
[HM] A "linear algebra" result
Posted:
Apr 19, 2002 10:09 AM


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 noncoplanar,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.
S D Agashe, from a humid Mumbai

