Intuitive meaning of high order Fréchet derivative $D^k f_p(v_1, \cdots, v_l)$

Solution 1:

The fundamental theorem of polynomials asserts that every $k$-linear symmetric mapping $V^k \to W$ is associated to one, and just one, $k$-th order homogeneous function $V \to W.$ This is proven in induction on $k$ and there is somewhere an "explicit" formula (quite similar to that of the determinant). So, by virtue of this theorem, the $k$-th derivative at $p,$ which is a $k$-linear symmetric mapping, is no other than the $k$-th order homogeneous component in the Taylor expansion of the function. I am afraid there seems to be no more interpretation for such symmetric mapping. And, as all the derivatives are symmetric, one is then forced to study the "mirror" object of antisymmetric mappings; this gives rise to the so called "differential varieties" and "differential forms."

All of this can be found in Cartan's wonderful gems "Differential Calculus" and "Differential forms" (in Banach spaces). I am afraid that finding the books is quite impossible (the first one at least) nowadays.