I came across this post lying dormant on some online forum. I am putting it here verbatim, it seems to me worth a lot.
By Prof. S. D. Agashe, IIT Bombay
(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 $c\times r$, $a$, $b$, $c$ being constant unit vectors. Since $r$, $(a.r)b$ and $c\times r$ 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 $\lambda . r + \mu (a.r)b + \nu (c\times r)$, 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 ($a \times b + 2c$). 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.