Taylor series for multiple output functions

Question

So far, I have seen the Taylor series expansion for functions of $\mathbb{R}^n \rightarrow \mathbb{R}$. Now, let’s suppose that there exists a differentiable function $\mathbf{f}: \mathbb{R}^n \rightarrow \mathbb{R}^m$, where $m > 1$ (and can be greater than or equal to $n$). Would the Taylor series expansion for functions of $\mathbb{R}^n \rightarrow \mathbb{R}$ apply for each element of $\mathbf{f}$? If not, what would be the correct formula(s)?