Let $M$ be a compact subset of $\mathbb{R}^3$ with smooth boundary $S=\partial M$. Consider M with the standard orientation $\mu=\mu_{0}$ from $\mathbb{R}^3$ and $S$ with the boundary orientation $\partial \mu$ induced from M.
Prove that there exists a smooth unit normal field $n(x)=(n_{1}(x),n_{2}(x),n_{3}(x))$ ($x \in S$) on $S$ such that for any $x \in S$, if $\{v_{1},v_{2}\}$ is a basis for $T_{x}S$ with $\partial \mu(x)=[v_{1},v_{2}]$, then $[v_{1},v_{2},n(x)]=\mu_{0}.$
Where $[v_{1},v_{2}]$ denotes the equivalence class of $\{v_{1},v_{2}\}$ under the equivalence relation $\{v_{1},v_{2}\}\sim \{w_{1},w_{2}\}$ iff $\{v_{1},v_{2}\}$ and $\{w_{1},w_{2}\}$ have the same orientation.
I'm really unsure on how to do this, so any help would be greatly appreciated.
Although in some sense the Gram-Schmidt process works for this problem, there is a shortcut.
Generally speaking, given $n-1$ linearly independent vectors $v_1,...,v_{n-1}$ in $\mathbb{R}^n$, there is a unique unit vector $v_n$ which is orthogonal to each of $v_1,...,v_{n-1}$ and such that the basis $v_1,...,v_{n-1},v_n$ is positively oriented. This vector $v_n$ can be written as a function of the coordinates of $v_1,...,v_{n-1}$. Furthermore, each coordinate of $v_n$ is a smooth function of the coordinates of $v_1,....,v_{n-1}$.
For example, in dimension 2, if $v_1=\langle a,b \rangle$ then $v_2 = \langle -b,a \rangle \, / \, \sqrt{a^2+b^2}$. You can check the positive orientation condition by making a matrix with first row $v_1$ and second row $v_2$ and then computing the determinant to verify that it is positive.
In dimension 3, one can do the same with the cross product, taking $v_3 = (v_1 \times v_2) \, / \, \| v_1 \times v_2 \|$.
So then, near each point of $\partial M$ choose a coordinate patch for $\partial M$ which is positively oriented (with respect to the boundary orientation). As long as the coordinate functions of your coordinate patch are at least $C_1$, you obtain a positively oriented field of bases $v_1(p),...,v_{n-1}(p) \in T_p \partial M$ for $p$ in that patch, and the coordinate functions of each vector $v_i(p)$ are smooth. Thus the coordinte functions for $v_n(p)$ are smooth.
Now although $v_n(p)$ has been constructed patch-by-patch, where two patches overlap the values of $v_n(p)$ will be unique, because of the uniqueness of the construction of the unit orthogonal vector.