dn

In mathematics and the physical sciences, dN denotes an infinitesimal change in the quantity N. The symbol d represents the differential, indicating a small, but finite, change used to describe rates, linearizations, and integrals. The precise meaning of dN depends on what N represents and the variables on which N depends.

In one-variable calculus, if N is a function of x, then dN = (dN/dx) dx. This expresses the differential of N as the product of the derivative and a small change in x. For multivariable functions N = N(x1, x2, ..., xk), the differential is dN = ∑i (∂N/∂xi) dxi, giving the first-order change in N due to small changes in all independent variables.

As an approximation, dN provides the linearization of N about a point and serves as the basis for the differential form in integration: the total change in N over a finite path or interval satisfies N(final) − N(initial) = ∫ dN.

In physics and engineering contexts, dN commonly represents the differential change in the number of discrete objects, such as particles, photons, or events, counted in a region of parameter space. For example, dN/dE might describe the energy spectrum of particles, the rate of events per energy interval, and dN can be integrated over a domain to obtain the total count N.

In probability and statistics, dN is sometimes treated as a random differential representing the expected infinitesimal count in an interval, with actual counts modeled by stochastic processes such as Poisson processes where N is a counting variable.

Because dN is context-dependent, its precise meaning hinges on the definitions of N and the variables involved. It is a differential and not a standalone quantity; it becomes meaningful only within equations describing rates, linearizations, or integrals.