I am doing an independent study at my school and have been studying the book Calculus on Manifolds by Spivak. There are a couple of details Spivak states are "easy to check" but even with the help of our professor we were not able to do so.
Let $M\subset \mathbb{R}^n$ be a $k$-dimensional manifold with boundary. Then $\partial M$ is a $(k-1)$-dimensional manifold. Let $x\in \partial M$, so $(\partial M)_{x} \subset M_{x}$ is a $(k-1)$-dimensional subspace, and thus there are exactly two unit vectors in $M_{x}$ that are orthogonal to $(\partial M)_{x}$. Let $f: W \to \mathbb{R}^n$ be a coordinate system with $W \subset H^k$ and $f(0)=x$. $f$ induces a linear transformation $f_{*}: \mathbb{R}_{0}^{k}\to \mathbb{R}^n_{x}$ defined by $f_{*}(v_0)=(f^{\prime}(0)v)_{x}$. Then only one of these unit vectors can be written as $f_{*}(v_{0})$ for some $v\in \mathbb{R}^k$ with $v_k < 0$. Spivak states it can be checked that this definition does not depend on choice of coordinate system $f$, but how can I show this? Any help is much appreciated.
