Weights and Measures

Several times throughout this book we've hinted at a physical basis for some of our notation. It's time that we made this more precise; a simple calculus problem will help explain.

Consider a thin rod, or wire, positioned along the interval [a, b] on the x-axis and having a nonuniform distribution of mass. For example, the rod might vary slightly in thickness or in density (mass per unit length) as x varies. Our job is to compute the density (at a point) as a function f(x), if at all possible.

What we can measure effectively is the distribution of mass along the rod. That is, we can easily measure the mass of any segment of the rod, and so we know the mass of the segment lying along the interval [a, x] as a function F(x). Said in slightly different terms, we are able to measure small, discrete “chunks” of mass as dm = F(x + dx) − F(x) = dF, and so we're led to define the density f(x) = dm/dx = F′(x) as the derivative of the distribution F(x), provided that F is differentiate, of course.

But F is an arbitrary increasing function – is every such function differentiate? And, if not, can we say anything meaningful about this problem? Could we, for example, still find the center of mass (the line x = µ through which the rod balances) when F is not differentiable?