Problem 49 - Calculus 3

Question

Find the level surface for the functions of three variables and describe it. $$ w(x, y, z)=x^{2}+y^{2}+z^{2}, c=9 $$

Step-by-Step Solution

Answer

The level surface is a sphere centered at the origin with radius 3.

Step 1: Understanding Level Surface

A level surface for a function of three variables is the set of points \(x, y, z\) that satisfy the equation \(w(x, y, z) = c\). For the given function \(w(x, y, z) = x^2 + y^2 + z^2\), we need to find the set of all \(x, y, z\) such that \(x^2 + y^2 + z^2 = 9\).

Step 2: Recognize the Equation

The equation for the level surface is \(x^2 + y^2 + z^2 = 9\). This is the standard form of the equation of a sphere centered at the origin \((0, 0, 0)\) in three-dimensional space.

Step 3: Identify Surface Characteristics

The equation \(x^2 + y^2 + z^2 = 9\) represents a sphere with radius \( ext{radius} = \sqrt{9} = 3\) units. It is centered at the origin \( (0, 0, 0) \).

Step 4: Final Description of Level Surface

Therefore, the level surface \(w(x, y, z) = 9\) is a sphere centered at the origin with a radius of 3. In other words, it includes all the points whose distance from the origin is 3.

Key Concepts

Function of Three Variables

When dealing with functions of three variables, we come across expressions like \( w(x, y, z) \), which depend on the three independent variables: \( x \), \( y \), and \( z \). These functions are used to describe a variety of surfaces in three-dimensional space. For example, the given function is \( w(x, y, z) = x^2 + y^2 + z^2 \). This represents a combination of the three coordinates squared and then summed up.

• In such functions, the output \( w \) represents a scalar value or a constant level.
• The level surface is the collection of all points \( (x, y, z) \) that satisfy the condition \( w(x, y, z) = c \).

Here, the level surface is given by setting \( w(x, y, z) \) equal to a constant, which helps to identify specific geometric shapes in space. This exercise focuses on understanding how these shapes are described mathematically and offers a method to visualize them effectively.

Sphere Equation

The sphere equation arises naturally in this context. In three-dimensional space, the equation \( x^2 + y^2 + z^2 = c \) suggests a sphere's presence. Notably, \( c \) here acts as a important constant determining the sphere's size.

• The sphere is a symmetrical object where every point on its surface is equidistant from the center.
• In our example, setting \( w(x, y, z) = 9 \) forms the sphere equation \( x^2 + y^2 + z^2 = 9 \).

This equation is crucial and widely used in mathematics and physics. Knowing how to recognize and manipulate this equation helps in solving various problems involving spatial dimensions and shapes.

Radius of a Sphere

The radius of a sphere is a key component in understanding its geometric properties. It represents the distance from the center of a sphere to any point on its surface.
Consider the sphere equation \( x^2 + y^2 + z^2 = r^2 \). Here, \( r \) is the radius of the sphere. Given the equation \( x^2 + y^2 + z^2 = 9 \), we identify that the sphere's radius is \( r = \sqrt{9} \), which simplifies to \( r = 3 \).

• The radius directly influences the size of the sphere. Larger radii mean larger spheres.
• Understanding the radius is essential for calculating other properties, such as surface area and volume.

By incorporating the concept of radius, we calculate distances and dimensions, offering a complete understanding of three-dimensional shapes involving spheres.