  The Parzen window density is a well-known technique, associating Gaussian
kernels with data points. It is a very useful tool in data exploration, with
particular importance for clustering schemes and image analysis. This method is
presented here within a formalism containing scalar fields, such as the density
function and its potential, and their corresponding gradients. The potential is
derived from the density through the dependence of the latter on the common
scale parameter of all Gaussian kernels. The loci of extrema of the density and
potential scalar fields are points of interest which obey a variation condition
on a novel indicator function. They serve as focal points of clustering methods
depending on maximization of the density, or minimization of the potential,
accordingly. The mixed inter-dependencies of the different fields in d-dim
data-space and 1-d scale-space, are discussed. They lead to a Schr\H{o}dinger
equation in d-dim, and to a diffusion equation in (d+1)-dim
