2

Pressley's Differential Geometry book supposes that the angle between the tangent vector to a curve and any fixed vector is smooth (i.e. all orders of differentiablity exits and are continuous) and in order to prove that I approached a few ways and two of them blocked by the following two questions I need to prove so completing the proof for the mentioned problem:

1- Suppose that $t=(t_1,t_2)$ is a smooth function and $\langle t,n\rangle=0$ where $n$ is the unit vector perpendicular to $t$ formed by rotation of $t$ counterclockwise with angle $\pi/2$; how to prove that $n$ is smooth? (by intuition when t varies slowly so does n which is not only a rigorous proof but also suggests for just continuity).

2- $t=(t_1,t_2)=(\cos \phi (s), \sin \phi (s))$. Since $t$ is smooth so is $t_1$ and $t_2$. Since composition of a continuous function (here : sin and cos) and a discontinuous function (here: $\phi(s)$) still maybe continuous so much not possible to say about $\phi$; also arcsin and arccos are not smooth everywhere like sin and cos are; so is it is possible to deduce from smoothness of $t=(t_1,t_2)=(\cos \phi (s), \sin \phi (s))$, the smoothness of $\phi (s)$ at all?

Added: 3- If non of the two questions is answerable, so how to prove that that the angle between the tangent vector to a curve and any fixed vector is smooth, if the tangent vector is smooth?

  • $\langle t ,n \rangle=0$ does not imply that $n$ is smooth and $(\cos \phi(s), \sin \phi (s))$ continuous does not imply that $\phi$ is smooth. – Kavi Rama Murthy Sep 19 '18 at 06:06
  • @KaviRamaMurthy, so how to prove that that the angle between the tangent vector to a curve and any fixed vector is smooth, if the tangent vector is smooth? or how to prove that if the tangent vector to a curve is smooth so is the perpendicular vector to it? Thanks. –  Sep 19 '18 at 13:42
  • @KaviRamaMurthy gave very strong hints to both of your questions. Basically it was pointed out that you need additional assumptions. In terms of your two questions in your original post, for the first you need $n$ is continuous and that it has unit length. (Can you come up with counter examples where exactly one of the either is violated?) For your second the question should be: given $t_1, t_2$ smooth, with $t$ unit length, can one find a smooth function $\phi(s)$ such that $t_1 = \cos(\phi(s))$ and $t_2 = \sin(\phi(s))$. (Do you see how the question you asked is different from this?) – Willie Wong Sep 19 '18 at 13:53
  • @WillieWong, I can't prove anything:( if $n_1/n_2$ is smooth how can both $n_1$ and $n_2$ be smooth? - intuitively $n$ is continuous because it is always perpendicular to a continuous vector. But I can't prove it and even after I know the proof it is far from being smooth. –  Sep 19 '18 at 13:59

1 Answers1

0

Some hints:

  1. You have an explicit formula for $n$. The components of $n$ are $$ (n_1, n_2) = \left( \frac{-t_2}{\sqrt{t_1^2 + t_2^2}}, \frac{t_1}{\sqrt{t_1^2 + t_2^2}} \right) $$

  2. If $\phi(s)$ is a continuous function and you know $\cos(\phi(s))$ and $\sin(\phi(s))$ is smooth. Then in an interval where $\cos(\phi(s)) \neq 0$ you have by implicit differentiation $\phi'(s) = \dfrac{1}{\cos(\phi(s))} \dfrac{d}{ds} \sin(\phi(s))$ which is a ratio of two smooth functions and hence smooth.

    Now, away from the points $\cos(\phi) = \pm 1$, you have $\cos$ is (smoothly) invertible. And away from the points $\sin(\phi) = \pm 1$, you also have $\sin$ is (smoothly) invertible. Those two sets of points don't overlap.

Willie Wong
  • 73,139