Problem 27 - APEX Calculus

Question

In Exercises \(27-30,\) describe the level surfaces of the given functions of three variables. $$ f(x, y, z)=x^{2}+y^{2}+z^{2} $$

Step-by-Step Solution

Answer

The level surfaces are spheres centered at the origin with radius \( \sqrt{c} \) for \( c > 0 \), a point at the origin for \( c = 0 \), and non-existent for \( c < 0 \).

Step 1: Introduction to Level Surfaces

The problem requires describing the level surfaces of the function \( f(x, y, z) = x^2 + y^2 + z^2 \). Level surfaces are obtained by setting the function equal to a constant \( c \), that is, solving \( x^2 + y^2 + z^2 = c \).

Step 2: Identify the Level Surface Equation

The equation \( x^2 + y^2 + z^2 = c \) represents the level surfaces for different values of \( c \). This is the equation of a sphere centered at the origin \((0,0,0)\) with radius \( \sqrt{c} \).

Step 3: Determine the Nature of Level Surfaces

Since the level surface equation is \( x^2 + y^2 + z^2 = c \), for each positive value of \( c \), the surface is a sphere. If \( c = 0 \), the level surface is just a point at the origin. If \( c < 0 \), there are no real surfaces, as the sum of squares cannot be negative.

Step 4: Final Description of Level Surfaces

For \( c > 0 \), the level surfaces are spheres with radius \( \sqrt{c} \) centered at the origin. For \( c = 0 \), the level surface is a single point \((0,0,0)\). There are no level surfaces for \( c < 0 \).

Key Concepts

Three-Dimensional Geometry

Three-dimensional geometry enables us to understand shapes and surfaces in a space that has three dimensions, namely length, width, and height. This is different from two-dimensional geometry, which only considers length and width, like a flat surface. In three-dimensional geometry, we describe objects and their properties, such as points, lines, planes, and solids, with a special emphasis on how these objects relate to each other in space.

The function given in the exercise, \(f(x, y, z) = x^2 + y^2 + z^2\), defines a surface in a three-dimensional space. Each point \((x, y, z)\) on this surface satisfies a specific equation, contributing to a complete geometric shape in space. Understanding these principles is essential for solving problems involving volumes and areas of different figures, as well as for visualizing complex shapes that three-dimensional geometry offers. This foundation prepares students for more advanced topics such as multivariable calculus.

Sphere Equation

The equation \(x^2 + y^2 + z^2 = c\) seen in the exercise represents a sphere, which is a perfectly symmetrical shape in three-dimensional space. The sphere is centered at the origin \((0,0,0)\), and the value of \(c\) determines the radius of the sphere. The radius can be found by taking the square root of \(c\).

• If \(c > 0\), the equation describes a sphere with radius \(\sqrt{c}\). This sphere is symmetric around the origin, meaning that every point on the sphere is equidistant from the center.
• If \(c = 0\), the surface described is not a sphere but rather a single point at the origin, as there is no distance away from the center.
• If \(c < 0\), the equation does not describe any real surface because the sum of squares of real numbers can never be negative.

Visualizing and understanding spheres within this coordinate system is crucial, particularly when exploring phenomena or doing calculations in physics and engineering that rely on spherical symmetry.

Multivariable Calculus

Multivariable calculus expands the concepts of calculus to functions of more than one variable, allowing us to explore changes over surfaces in a three-dimensional space. It is a powerful field that deals with varying functions based on changes in each variable—in this case, the variables are \(x\), \(y\), and \(z\).

Level surfaces, like those described by \(f(x, y, z) = x^2 + y^2 + z^2\), are an important concept in multivariable calculus. These surfaces show us how a function behaves when all variables are adjusted to keep the output constant. In this exercise's context, each level surface represents values of \(c\), forming different spheres.

Understanding multivariable functions helps in various practical applications, such as optimizing several parameters simultaneously, analyzing vector fields, or even computing changes in physical phenomena over time and space. Mastering multivariable calculus principles not only aids in solving mathematical problems but also enhances the ability to apply mathematics in real-world scenarios.