Derivative relationships between volume and surface area of compact regions in R d

Abstract

We explore the idea that the derivative of the volume, V, of a region in R d with respect to r equals its surface area, A, where r = d(V/A). We show that the families of regions for which this formula for r is valid, which we call homogeneous families, include all the families of similar regions. We determine equivalent conditions for a family to be homogeneous, provide examples of homogeneous families made up of non-similar regions and offer a geometric interpretation of r in a few cases. Copyright © 2007 Rocky Mountain Mathematics Consortium.