Linear linear everywhere....

We had the distinct pleasure of being lectured by Dr. Ron Gentle last week as our regular Calc IV(multi-variable) professor was out of town for two days. Dr. Gentle is known for viewing the entire universe as a linear construct of some sort or another and the subject matter for his lectures, multi-variable chain rule differentiation, was no exception, being distilled down to a simple vector multiplication for any possible type of differentiation out there. I'll probably talk more about that later, but the main point is that Dr. Gentle is right, linear algebra is *everywhere*.

A case in point was today's lecture (from the usual guy, Dr. Dale Garraway) on directional derivatives. To make a long story short, we have already worked through partial derivatives where given $f(x,y)=x^2+y^3$, the first partial derivative in the $x$ direction is $f_x(x,y)=2x$ and the first partial in the $y$ direction is $f_y(x,y)=3y^2$.

Great, but what if you want to know the derivative in some other direction, like, say, in the direction of the vector $\vec{v}=(a,b)$? The general answer is this: $f_{\vec{v}}(x,y)=f_x(x,y)a + f_y(x,y)b$. The astute reader will notice that this is just a linear combination of the two first derivatives. Could it be that the first derivatives just form a basis for a function space defining all the derivatives of this multi-variable function? It sure looks like it.

In fact it looks like essentially a basis conversion from a spatial basis (the vector in $\mathcal{R}^2$ or something like that) to a functional basis giving the derivative. Neat.