Let $M\subset \mathbb{R}^n$ be a closed compact smooth oriantable manifold. Let $E\to M$ be the normal bundle of the embedding and $N\subset \mathbb{R}^n$ a (closed) tubular neighbourhood. $N$ is an $n$-manifold with boundary and $\partial N$ an $(n-1)$-manifold.
The Gauss map $G:\partial N\to S^{n-1}$ sends an element $v$ of $\partial N$ to the normalized normal vector pointing from $v$ outwards (image).
How can I define this map more rigorously? I.e., how can I make ''pointing from $v$ outwards'' more precise?
Now $\partial N$ is homotopic to the total space $E'$ of the sphere bundle $E'\to M$ of $E$.
Let $T$ be the tangent bundle of $M$ and $E\oplus T\cong \mathbb{R}^n\times M$ the trivial bundle on $M$. The inclusion $E\subset E\oplus T\cong \mathbb{R}^n\times M$ of vector bundles on $M$ restricts to sphere bundles and composing with the projection is $$ G':\partial N\cong E'\to (S^{n-1}\times M)\to S^{n-1} $$
How can I see that $G'$ is homotopic to the Gauss map $G$?