Problem 48 - University Calculus: Early Transcendentals

Question

Write inequalities to describe the sets. The upper hemisphere of the sphere of radius 1 centered at the origin

Step-by-Step Solution

Answer

The inequalities are \\(x^2 + y^2 + z^2 \\leq 1\\) and \\(z \\geq 0\\).

Step 1: Recognize the Sphere Equation

A sphere of radius 1 centered at the origin can be described by the equation \(x^2 + y^2 + z^2 = 1\). The equation represents all points in 3D space that are exactly 1 unit away from the origin, forming a complete sphere.

Step 2: Identify the Upper Hemisphere

The upper hemisphere is the part of the sphere where the \(z\)-coordinate is positive. This means we should only consider points where \(z \geq 0\). This restriction ensures that we only have the upper half of the sphere.

Step 3: Write the Inequalities

To describe the upper hemisphere using inequalities, combine the equation of the sphere with the condition for the upper hemisphere. The inequalities become \(x^2 + y^2 + z^2 \leq 1\) and \(z \geq 0\). The first inequality represents all points contained within or on the boundary of the sphere, while the second ensures we're only considering the top half of the sphere.

Key Concepts

Sphere Equation

Understanding the sphere equation is crucial in describing various aspects of 3D geometry. A sphere in mathematical terms is a three-dimensional object where all points on the surface are equidistant from a central point. This distance is the radius. For a sphere centered at the origin (0,0,0) with a radius of 1, the equation is simply:

• \(x^2 + y^2 + z^2 = 1\)

This equation signifies that any point \((x, y, z)\) satisfying it lies exactly on the surface of the sphere, precisely 1 unit away from the center, which is the origin in this case.
It’s essential to know that we use squared terms to express the distances, capturing the notion of a sphere as an entity where distances from the center are equal in all directions.

Coordinate Geometry

Coordinate geometry allows us to analyze geometric shapes using a coordinate system. Particularly in 3D space, every point is defined using three coordinates: x, y, and z.
These coordinates help us locate points in a three-dimensional plane, a step further than the usual two-dimensional plane of x and y. In the example of the sphere, the points on its surface are expressed using coordinate geometry principles.
By assigning an x, y, and z value to each point, we can analyze and express complex shapes and solids, like spheres and hemispheres, via equations and inequalities derived from their geometric properties. The upper hemisphere of a sphere, for instance, can be visualized and described using coordinate systems, applying simple restrictions like ensuring the z-coordinate is non-negative.

3D Space

Three-dimensional space is an extension of the two-dimensional plane, filled with volume incorporating length, width, and height. This extra dimension, z, transforms our space from a flat plane to a volume. Dealing with 3D space involves vectors and geometric shapes such as cubes, spheres, and cylinders.
In considering spheres in 3D space, every point is determined by x, y, and z coordinates, which tells us where the point lies in reference to the three axes. The additional complexity of 3D space allows for comprehensive modeling and visualization of objects like spheres.
For instance, when describing just the upper part of a sphere or its upper hemisphere, we introduce conditions such as \(z \geq 0\). These conditions help narrow down the part of space we are analyzing, reflecting its three-dimensional nature succinctly.