Поверхностные интегралы векторных полей: поток

Analyzing how weather systems change over time begins with finding the net mass of air flowing into or out of a specific region. This requires defining an imaginary boundary surface, oriented using a unit normal vector.

Airflow is described by a velocity vector field and a density field that varies with position. Their product gives a new vector field that shows the mass flow per unit area.

To compute net mass flow, the surface is divided into small, nearly flat patches. For each patch, the dot product of ρv and the unit normal vector gives the mass flow per unit area through the surface. This value is multiplied by the area of the patch.

Summing these quantities and taking the limit as the patch size approaches zero gives the surface integral, called the net flux or net mass flow. By defining ρv as a new vector field F, the net flux becomes the integral of the dot product of F and n over the surface S.

This result, called the flux of F across S, gives the net mass flow through the boundary, and helps meteorologists track the movement of air masses during weather modeling.