Imagine I have a 2-variable 2-dimensional vector function $\mathbf{F}(\mathbf{x})=(F_1(\mathbf{x}),F_2(\mathbf{x}))$, where $\mathbf{x}=(x_1,x_2)$. How do I Taylor-expand such function around $\mathbf{x}=\mathbf{0}$ to third order terms?
My approach: Up to second order, we may write $$ \mathbf{F}(\mathbf{x})=\mathbf{F}(\mathbf{0})+\mathbf{J}(\mathbf{0})\mathbf{x}+\frac12 \begin{pmatrix} \mathbf{x}^T \mathbf{H}_1(\mathbf{0})\mathbf{x}\\ \mathbf{x}^T \mathbf{H}_2(\mathbf{0})\mathbf{x} \end{pmatrix}+h.o.t. $$ where $\mathbf{J}$ is the Jacobian matrix of $\mathbf{F}$, and $\mathbf{H}_1$ and $\mathbf{H}_2$ are the Hessian matrices of functions $F_1$ and $F_2$, respectively. How does one write the third term in similar compact notation?
Following the general answer here and the formulation here, we have the following.
Consider a function $\mathbf{F}:\mathbb{R}^n\to \mathbb{R}^m$ given by $\mathbf{F}(\mathbf{x})=(F_1(\mathbf{x}),...,F_m(\mathbf{x}))$, where $\mathbf{x}=(x_0,...,x_n)$.
The general $k$th-order Taylor expansion of $\mathbf{F}(\mathbf{x}+\mathbf{x}_0)$ about $\mathbf{x}_0$ is given by $$ \mathbf{F}(\mathbf{x}+\mathbf{x}_0)\simeq T_{\mathbf{F},\mathbf{x}_0,k}(\mathbf{x}):=\sum_{j=0}^k\frac{(D^j\mathbf{F})_{\mathbf{x}_0}[(\mathbf{x})^j]}{j!} $$ where the Frechet-derivative terms $(D^j\mathbf{F})_{\mathbf{x}_0}[(\mathbf{x})^j]$ may be written in the vector form as $$ \begin{align}\label{eqfrechetd} (D^j\mathbf{F})_{\mathbf{x}_0}[(\mathbf{x})^j]= \begin{pmatrix} \sum_{i_1,...,i_j=1}^n\frac{\partial^jF_1}{\partial x_{i_1}\cdots \partial x_{i_j}}(\mathbf{x}_0) (x_{i_1}\cdots x_{i_j})\\ \vdots\\ \sum_{i_1,...,i_j=1}^n\frac{\partial^jF_m}{\partial x_{i_1}\cdots \partial x_{i_j}}(\mathbf{x}_0) (x_{i_1}\cdots x_{i_j}) \end{pmatrix} \end{align} $$ where we used the notation $$ \begin{align} \sum_{i_1,...,i_j=1}^n&=\sum_{i_1=1}^n\cdots \sum_{i_j=1}^n \end{align} $$ In my case, $n=m=2$ and so the third term of the Taylor expansion is given by $$ \frac16(D^3\mathbf{F})_{\mathbf{x}_0}[(\mathbf{x})^3]= \frac16\begin{pmatrix} \sum_{i_1,i_2,i_3=1}^2\frac{\partial^3F_1}{\partial x_{i_1}\partial x_{i_2}\partial x_{i_3}}(\mathbf{x}_0) x_{i_1}x_{i_2}x_{i_3}\\ \sum_{i_1,i_2,i_3=1}^2\frac{\partial^3F_2}{\partial x_{i_1}\partial x_{i_2}\partial x_{i_3}}(\mathbf{x}_0) x_{i_1}x_{i_2}x_{i_3} \end{pmatrix} $$