I came across the following "linear algebra" result recently:
(Source: Vector Calculus, by Durgaprasanna Bhattacharyya, University Studies Series,Griffith Prize Thesis, 1918, published by the University of Calcutta, India, 1920, 90 pp)
Chapter IV: The Linear Vector Function, article 15, p.24:
"The most general vector expression linear in r can contain terms only of three possible types, r, (a.r)b and cxr, a, b, c being constant unit vectors. Since r, (a.r)b and cxr are in general non-coplanar,it follows from the theorem of the parallelepiped of vectors that the most general linear vector expression can be written in the form
lambda r + mu (a.r)b + nu cxr
where lambda, mu, nu are scalar constants".
Bhattacharyya does not prove this. Has anyone seen a similar result and its proof?
Bhattacharyya uses this to show that the divergence of the linear function is (3 lambda + a.b), that the curl is (axb + 2c). He goes on to define div and curl of a differentiable function as the div and curl of the (linear) derivative function. The div and curl of a linear function are defined in terms of certain surface integrals.
I am excited about this result because it seems to provide an excellent route to div and curl, as Bhattacharyya himself remarks.
Sorry for a rather long and "technical" communication.