So far I've only seen examples of high-order (ie: second, third) partial derivatives of functions $f:\mathbb{R^n} \to \mathbb{R}$, but not any with $f:\mathbb{R^n} \to \mathbb{R^m}$.
Is there a reason for this? Is this because the definition of a partial derivative is a function $f: \mathbb{R^n} \to \mathbb{R}$?
Merry Christmas!
Yeah, it is defined for real-valued functions. However, given a function mapping to $\mathbb{R}^m$, you are able to consider component functions $f_i = \pi_i \circ f$, and then consider their partial derivatives.
In fact, in the usual multivariate calculus you probably considered $X: D \rightarrow \mathbb{R}^3$ where $D$ is the parameter domain and typically we denote $(u,v) \in D$. You may recall, we calculate the normal vector field $N(u,v) = X_u \times X_v$ where $$ X_u = \frac{\partial X}{\partial u} = \left< \frac{\partial x}{\partial u}, \frac{\partial y}{\partial u}, \frac{\partial z}{\partial u} \right> \qquad \& \qquad X_v = \frac{\partial X}{\partial v} = \left< \frac{\partial x}{\partial v}, \frac{\partial y}{\partial v}, \frac{\partial z}{\partial v} \right>$$ So, I contend, these sort of partial derivatives are quite common. The reason we don't talk much about the derivative of $F: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is that it is not a simple scalar function. However, it is naturally connected to the linearization of $F$ at a given point. Similarly, higher derivatives are connected with the best multinomial expansions and the higher derivatives are described properly by symmetric multilinear functions which are derived from iterated Frechet derivatives. This is all in the 2nd volume of Zorich, or, many other good advanced calculus texts.
