sphere in a 3D coordinate system is a special sphere. It is centered at the origin \((0, 0, 0)\) and has a radius of 1. The equation of a unit sphere is \[ x^2 + y^2 + z^2 = 1\]This equation tells us that the sum of the squares of the coordinates \(x\), \(y\), and \(z\) always equals 1. This sum ensures that every point on the surface of this sphere is exactly one unit away from the origin.
The unit sphere is a fundamental concept in geometry because it provides a simple model for understanding spherical shapes and distances from a central point. It's frequently used in exercises to build an intuition about spheres and the concept of radius. When we look at subsets of a unit sphere, like only the upper hemisphere where \(z \geq 0\), we apply additional conditions to focus our view on specific sections of this sphere.

Inequalities in Space

Inequalities are essential for describing regions in space rather than just surfaces or lines. For example, the inequality \[x^2 + y^2 + z^2 \leq 1\]defines the volume of a sphere, including all points inside it along with its surface. This inequality specifies that the distance from the origin to any point \((x, y, z)\) is less than or equal to 1.
When combined with other conditions, such as \(z \geq 0\), we can describe even more specific areas in space. Here, the inequality and the condition come together to define a solid upper hemisphere. We include both the interior volume and the surface of the hemisphere, allowing for the description of a more complex, filled 3D region.
Such inequalities are vital tools in geometry and calculus, as they let us express and solve for bounded regions in space, enabling a deeper understanding of spatial volumes and surfaces.