In a textbook im reading about manifolds they write:
Let $W \subset \mathbb{ R } ^n $ be an open set and $ F : W \rightarrow \mathbb{ R } ^ m $ a $C ^ k $ function such that for every point $ p \in M := F ^ { -1 }( 0 $ the derivative $DF( p ): \mathbb{ R } ^ n \rightarrow \mathbb{ R } ^ m$ has rang $m $. Then $ M $ is a $C ^ k $-manifold with dimension $d = n - m$.
Shouldnt the derivate be a function from the type $ DF(p) : \mathbb{R}^n \rightarrow \mathbb{R } ^ { m \times n }$ ?
The first time I saw this I gussed it was a mistake in the textbook but it seems to appear multiple times afterwards. Do I get something wrong there?