Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: [HM] possible Hilbert quote?
Replies: 39   Last Post: Apr 19, 2002 10:09 AM

 Messages: [ Previous | Next ]
 Prof. S.D. Agashe Posts: 30 Registered: 12/3/04
[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
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.