>>13070633Now it gets more interesting.
We say that is totally differentiable at if there exists a matrix/linear operator such that
.
What is going on here? What is going on is that in the point x the matrix M is the best linear approximation to f. To exemplify this:
f is totally differentiable in x if and only if there exists an "error function" such that
which fulfills , so the error goes to zero in O(h)!
So then you say that is the total derivative of f in x. This is a matrix/linear operator!
So then you say that f is everywhere totally differentiable if f has a total derivative in every point x.
What does the total derivative represent? Easy - given a normalized vector v, we have
, so you apply the matrix to the vector and get the partial derivative.
Similarly if you apply the total derivative to the coordinate directions you see that its columns are the partial derivatives.
Existence and continuity of partial derivatives in all coordinate directions at a point
implies
existence of a total derivative at that point
implies
existence but maybe not continuity of all partial derivatives at that point.
Surjectivity of Df(x) here refers to surjectivity of the matrix/linear operator Df(x) and is interesting if you want to know how the tangent space of the graph of your function looks.
If you're doing DiffGeo you better learn about tangent spaces or you're indeed fukd.
