I've problem understanding the proof of the following Theorem:
Theorem [Whitney]: Let $M\subset \mathbb{R}^{2n}$ be a compact oriented $n$-dimensional submanifold without boundary. Then $M$ has non vanishing normal vector field.
The proof given in Hirsch's book is very short (page 138). It says that it's equivalent to prove that the Euler number of the normal bundle $\chi(\nu)=0$ and therefore he writes: $$ \chi(\nu)=\sharp(M,M;W)\overset{(*)}{=}\sharp(M,M;\mathbb{R}^{2n})\overset{(**)}{=}0$$ where $W$ is a neighbourhood of the zero section of $\nu$ identified with a neighbourhood of $M$ in $\mathbb{R}^{2n}$.
What I don't understand are the equalities $(*)$ and $(**)$ and where the hypothesis of orientability and the fact that $M$ is embedded in $\mathbb{R}^{2n}$ are needed in this reasoning.
$(**)$ could be explained by the fact that if the bundle is trivial $(\mathbb{R}^{2n})$ then clearly the intersection number should be zero. But I don't know how to pass from $W$ to $\mathbb{R}^{2n}$