78 7 Fields, Spatial Differential Operators
7.1 Scalar Fields, Gradient.
In physics, potential functions are important examples for scalar fields. The math-
ematical considerations presented in the following for potentials apply just as well
for other scalar fields, such as the number density, the mass density and the charge
density, or concentration and temperature fields.
7.1.1 Graphical Representation of Potentials.
In the case of a two-dimensional space, the value of a potential field can be plotted
into the third dimension, i.e. as the height above a plane. It can be visualized just
like a panoramic map of a landscape, or in a 3D graphics plot. Alternatively, the
information about the potential can be shown in equipotential lines, just like the
lines of equal height in maps used for hiking.
The panoramic view of a potential function depending on a three-dimensional
vector requires a four-dimensional space, which is beyond our visual experience.
The surfaces of equal potential energy, also calledequal potential surfacescan be
visualized in 3D graphics. The surface where the value of the potential is equal toc,
is determined by
Φ(r 1 ,r 2 ,r 3 )=c.The solution of this equation forr 3 yields
r 3 =z(r 1 ,r 2 ,c),wherecis a curve parameter. In this way, equipotential surfaces of 3D fields can be
represented geometrically with the same tools used for 2D potential functions.
Three simple special cases associated with planar, cylindrical and spherical geom-
etry are discussed next.
- Planar Geometry. Let a preferential direction be specified by the constant unit
vectore, and a special potential depending onrin the formΦ=Φ(e·r)=Φ(x),
wherex=e·r. In this case the equipotential surfaces are planes perpendicular
toe. The information in the variation of the potential is contained in the one-
dimensional functionΦ(x). - Cylindrical Geometry. Let the direction of a preferential axis be parallel to the
constant unit vectore. Now the case is considered, where the potential depends
just on the components ofrperpendicular toe. Then one hasΦ =Φ(r⊥),
withr⊥ =r−e·re. Or, in other words, one hasΦ=Φ(x,y), where the
components ofr⊥are denoted byxandy. When, even more special, the potential
just depends on distanceρ=
√
r⊥·r⊥=√
x^2 +y^2 of a point from the axis,