Sylvester´s criterion applied to Poisson´s equation. A quantitative proof of Earnshaw´s theorem

Without solving Poisson´s (or Laplace´s) equation, we use Sylvester´s criterion to analyse the conditions for an extremum of electric potential. Particular attention is given to Earnshaw´s theorem and to its treatment by various authors. A quantitative demonstration of the theorem is given, and qualitative aspects of unstable points that are, however, relative maxima or minima, are discussed.