# 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>>

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 $F=⟨-y,x⟩$
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diskusje5
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) $F\left(x,y\right)=⟨x,y⟩$
Here the curve C is the unit circle ${x}^{1}+{y}^{2}=1$
The parametric equations are
$x=\mathrm{cos}t,y=\mathrm{sin}t$
$dx=-\mathrm{sin}tdt,dy=\mathrm{cos}tdt$
the flux of $\stackrel{\to }{F}$ across the curve C is
${\int }_{C}\stackrel{\to }{F}\stackrel{^}{n}ds={\int }_{C}pdy-Qdx$
${\int }_{C}\stackrel{\to }{F}\stackrel{^}{n}ds={\int }_{C}xdy-ydx$
${\int }_{C}\stackrel{\to }{F}\stackrel{^}{n}ds={\int }_{0}^{1}\left(\mathrm{cos}t\right)\mathrm{cos}tdt-\mathrm{sin}\left(-\mathrm{sin}t\right)dt$
${\int }_{C}\stackrel{\to }{F}\stackrel{^}{n}ds={\int }_{0}^{1}1dt=1$
b) $\stackrel{\to }{F}=⟨-y,x⟩$
Here
p=-y, Q=x
${\int }_{C}\stackrel{\to }{F}nds={\int }_{C}-ydy-xdx$
${\int }_{C}\stackrel{\to }{F}nds=\in i{t}_{0}^{1}-\mathrm{sin}t\mathrm{cos}tdt-\mathrm{cos}t\left(-\mathrm{sin}tt\right)dt$
${\int }_{C}\stackrel{\to }{F}nds={\int }_{0}^{10}dt=0$