Problem 8 - Advanced Engineering Mathematics

Question

Describe the level surfaces but do not graph. $$ F(x, y, z)=x^{2}+y^{2}+z^{2} $$

Step-by-Step Solution

Answer

Level surfaces are spheres centered at origin with radius \( \sqrt{c} \), where \( c \geq 0 \).

Step 1: Understanding Level Surfaces

The level surface of a function of three variables, like \( F(x, y, z) = x^2 + y^2 + z^2 \), is a set of points \((x, y, z)\) where the function equals a constant. To describe the level surfaces, we consider \( F(x, y, z) = c \), where \( c \) is a constant.

Step 2: Substituting in Level Surface Equation

Substitute \( F(x, y, z) = c \) into the equation to get \( x^2 + y^2 + z^2 = c \). This equation is that of a sphere, centered at the origin \((0, 0, 0)\), with radius \( \sqrt{c} \).

Step 3: Analyzing the Value of c

Since \( x^2 + y^2 + z^2 \) is always nonnegative (because they are squares), \( c \) must be a nonnegative real number. When \( c = 0 \), the level surface is a single point, the origin itself. When \( c > 0 \), the level surface is a sphere with radius \( \sqrt{c} \).

Step 4: Description of Level Surfaces

For different values of \( c \), we obtain spheres of different radii centered at the origin. Specifically, for each \( c > 0 \), \( x^2 + y^2 + z^2 = c \) describes a sphere of radius \( \sqrt{c} \). For \( c = 0 \), the level surface is just the origin \((0, 0, 0)\).

Key Concepts

Multivariable Calculus

Multivariable calculus is an extension of calculus that studies functions of more than one variable. Unlike single-variable calculus, which deals with functions that map from the real numbers to the real numbers, multivariable calculus works with functions that map from a number of variables to a single output. In this case, the function is defined as \[F(x, y, z) = x^2 + y^2 + z^2\]which takes three inputs and produces one output.

A key aspect of multivariable calculus is the concept of a level surface. A level surface represents a set of points in three-dimensional space where the function equals a specific value. For example, \[F(x, y, z) = c\]does not yield a single output but rather a surface (or in this context, a sphere) of equal function values. These surfaces are useful in visualizing how values change, showing contours similar to topographic maps but in three dimensions.

Spherical Coordinates

In three-dimensional geometry, spherical coordinates are an alternative to Cartesian coordinates. These coordinates specify a point by its distance from the origin and two angles. Spherical coordinates are particularly useful when dealing with problems involving spheres, like the level surfaces seen in \[x^2 + y^2 + z^2 = c\]

The conversion between Cartesian and spherical coordinates helps simplify calculations:

• \( x = ho \sin \phi \cos \theta \)
• \( y = ho \sin \phi \sin \theta \)
• \( z = ho \cos \phi \)

Here, \( \rho \) is the radial distance from the origin (equivalent to the sphere's radius \( \sqrt{c} \)), \( \theta \) is the azimuthal angle measured from the x-axis, and \( \phi \) is the polar angle from the z-axis. This representation aligns perfectly when describing the level surfaces of a sphere centered at the origin, making spherical coordinates a natural choice for such geometry.

Three-dimensional Geometry

Three-dimensional geometry investigates the properties and relations of points, lines, surfaces, and solids in space. In the context of the provided exercise, understanding the equation \[x^2 + y^2 + z^2 = c\]provides insight into three-dimensional shapes, specifically spheres.

The equation forms spheres centered on the origin. The variable \( c \) represents the square of the radius of these spheres. As \( c \) varies, so does the size of the sphere, reflecting how a level surface can morph depending on constants in the equation.

Three-dimensional geometry is key to visualizing and solving problems with spatial relationships. Recognizing geometric shapes, such as spheres and planes, in the context of these equations is crucial for understanding the physical and analytical properties of different surfaces and forms in 3D space.