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

Question

Let $f:V \to W$ be a map between two Banach spaces $V$ and $W$. Let's denote the $k$-th Fréchet derivative of $f$ at $p$ as $D^kf_p$. Then $D^kf_p(v_1, v_2, \cdots, v_l)$ is a $(k-l)$-linear map from $l$ product of $V$ to $W$. Are there any intuitive meaning of the map? If $l = 1$, $D^kf_p(v)$ is a linear approximation of $D^{k-1}f_{p+v} - D^{k-1}f_{p}$ by definition. If $l = k$ and $v_1 = \cdots = v_k = v$, then it gives the order $k$ term of the Taylor's expansion of $f(p +v)$. However, for other cases, what meaning can we give to the quantity in general?

Answer 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.

Answer 2

It depends, what is meant by "meaning". I tend to think of the $k$-th derivative as the infinite-dimensional analogon of iterated partial derivatives. If $V=\mathbb R^d$ and $v_i = e_{n_i}$ are some canonical basis vectors, then $$ D^k f_p(v_1,\dots,v_k) = \frac{\partial^k}{\partial x_{n_1}\dots\partial x_{n_k}} f(p) \,.$$ If the $v_i$ are not the canonical basis vectors then we have to take the iterated directional derivatives.

You mentioned the interpretation with the Taylor series. Just as $D^k f_p(v,\dots,v)$ is the $k$-th term in the Taylor expansion of $f$, in the same manner $$ D^k f_p(v_1,\dots,v_l)$$ evaluated at $v,\dots,v$ is the $k-l$-th term of the Taylor expansion of $D^l f_p(v_1,\dots,v_l)$.

Another way of looking at it is to consider the $k$-linear map $$ (v_1,\dots,v_k) \mapsto D^k f_p (v_1,\dots,v_k) $$ as the basic object and the $k-l$-linear map $$ (v_{l+1},\dots,v_k) \mapsto D^k f_p(v_1,\dots,v_l)(v_{l+1},\dots,v_k) $$ as the result of fixing the first $l$ values.

I realize that my answer does to some extent restate the obvious. But beyond the statement that $D^k f_p$ is the $k$-th order term in the Taylor approximation, I do not have much to add.