For any vector field on R^n it is possible to find a single function that will act as an integrating factor for the vector field in the sense that the new "renormalized" vector field will have zero divergence. This result has extraordinary implications for in physics a zero divergence implies some sort of conservation law.

As a way to construct (an infinite number of) such "renormalization" consider the function constructed from the functions Vk that are the components of the vector field:

Lamda:=(a*V1^p + b*V2^p + ...)^(n/p)

The function lambda is homogeneous of degree n.

When n = 1, any p, and any signature, a = plus or minus 1, b= plus or minus 1,...
this Holder function, when divided into the original vector field, will create a "renormalized" vector field which has zero divergence with respect to the initial variables!

The classic case is when a=b=...=1 and p=2. Then lambda is the Euclidean norm of the vector field and division of the vector field by this norm is defined as the Gauss Map. If by chance the divergence of this renormalized vector vanishes, then (in three dimensions) the normal resides on a minimal surface.

However, by forming the Jacobian matrix of the normal field, it is always possible to construct a set of vector "currents" which always have zero divergence (if n = 1). These vector currents are obtained by premultiplying the original scaled vector by the matrix adjoint to the Jacobian matrix. The elements of the Jacobian matrix are the partial derivatives of each of the components of the original vector Vk. The zero divergence condition generates a "Maxwell-Ampere" conserved current. Given a Vector potential, A, the induced conserved current, J, (when n = 1) is not unique, as any value of p, or a,b and c yields the zero divergence states.

When p = 2 and n = 1, is state in some sort of lowest configuration? Are the other surfaces ( p > 2) excited states in some sense? More on this idea later.

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