Section6.4Using Technology to Visualize Potentials

If a scalar field such as temperature or electrostatic potential or gravitational potential represents a scalar-valued physical quantity at each point in space, then, since space is three-dimensional, a scalar field is effectively a function of three variables. How can we graph such functions? For functions of one or two variables, if we use one or two dimensions to represent the domain of the scalar field, we have at least one physical dimension left to represent the value of the function. Thus, functions of one variable are plotted in two dimensions and functions of two variables are plotted in three dimensions, as shown in Figure Figure 6.4.1.

To plot a function of three variables in an analogous way, we would need to live in four physical dimensions.

(a)

You should take some time to brainstorm some alternative methods of representing a function of three variables.

(b)

After you have brainstormed some ideas yourself, you can use the Sage code below to explore several different mechanisms for visualizing scalar fields in three dimensions for the particular case of electrostatic potentials for several discrete charges. Alternatively, if you have access to Mathematica, you can use this Mathematica notebook for the same purpose.

The code in the first box defines the scalar potential $V\text{.}$

This Sage code plots a contour diagram of the scalar potential $V$ (on a horizontal slice).

This Sage code graphs the scalar potential $V$ on a horizontal slice.

This Sage code plots a single, three-dimensional contour of the scalar potential $V\text{.}$