Corollaries on Boundary Theorems.

There are two corollaries to the boundary theorems.


1. Cor 1: Consider the circulation over a closed curve Γ of a  vector field C ( x, y, z) which has Curl C =0 everywhere. Then the RHS of Stokes Theorem is a surface integral of the normal component of the Curl C over an open surafce S which has Γ as its boundary is equal to ZERO as C has zero curl everywhere. So also the LHS of the Stokes theorem which is the circulation of C over Γ is equal to ZERO. Now break the closed curve Γ into two paths from point A to B and then back to A. Since this is zero this can only happen if the line integral of C is path independent
so that its value from A to B is equal to negative of its value from B to A. Now the gradient theorem tells us that if the line integral of the vector field C is path independent then C=∇ φ ( x, y, z) where φ is a scalar field. So we have the corollary that if Curl C=0 then C=grad φ Or Curl C is zero means that the vector field is a CONSERVATIVE field. It also shows that Curl ( grad φ)=0 identically.


2. The second corollary involves the surface integral of a vector field C over an open surface S which has the closed curve Γ as its boundary. Now let the closed curve Γ PINCH OFF to a point. In this limit of the curve pinching to a point the open surface S becomes a closed surface. Now consider circulation of a vector field C over Γ . In the PINCHING LIMIT the circulation is ZERO as the curve pinches off to a point. The corresponding surface integral of the normal component of Curl C by stokes theorem on the RHS is now over a closed surface S in the pinching limit and this is also zero.


Now since the surface integral of Curl C is over a closed surface S in the pinching limit we can apply the Gauss Theorem to the surface integral and this is equal to the volume integral of the Div ( Curl C)
over the volume enclosed by the surface S ( which is CLOSED in the PINCHING LIMIT) and this is
also equal to ZERO. Now since the volume is arbitrary this means that DIv ( Curl C) is identically zero. So the Cor is that if Div v=0 of a vector field v then it is possible to express v=Curl C.