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

• 6.28318531
In summary, the vector field F and its components, P and Q, have specific relationships in all four quadrants, which can be used to determine the possible values of circulation and show that the flux across C is non-zero. This is demonstrated by evaluating the integrals in each quadrant and showing that they do not cancel out, resulting in a 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?

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?

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.

What is flux?

Flux is a measure of the flow of a vector field through a given surface. It is represented by the symbol Φ and is calculated by taking the dot product of the vector field and the surface normal multiplied by the surface area.

What is circulation?

Circulation is a measure of the flow of a vector field along a closed path. It is represented by the symbol C and is calculated by integrating the dot product of the vector field and the tangent vector along the path.

How do you find the flux through a surface?

To find the flux through a surface, you first need to determine the vector field and the surface normal. Then, you can calculate the dot product of these two vectors and multiply it by the surface area. The resulting value is the flux through the surface.

How do you prove non-zero flux?

To prove that the flux through a surface is non-zero, you can use the Divergence Theorem. This theorem states that the flux through a closed surface is equal to the volume integral of the divergence of the vector field over the enclosed volume. If the divergence is non-zero, then the flux is also non-zero.

What are some real-life applications of flux and circulation?

Flux and circulation have many real-life applications, such as in fluid dynamics, electromagnetism, and thermodynamics. They are used to calculate things like fluid flow rates, electric and magnetic fields, and heat transfer. They are also important in the study of weather patterns and ocean currents.

• Calculus and Beyond Homework Help
Replies
12
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
24
Views
918
• Calculus and Beyond Homework Help
Replies
1
Views
989
• Calculus and Beyond Homework Help
Replies
6
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
417
• Calculus and Beyond Homework Help
Replies
11
Views
2K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
9
Views
1K