Question

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 \(\displaystyle{F}={\left\langle{x},{y}\right\rangle}\)
b. The rotation vector field \(\displaystyle{F}={\left\langle-{y},{x}\right\rangle}\)

Answer 1

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) \(\displaystyle{F}{\left({x},{y}\right)}={\left\langle{x},{y}\right\rangle}\)
Here the curve C is the unit circle \(\displaystyle{x}^{{1}}+{y}^{{2}}={1}\)
The parametric equations are
\(\displaystyle{x}={\cos{{t}}},{y}={\sin{{t}}}\)
\(\displaystyle{\left.{d}{x}\right.}=−{\sin{{t}}}{\left.{d}{t}\right.},{\left.{d}{y}\right.}={\cos{{t}}}{\left.{d}{t}\right.}\)
the flux of \(\displaystyle\vec{{F}}\) across the curve C is
\(\displaystyle\int_{{C}}\vec{{F}}\hat{{n}}{d}{s}=\int_{{C}}{p}{\left.{d}{y}\right.}-{Q}{\left.{d}{x}\right.}\)
\(\displaystyle\int_{{C}}\vec{{F}}\hat{{n}}{d}{s}=\int_{{C}}{x}{\left.{d}{y}\right.}-{y}{\left.{d}{x}\right.}\)
\(\displaystyle\int_{{C}}\vec{{F}}\hat{{n}}{d}{s}={\int_{{0}}^{{1}}}{\left({\cos{{t}}}\right)}{\cos{{t}}}{\left.{d}{t}\right.}-{\sin{{\left(-{\sin{{t}}}\right)}}}{\left.{d}{t}\right.}\)
\(\displaystyle\int_{{C}}\vec{{F}}\hat{{n}}{d}{s}={\int_{{0}}^{{1}}}{1}{\left.{d}{t}\right.}={1}\)
b) \(\displaystyle\vec{{F}}={\left\langle-{y},{x}\right\rangle}\)
Here
p=-y, Q=x
\(\displaystyle\int_{{C}}\vec{{F}}{n}{d}{s}=\int_{{C}}-{y}{\left.{d}{y}\right.}-{x}{\left.{d}{x}\right.}\)
\(\displaystyle\int_{{C}}\vec{{F}}{n}{d}{s}=\in{i}{{t}_{{0}}^{{1}}}-{\sin{{t}}}{\cos{{t}}}{\left.{d}{t}\right.}-{\cos{{t}}}{\left(-{\sin{{t}}}{t}\right)}{\left.{d}{t}\right.}\)
\(\displaystyle\int_{{C}}\vec{{F}}{n}{d}{s}={\int_{{0}}^{{10}}}{\left.{d}{t}\right.}={0}\)