In the lecture notes I'm reading, an immersion is defined as a $C^{\infty}$ map $f:U\rightarrow V$ between open subsets $U\subseteq \mathbb{R}^m$ resp. $V\subseteq\mathbb{R}^n$, with $m\leq n$, such that for all $x\in U$ the rank of $Df(x)$ be maximal, i.e., be equal to $m$.
My question is: 1) Does this mean that the $Df(x)$ is constant as long as $x$ varies of a connected component of $U$? If yes, how can I prove this?
2) Is this definition standard? Wikipedia only states this definition for manifolds and that is too advanced for me, so that I'd be able to see if that definition matches this one.
1) No. An example would be the map $f\colon \mathbb{R} \to \mathbb{R},~~x\mapsto (x,x^2)$. It maps the real line onto the standard parabola. This is an immersion, because $Df(x) = \begin{pmatrix}1\\2x\end{pmatrix}$ and this matrix of size $2\times1$ has rank $1$, which is maximal. But you can also see that $Df(x)$ is nowhere constant (and $U=\mathbb{R}$ has just one component).
2) This is a standard definition, a special case of the definition on wikipedia. Quote:
You don't know what a manifold is, but in your case, $D_p f$ is nothing else then $Df(p)$ and the manifold is the open set $U$, which has dimension $m$. Note that you wrote $m\leq n$, so $\min(m,n)=m$