TITLE: Taylor expansion for vector-valued function?
QUESTION [15 upvotes]: Let $f:\mathbb{R}^m \to \mathbb{R}^n$. Is it possible to do a Taylor expansion of $f$ around $\theta\in\mathbb{R}^m$? I am hoping for something like
$$f\left(\theta\right) = f\left(\theta_0\right) + A \left(\theta - \theta_0\right) + \left(\theta - \theta'\right)^T \text{something} \left(\theta - \theta'\right)$$
where $A$ is a $n\times m$ matrix, and its rows are gradient of $f_i$ ($i$-the entry of vector $f$) with respect to vector $\theta$. But what should "something" be?
For example, let us consider the simple case where $f :\mathbb{R}^m \to \mathbb{R}^2$. Denote $f = \left(f_1,f_2\right)$. Then
$$f_1\left(\theta_n\right) = f_1(\theta_0) + \nabla f_1\left(\theta_0\right)(\theta_n - \theta_0) + (\theta_n - \theta_0)^T H_1(\theta') (\theta_n - \theta_0) $$
$$f_2\left(\theta_n\right) = f_2(\theta_0) + \nabla f_2\left(\theta_0\right)(\theta_n - \theta_0) + (\theta_n - \theta_0)^T H_2(\theta') (\theta_n - \theta_0) $$
If I try to put this in matrix form, I will get something like
$$f\left(\theta_0\right) = f(\theta_0) + \begin{pmatrix} \nabla f_1(\theta_0)\\ \nabla f_2(\theta_0)\end{pmatrix} (\theta_n - \theta_0) + ???$$

REPLY [3 votes]: Just for the sake of completeness, Wikipedia knows such expansions under the name jet. The form of the remainder has also been discussed on this site.