Some help with question 8.b, chapter 2.3, O'Neil's Intro to Dif. Geometry

This exercise states the folowing:

" For a unit-speed curve $\beta(s) = (x(s),y(s)) \in \mathbb{R}^2$, the unit tangent is T = $\beta' = (x',y')$ as usual, but the unit normal N is defined by a rotation of T 90-degrees (i.e.: N =(-y',x')), so, N and T are collinear, and the plane curvature $\kappa$ of $\beta$ is defined by $T' = \kappa\ N$. The $\textit{slope angle } \phi(s)$ of $\beta$ is the differentiable function such that:

$$T = (\cos(\phi), \sin(\phi)) = \cos(\phi)U_x + \sin(\phi)U_y$$

Prove the existance of $\phi$. (Consider $f,g:I\subset\mathbb{R}^{n}\longrightarrow\mathbb{R}$ differentiable functions on $I$, assume $f^{2}+g^{2}=1$ and the existence of $\theta_{0}$ s.t. $f(0)=\cos\theta_{0}$, $g(0)=\sin\theta_{0}$. If $\theta$ is the function s.t. $\theta(t)=\theta_{0}+\int_0^t(fg^{\prime}-gf^{\prime})du$. Prove $f=\cos\theta$, $g=\sin\theta$, showing $\theta$ is differentiable, unambiguously defined at $I$)."

I'm lost. Can someone give me a hint? What's the point of the suggestion between parentheses? Thank you!

I take the domain of $\beta$ to be the interval $I$. The function $T:I\rightarrow \mathbb{R}^2$ is differentiable. As $\beta$ is unit speed, thus $Im(T)\subseteq \mathbb{S}^1$. Hence, for every $s\in I$, we will have $T(s)\in \mathbb{S}^1$ and so there must exist some angle $\phi_s$ (depending on $s$, that's why I added a subscript _s) such that $T(s)=e^{i\phi_s}$. The question now is can we choose the angle $\phi_s$ so that it depends differentiably on $s$. This is what the book wants you to prove. The book's hint is to prove the claim below:

Claim: Let $f,g: I\rightarrow \mathbb{R}^2$ be differentiable such that $f^2+g^2=1$ holds on all $I$, then there exists a differentialbe angle function $\theta:I\rightarrow \mathbb{R}$ such that $(f+ig)=e^{i\theta}$

Proof:

1. Take $\theta$ as defined in the hint given by your book.

2. Prove that $(f+ig)$, $e^{i\theta}$ are both solutions to the linear ODE $x'=i(fg'-gf')x$ (Where $x$ is a function with domain $I$ and codomain $\mathbb{C}$, multiplication denotes complex multiplication). $(f+ig),e^{i\theta}$ have the same inital condiiton so they must be equal by the uniquness theorem of ODEs