PX275 - D3g - the physical significance of green's theorem in the plane

green's theorem in the plane:

\oint \vec{F} \cdot d \vec{r} = \iint_{R} (\vec{\nabla} \times \vec{F}) \cdot d \vec{A} = \iint_{R} (\vec{\nabla} \times \vec{F})_{z} d A

considering a field, $\vec{F}$ , describing the laminar flow of water
at $A$ and $B$ , $\vec{F} \cdot d \vec{r}$ do not cancel, but they do at diametrically opposite points, ie: $A$ and $B$ , or $B$ and $C$ , therefore:

\oint_{C} \vec{F} \cdot d \vec{r} = 0

comparing to the whirlpool
$\vec{F}$ and $d \vec{r}$ are always parallel to each other, so:

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

considering the magnetic field around a current carrying wire:

\vec{B} = \frac{μ_{0} I}{2 π r} \hat{ϕ}

in the radial direction: $\vec{B} ⊥ d \vec{r}$ , so, $\vec{B} \cdot d \vec{r} = 0$
for the loop:

\oint_{C} \vec{B} \cdot d \vec{r} = \int_{l i n e} {\vec{B}}_{r 1} \cdot d \vec{r} \int_{l i n e} {\vec{B}}_{r 2} \cdot d \vec{r}

using the relation for the length of arc, $l = θ r$ , along the arc, $\int d r = θ r_{n} :$

\begin{array}{r} \int_{l i n e} {\vec{B}}_{r 1} \cdot d \vec{r} = \frac{μ_{0} I}{2 π r_{1}} = \frac{μ_{0} I θ}{2 π} \\ \int_{l i n e} {\vec{B}}_{r 2} \cdot d \vec{r} = \frac{μ_{0} I}{2 π r_{2}} = - \frac{μ_{0} I θ}{2 π} \end{array}

therefore the integral around the loop becomes:

\oint_{C} \vec{B} \cdot d \vec{r} = 0