From some lecture notes I am trying to puzzle through ....

"... the derivative or Jacobian of a smooth map $f: \mathbb{R}^m \rightarrow \mathbb{R}^n$ at a point $x$ is a linear map $Df: \mathbb{R}^m \rightarrow \mathbb{R}^n$. In terms of partial derivatives, $Df\_x(X) = (\sum\_j\partial\_{x\_j}f\_1 \cdot X\_j,
\sum\_j \partial\_{x\_j}f\_2\cdot X\_j, ...)$ ... "

I'm so confused I'm not even sure where to begin. Well, first, shouldn't the derivative be a map $Df:\mathbb{R}^m\rightarrow \mathbb{R}^m\times\mathbb{R}^n$? Third, I am familiar with 3D integral calculus, and the only Jacobian I heard discussed there doesn't look like this at all, except, of course, that they both involve partial derivatibes. Also, I don't even know what $f\_1 \cdot X\_j$ means.

Thanks.