# Counting what crosses a boundary

Authored explanation; editorial review pending.

Take a short interval. If more tracers enter through its boundaries than leave, its count increases. Flux counts signed crossings per area per time; number density counts particles per volume. They are not the same quantity.

$$
\frac{\partial c}{\partial t}=-\frac{\partial J}{\partial x}
$$

Density changes at the negative spatial derivative of the flux.

In the one-dimensional description, the same balance applies per unit transverse area. The minus sign says an increasing rightward flux removes more particles at the right boundary than arrive at the left.

## Worked example

If seven particles enter and five leave during a time interval, the count rises by two. If both flows are five, the count is unchanged even though particles keep crossing.

Conservation counts net crossings; it does not require each individual particle to stop.