Functions of Three or More Variables

A function of three variables assigns a single real number—a scalar value—to every point in three-dimensional space. These points are defined by their coordinates: x, y, and z.

A practical example is the electric potential surrounding a point charge. The potential at any spatial point depends on its distance from the charge.

At each point, the function assigns a specific voltage, mapping a three-dimensional location to a corresponding scalar value.

These functions can be visualized by identifying the set of points where the output remains constant.

These points form a level surface—a two-dimensional surface within a three-dimensional region where the function remains unchanged.

In this example, level surfaces are known as equipotential surfaces. Moving along a surface changes the position, but the potential remains constant.

Consider an example of a mathematical function that generates spherical level surfaces. Here, the output depends solely on the distance from the origin.

Each sphere has a constant value at every point on its surface.

Level surfaces provide essential insight into the shape and behavior of multivariable functions in three dimensions and beyond.