Problem 32 - Thomas Calculus

Question

Write inequalities to describe the sets in Exercises \(29-34\) The upper hemisphere of the sphere of radius 1 centered at the origin

Step-by-Step Solution

Answer

The set is described by the inequalities \(x^2 + y^2 + z^2 \leq 1\) and \(z \geq 0\).

Step 1: Understand the Problem

We need to describe the inequalities that define the upper hemisphere of a sphere. A sphere centered at the origin with radius 1 is defined by the equation \(x^2 + y^2 + z^2 = 1\). The upper hemisphere includes all the points where the z-coordinate is non-negative.

Step 2: Sphere Equation

The equation \(x^2 + y^2 + z^2 = 1\) represents the entire sphere with radius 1 centered at the origin. Points on this sphere satisfy the condition that the sum of the squares of their coordinates equals 1.

Step 3: Restrict to Upper Hemisphere

To describe only the upper hemisphere, we need to add the condition that \(z \geq 0\) to ensure we only include the upper part of the sphere where the z-coordinate is non-negative.

Step 4: Write the Inequalities

Combine the conditions to form the inequalities that describe the set. The inequalities are: \(x^2 + y^2 + z^2 \leq 1\) and \(z \geq 0\). This restricts the set to the upper hemisphere of the sphere.

Key Concepts

Sphere centered at origin

A sphere centered at the origin in a 3D coordinate system is a perfect representation of symmetry in geometry. It is a set of all points that are equidistant from a central point—in this case, the origin of the coordinate system. The center of the sphere being at the origin means that its coordinates are (0, 0, 0). This is a convenient position because it simplifies many computations and visualizations. To better understand, visualize a globe where the center is the core of the Earth. Every point on the surface of the globe can be described using a simple equation when the globe is correctly aligned with the coordinate system. This simplicity provides a foundation for exploring more complex shapes and forms in three-dimensional space. When dealing with a sphere centered at the origin, its mathematical description is streamlined, making it easier to identify properties such as symmetry and uniformity. This spherical shape is also used to define various restrictions, such as hemispheres or specific segments, using additional conditions.

Radius of a sphere

The radius of a sphere is a fundamental concept that defines the extent of the sphere in all directions from its center. The radius is always a positive real number and acts as the "length of a spoke" from the center to any point on the surface of the sphere. In the context of the equation for a sphere, which is given by \[x^2 + y^2 + z^2 = R^2,\]where \(R\) is the radius, it clearly demonstrates how far the sphere extends into space. Here, every point that satisfies this equation lays exactly on the surface of the sphere. For our case, a sphere with radius 1 at the origin, the equation becomes\[x^2 + y^2 + z^2 = 1.\]Some key points about the radius:

• The radius determines the size of the sphere. Larger radii mean larger spheres.
• It is crucial for calculating the volume and surface area of the sphere.
• Since the radius represents the distance from the center, it also defines all aspects of the geometry and symmetry involved.

3D coordinate system

A 3D coordinate system is an extension of the more familiar 2D coordinate system. It allows us to describe locations and shapes in three-dimensional space, rather than the flat plane of 2D. This system uses three axes: the X-axis, Y-axis, and Z-axis. Each axis is typically perpendicular to the other two, establishing a three-dimensional grid that helps us specify the location of any point precisely by a triplet of numbers: (x, y, z). For example:

• The x-coordinate denotes the position along the X-axis.
• The y-coordinate indicates the position along the Y-axis.
• The z-coordinate marks the position along the Z-axis.

Within this framework, the concept of position and distance becomes easier to manage, especially when dealing with objects like spheres, cylinders, or cubes. The origin point (0, 0, 0) acts as the central reference point in such a coordinate system. This makes it foundational in geometry, physics, engineering, and various aspects of computer graphics or spatial analysis. Understanding the 3D coordinate system is essential for extending basic geometric calculations into more complex scenarios involving volume, area, and distance in a three-dimensional world.