Functions of Three or More Variables

A function of three variables assigns a single real number to each point in three-dimensional space. Every point is identified by its Cartesian coordinates, x, y, and z, and the function maps this ordered triple to a scalar value. Such functions are commonly used to describe physical quantities that vary throughout space.

A representative example is the electric potential generated by a point charge. In this case, the potential at a given location depends only on the distance from the charge. If the charge is positioned at the origin, the potential decreases as the distance from the origin increases. The function thus assigns a specific voltage to every point in space, defining what is known as a scalar field. The value of the function changes continuously from point to point, reflecting how the potential varies with radial distance.

The functions of three variables can be visualized by identifying sets of points at which the function takes the same value. These sets form level surfaces, which are two-dimensional surfaces embedded within three-dimensional space. For the electric potential surrounding a point charge, such level surfaces are called equipotential surfaces. Moving along an equipotential surface changes the spatial position but does not alter the potential's value.

A mathematical function that produces spherical level surfaces is one whose output depends solely on the distance from the origin. In this situation, all points located at the same distance from the origin share the same function value. Each constant value corresponds to a sphere, and every point on that sphere has an identical output.

Level surfaces provide important geometric insight into the structure and behavior of multivariable functions, offering a clear way to interpret scalar fields in three dimensions and beyond.