Circulation of two-dimensional flows Let C be the unit circle with counterclockwise orientation. Find the circulation on C of the following vector fields. a. The radial vector field F = <<x, y>> b. The rotation vector field F = << -y, x>>

Similar to circle sphere is a two dimensional space where the set of points that are at the same distance r from a given point in a three dimensional space. In analytical geometry with a center and radius is the locus of all points is called sphere.
a) ${\displaystyle F(x,y)=⟨x,y⟩}$
Here the curve C is the unit circle ${\displaystyle x^1+y^2=1}$
The parametric equations are
${\displaystyle x=\cos t,y=\sin t}$
${\displaystyle dx=-\sin tdt,dy=\cos tdt}$
the flux of ${\displaystyle \overrightarrow{F}}$ across the curve C is
${\displaystyle {\int }_C\overrightarrow{F}\hat{n}ds={\int }_{C}pdy-Qdx}$
${\displaystyle {\int }_C\overrightarrow{F}\hat{n}ds={\int }_{C}xdy-ydx}$
${\displaystyle {\int }_C\overrightarrow{F}\hat{n}ds={\int }_0^1\left(\cos t\right)\cos tdt-\sin (-\sin t)dt}$
${\displaystyle {\int }_C\overrightarrow{F}\hat{n}ds={\int }_0^{1}1dt=1}$
b) ${\displaystyle \overrightarrow{F}=⟨-y,x⟩}$
Here
p=-y, Q=x
${\displaystyle {\int }_C\overrightarrow{F}nds={\int }_C-ydy-xdx}$
${\displaystyle {\int }_C\overrightarrow{F}nds=\in i{t}_0^1-\sin t\cos tdt-\cos t(-\sin tt)dt}$
${\displaystyle {\int }_C\overrightarrow{F}nds={\int }_0^{10}dt=0}$