Foundation · Explanatory preview
Counting what crosses a boundary
What accumulates in a region equals what enters minus what leaves, provided nothing is created or destroyed inside.
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Why does a difference of flows change the density?
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.
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.
One 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.
A stopping point: Conservation counts net crossings; it does not require each individual particle to stop.
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