Flux & Circulation Homework: Find Values & Prove Non-Zero Flux

[6.28318531]

Homework Statement

Let F(x,y) = ( P(x,y), Q(x,y)) be a vector field that is continuously differentiable along the closed smooth curve C : x²+y² = 1. Moreover let

-F(x,y) = F( -x, -y)≠ 0 and

P(x,y) = -P(-x,y) and Q(x,y) = Q(-x,y).

Determine all the possible values of the circulation around C, and argue why flux across C is non zero.

Homework Equations

Circulation = ∫_CF. T ds , T tangent vector
Flux = ∫_CF. N ds , N normal vector

The Attempt at a Solution

I'm not quite sure where to begin . Do we parametrize C by r(t) = (cos t, sin t) 0≤ t≤ 2\pi, and then use the definitions of flux and circulation?

 

[voko]

From the given relations among F, P and Q you can deduce how they are related in all four quadrants. What will that mean with regard to the integrals?

 

[6.28318531]

Ah now I see for the flux part;

Look at each quadrant {x,y>0} {x,y<0} {x>0, y<0} and {x<0,y>0}

When x, y> 0, The integral evaluates to a value.

Let ∫_CF.n ds = f

Then for x,y<0

the integral becomes ∫F( -P, -Q) .n ds = - f

Then for x> 0, y< 0
∫F(P(x,-y), Q(x,-y)).n ds= a ≠f


Then for x< 0, y> 0

∫F(P(-x,y),Q(-x,y)).n ds =∫ F( -P(x,y),Q(x,y)).n ds =b ≠a.

So you can see the total flux is non zero. The two regions x,y>0 and x,y<0 cancel out
and the other two are not equal.