Electrostatics and the index of vector fields
Let $\varphi$ be an electrostatic potential defined by $N$ stationary charged particles in R$\sp3$, and let $N\sb\varphi$ denote its set of critical points. What can be said about $N\sb\varphi$? This question attracted the interest of many scientists including a number of mathematicians.
J. C. Maxwell tried to describe $N\sb\varphi$, but, unfortunately, he made a few errors. A. Yanushauskas showed that if $K$ is a compact subset of R$\sp3$, then $K \cap N\sb\varphi$ consists of a finite number of isolated points and a finite number of pieces of analytic curves. Then, S. S. Cairns and M. Morse proved that if $\varphi$ is nondegenerate and the total charge is not zero, then $N\sb\varphi$ consists of isolated points whose number is at least $N$ $-$ 1.
In this thesis we use the law of vector fields and some of its properties, in addition to other ideas from different branches of mathematics, to find various results. First, we note two of Maxwell's mistakes and prove a theorem about the angles of intersection of level curves. Next, we answer two questions by Morse and Cairns, and correct their mistake. Then we describe $N\varphi$ when $\varphi$ is noncompletely-degenerate, the total charge is not zero, and the particles are on a straight line. Finally, we pose the following interesting question. Is there a singular point of an electrostatic field or its restriction to a plane, whose index is equal to a given integer n? We show a singular point with index $-$2 in the xy-plane and index 0 in the space, then we make several conjectures about other integers.