Flow

Flow represents the total work or circulation done by a particle in a vector field as it moves along a path. It measures how much the flow “pushes” in the direction of the path while traveling through it.

It is derived from integrating the vector field’s component in the direction of the surfaces tangential vector.

Flow Integrals and Circulation for Velocity Fields

Relying on the principle of Line Integrals if a vector field $F$ represents the velocity field of a flowing fluid in space the integral of $F\cdot T$ along a curve $C$ will yield the flow or circulation around the curve.

$$Flow=\int \_CF\cdot Tds$$

If the curve starts and ends at the same point $a=b$ the flow is called the circulation around the curve