Circulation per unit area

When discussing the " >idea behind Green’s theorem, I claimed that the “microscopic” circulation of a two-dimensional vector field F was

“microscopic circulation” = ∂F2-
∂x -∂F1-
∂y.
We can prove that the quantity
∂F2-
∂x -∂F1-
∂y.

does represent “circulation per unit area,” which is probably a better term for it than “microscopic circulation.”

Remeber that the circulation of F around the closed curve C is

CF ds.
so the “circulation per unit area” is simply the above integral divided by the area inside C:
   ∫
----C-F-⋅ ds--
Area  inside C.
It turns out if you let C shrink down to a point, this ratio becomes
   ∫
     F  ⋅ ds
----C---------
Area inside C ∂F2
----
∂x -∂F1
----
 ∂y,

assuming C is oriented counterclockwise.

We will sketch a proof of this for a rectangular curve C, oriented counterclockwise. Let the lower-left point of C be (a,b), its width be Δx, and its height be Δy. Label the edges of the rectangle by C1, C2, C3, and C4.

PIC.

We assume box small enough to approximate F as constant along each edge.

Along the curve C1, the value of y is constantly b, but the value of x changes from a to a + Δx. We assume we can ignore that change in x (since the box is small), and simply approximate F(x,y) as F(a,b) all along the segment C1.

Along the curve C2, the value of x is constantly a + Δx, but y changes from b to b + Δy. Since we assume we can ignore the change in y, we approximate F(x,y) as F(a + Δx,b) all along segment C2.

Similiarly, along C3, y = b + Δy, and we approximate x as a (even though it ranges between a and a + Δx). We approximate F(x,y) as F(a,b + Δy) all along segment C3.

Along C4, x = a, and we approximage y as b + Δy (even though it ranges between b and b + Δy). We approximate F(x,y) as F(a,b) all along segment C4.

We summarize these approximations in the following figure.

PIC.

Next, to compute the integral CF ds, we remember that it is the same thing as integrating the scalar-valued function F T, where T is the unit tangent vector along C:

CF ds = CF Tds.

We need to compute F T along each segment of C. The tangent vector T is constant along each segment, and we are approximating F as constant along each segment, so the dot product F T will be constant along each segment of C.

Along C1, the path is directed in the positive x direction, so the unit tangent vector is T = (1, 0). The dot product F T is simply the first component of F(a,b):

F T = F1(a,b).

Along C2, the path is directed in the positive y direction, so T = (0, 1), and F T is simply the second component of F(a + Δx,b):

F T = F2(a + Δx,b).

Along C3, the path is directed in the negative x direction, so the unit tangent vector is T = (-1, 0). The dot product F T is minus the first component of F(a,b + Δy):

F T = -F1(a,b + Δy).

Along C4, the path is directed in the negative y direction, so T = (0,-1), and F T is minus the second component of F(a,b):

F T = -F2(a,b).

We summarize these findings in the following figure.

PIC.

The integrals are now easy to compute. Along C1, F T is constant, so the integral is simply its value F1(a,b) times the length of the segment Δx:

C1F ds = C1F Tds = C1F1(a,b)ds = F1(a,bx.
Along C2, the integral is F2(a + Δx,b) times its length Δy:
C2F ds = F2(a + Δx,by.
Similarly,
C3F ds = -F1(a,b + Δyx,
and
C4F ds = -F2(a,by.

The integral around all of C is just the sum along the four segments

CF ds = C1F ds + C2F ds + C3F ds + C4F ds
= F1(a,bx + F2(a + Δx,by - F1(a,b + Δyx - F2(a,by.

The circulation per unit area is the integral divided by the area of the rectangle, which is ΔxΔy

∫  F ⋅ ds
--C------
  Δx Δy = F  (a + Δx,  b)Δy  - F  (a, b)Δy  - (F (a,b + Δy )Δx  - F (a,b)Δx )
--2------------------2-------------1-----------------1----------
                             Δx Δy,

where I simply rearranged the terms in the numerator.

Half of the numerator is multiplied by Δy and half is multiplied byΔx. If we separate these into two fractions, we can cancel the Δy in the first fraction with the Δy in the demoninator

F2-(a-+-Δx,-b)Δy----F2(a,-b)Δy--
            Δx Δy = F2(a-+-Δx,-b) --F2(a,b)-
          Δx.

In the second fraction, we can cancel the Δx,

F1-(a,-b +-Δy-)Δx---F1(a,-b)Δx--
            Δx Δy = F1(a,b-+-Δy-) --F1(a,b)-
          Δy.
Putting these back together, we have
∫ F ⋅ ds
-C-------
 Δx Δy = F  (a + Δx,  b) - F  (a, b)
--2---------------2-----
          Δx -F (a,b + Δy ) - F (a,b)
-1---------------1------
          Δy.

Now, we let the curve C shrink down to a point. This means that Δx 0 and Δy 0. In this limit, the two fractions become something familar: partial derivatives of F.

lim Δx,Δy0∫  F ⋅ ds
-C-------
 Δx Δy = lim Δx0F2(a-+-Δx,-b) --F2(a,b)-
          Δx - lim Δy0F1-(a,b +-Δy-) --F1-(a,-b)
          Δy
= ∂F
∂x--(a,b) -∂F
∂y--(a,b).

We have shown the the circulation per unit area around the point (x,y) = (a,b) is

∂F--
∂x(a,b) -∂F--
∂y(a,b).
This is exactly what we integrate over a region D to obtain the total circulation around the border of D, according to Green’s theorem
D( ∂F          ∂F      )
  ---(x,y ) - ---(x,y)
  ∂x          ∂ydA = ∂DF ds.

where C = ∂D is the path going counterclockwise around D.