# 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>>

Question
Analytic geometry
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 $$\displaystyle{F}={\left\langle-{y},{x}\right\rangle}$$

2021-02-06
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
$$\displaystyle\int_{{C}}\vec{{F}}{n}{d}{s}=\int_{{C}}-{y}{\left.{d}{y}\right.}-{x}{\left.{d}{x}\right.}$$
$$\displaystyle\int_{{C}}\vec{{F}}{n}{d}{s}=\in{i}{{t}_{{0}}^{{1}}}-{\sin{{t}}}{\cos{{t}}}{\left.{d}{t}\right.}-{\cos{{t}}}{\left(-{\sin{{t}}}{t}\right)}{\left.{d}{t}\right.}$$
$$\displaystyle\int_{{C}}\vec{{F}}{n}{d}{s}={\int_{{0}}^{{10}}}{\left.{d}{t}\right.}={0}$$

### Relevant Questions

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The work required to move an object around a closed curve C in the presence of a vector force field is the circulation of the force field on the curve.
b. If a vector field has zero divergence throughout a region (on which the conditions of Green’s Theorem are met), then the circulation on the boundary of that region is zero.
c. If the two-dimensional curl of a vector field is positive throughout a region (on which the conditions of Green’s Theorem are met), then the circulation on the boundary of that region is positive (assuming counterclockwise orientation).
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