Question

Find an equation for the family of level surfaces corresponding to \(f .\) Describe the level surfaces. $$f(x, y, z)=\frac{1}{x^{2}+y^{2}+z^{2}}$$

Short answer

Answer: The family of level surfaces for the function \(f(x, y, z) = \frac{1}{x^2+y^2+z^2}\) is given by the equation \(x^2 + y^2 + z^2 = \frac{1}{c}\), where c is a constant. These level surfaces represent spheres centered at the origin, with their radii equal to \(\sqrt{\frac{1}{c}}\).

Step-by-step solution

Write the function as an equation

Set \(f(x, y, z)\) equal to a constant. Let the constant be represented by \(c\): $$f(x, y, z) = \frac{1}{x^2+y^2+z^2} = c$$

Simplify the equation

To simplify the equation, multiply both sides by the denominator \((x^2+y^2+z^2)\): $$1 = c(x^2 + y^2 + z^2)$$ Now, isolate the expression \((x^2+y^2+z^2)\) by dividing both sides by \(c\): $$\frac{1}{c} = x^2 + y^2 + z^2$$

Identify the geometric shape

Observe and analyze the equation \(\frac{1}{c} = x^2 + y^2 + z^2\). This equation represents the equation of a sphere in 3D space, with its center at the origin (0, 0, 0) and a radius of \(\sqrt{\frac{1}{c}}\). This means that the family of level surfaces corresponding to the function \(f(x, y, z)\) comprises spheres centered at the origin, with varying radii as the constant \(c\) changes. In conclusion, the equation for the family of level surfaces is given by: $$x^2 + y^2 + z^2 = \frac{1}{c}$$ And these level surfaces represent spheres centered at the origin, with their radii equal to \(\sqrt{\frac{1}{c}}\).

Key concepts

Multivariable Functions

Multivariable functions depend on more than one variable, making them a cornerstone of calculus and higher-dimensional analysis. Specifically, a function such as \(f(x, y, z)\) defines a relationship between three variables. Unlike single-variable functions which might define a simple line or curve, multivariable functions map complex surfaces or shapes in space.
For instance, the function \(f(x, y, z) = \frac{1}{x^2+y^2+z^2}\) relies on three variables: \(x, y,\) and \(z\). Each combination of these values can produce a distinct output for \(f\), creating a vast family of possible outputs.
In the realm of calculus, these functions allow us to explore rich geometric structures and compute volumes, gradients, and many other operations linked to spatial phenomena.

• These functions are invaluable in fields like physics, engineering, and economics.
• They can be visualized as surfaces, where certain outputs correspond to specific geometric shapes.
• Analyzing level surfaces or sets where the function equals a constant, sheds light on the 3D structure formed by the multivariable function.

Spheres in 3D Space

Spheres are a basic but integral part of geometry in 3D space. A sphere is the 3D equivalent of a circle in 2D space. Every point on a sphere's surface is equidistant from a central point known as the center.
The standard equation of a sphere with a center at the origin (0, 0, 0) is given by the formula \(x^2 + y^2 + z^2 = r^2\), where \(r\) is the radius of the sphere.
Understanding spheres in 3D space helps us visualize level surfaces formed by certain multivariable functions. These connections emphasize the innate beauty and symmetry found in mathematics.

• Key characteristics include symmetry in all directions and uniform surface curvature.
• Spheres are central to physics problems, given their simple shape and constant radius.
• Applications of spheres extend to areas like telecommunications, where spheres simulate signal ranges.

Equation of a Sphere

In the study of geometry, the equation of a sphere allows us to model objects and concepts in three dimensions. The general form of a sphere’s equation is \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\), where \((h, k, l)\) represents the center, and \(r\), the radius.
Specifically, for spheres that are centered at the origin \((0, 0, 0)\), this equation simplifies to \(x^2 + y^2 + z^2 = r^2\). By changing the value of \(r\), we can obtain spheres of different sizes.
The equation \(x^2 + y^2 + z^2 = \frac{1}{c}\) represents a family of spheres derived from a given multivariable function when set equal to a constant \(c\). This equation showcases various sphere radii determined by \(\sqrt{\frac{1}{c}}\), illustrating how changes in constants affect geometric properties.

• This form is fundamental in defining the geometry of objects in space.
• The radius is an important factor, determining the sphere's size and volume.
• Sphere equations enable the study and understanding of level surfaces, which are crucial in fields like meteorology and fluid dynamics.