PX275 - D3f - physical significance of the divergence theorem

example 1

looking at the velocity field of a flow of water, $\vec{F}$ , which is uniform
at a point, $A$ , $\vec{F} \cdot d \vec{n} \sim + v e$ , and at a point, $B$ , $\vec{F} \cdot d \vec{n} \sim - v e$ , but they have the same magnitude
therefore, the contributions cancel each other out:

(\vec{F} \cdot d \vec{n})_{A} + (\vec{F} \cdot d \vec{n})_{B} = 0

this happens for all points on the loop, and therefore:

\oint_{C} \vec{F} \cdot d \vec{n} = 0 ⟹ \iint_{R} \vec{\nabla} \cdot \vec{F} d A = 0

therefore, this is a divergence free field

example 2

at any point on the curve, $\vec{F}$ and $d \vec{n}$ are parallel to each other, and $\vec{F} \cdot d \vec{n}$ is always positive:

\oint_{C} \vec{F} \cdot d \vec{n} > 0

therefore, $\vec{F}$ is not a divergence free field