# Are there more “types” of derivatives than “symmetric” or “alternating”?

The $kth$ derivative of a function $f:\mathbb{R}^n \to \mathbb{R}$ can be thought of as a symmetric $k$-tensor. (Well, almost. It is not invariant under coordinate changes, and should really be thought of as part of a jet bundle. But we have nice coordinates on $\mathbb{R}^n$, so lets roll with it)

The exterior derivative maps alternating $k$-forms to alternating $k+1$-forms.

These two "types" of derivative are really the only ones I have met.

Is there a unified notion of "derivative" which contains both of these concepts? Assuming there is, are there other examples besides these two?

-

Basically, there are methods for defining and computing the invariants of a given geometric structure under the action of some specified (pseudo-)group of transformations, and they yield, as special cases, the exterior derivative of exterior forms, the covariant derivatives of functions with respect to an affine connection, the Schwarzian derivative of a function with respect to linear fractional transformations, and a host of others. Even the Levi-Civita connection of a Riemannian metric can be regarded as the natural first derivative of the metric, which gives a simple example of a (nonlinear) derivative that is not a tensor. Using Cartan's method of equivalence, for example, one can define the invariants to all orders of most $G$-structures and test them for equivalence under various pseudo-groups.