EQUIVALENCE OF GREEN'S THEOREM AND THE DIVERGENCE THEOREM IN THE PLANE

We derived Green's Theorem and the Divergence Theorem in the plane separately, but considering flux and circulation density.

But it is useful to realize that they are really the same thing and one statement can easily be obtained from the other.

Consider a simple closed path C, and let D be the domain that it encloses. Suppose that the orientation is such that the path goes around the domain D counterclockwise.

Consider a vector field F, and lets compute the outward flux integral for F and C.

The Divergecne Theorem says that

Now let's turn this into a statement about tangential line integrals -- work integrals.

The reason that we can do this is that if N is the outward normal, then

T = N^perp

Now, flux elements involve dot products. How does the perp operation interact with these? The answer is as good as possible! Since the perp operation is a rotation through pi/2 counterclockwise, it doesn't change the dot product at all when applied to both vectors

That is, if A and B are any two vectors,

dot(A,B) = dot(A^perp,B^perp)

Hence

dot(F,T) = dot(F^perp,T^perp) = -dot(F^perp,N)

Therefore,

Now clearly

-div(F^perp) = Q_x(x,y) - P_y(x,y)

which is exactly the curl of F!

Therefore, with C, D and F as above,

which is exactly Green's Theorem.

Of course the argument can be run backwards, so in one sense, the two theorems are just different ways of saying the same thing.

Practical Consequences

Suppose the path C isn't closed, and the vector field F has zero divergence. and we are asked to compute a flux integral along C. The above equivalence can be used to turn this flux integral into the work integral for a curl free vecotr field. This can be evaluated by finding the potential function.

In this way we can transfer our potential function methodology to flux integrals!

But