Taylor expansion of scalar-valued function with multiple vectors as variables

Question

If we have a scalar-valued functions with multiple vectors as variables: for example $f(\mathbf x ,\mathbf y)$ where $\mathbf x, \mathbf y \in \mathbb R^3 \times \mathbb R^3$

What is the Taylor expansion of $f$ about $\mathbf x$ and $\mathbf y$?

Answer 1

Well, assuming $f$ is very smooth, then we have \begin{align} f(x+h&, y+k) =\ f(x, y) + Df(x, y)(h, k) + \frac{1}{2!}D^2f(x, y)[(h, k), (h, k)]\\ &\ +D^3f(x, y)[(h, k), (h, k), (h, k)] +\ldots + \frac{1}{n!}D^nf(x, y)[\underbrace{(h, k), \ldots, (h, k)}_{n-\text{times}}]+\mathcal{O}(|(h, k)|^{n+1}) \end{align} where $D^kf(x, y)$ is a $(k, 0)$-tensors. This is probably of no good use for most people; nevertheless, it's easy to remember.