Positively oriented unit tangent

Question

$\{(x,y) \ | \ 1 \leq x^2+y^2 \leq 4\}$

The solution says that the unit tangent for the larger curve with radius $2$ is $\frac{1}{2}(-y,x)$ and the tangent for the smaller circle is $(y,-x)$.

Why this sign change ?

Answer 1

There is a classical reason (see figure): imagine that a small "tunnel" is digged into this set (an "annulus"), for example with coordinates $[\pm \cos(\varepsilon), \pm\sin(\varepsilon)]$ and $[\pm 2\cos(\varepsilon), \pm 2\sin(\varepsilon)]$ (for a small angle $\varepsilon$), then the global positive orientation you will give to the new curve (outside circle + inside circle + the 2 "sides" of the small tunnel) will provide in a natural way a negative orientation to the interior circle.

Remark: this kind of contour is classical in a branch of analysis called "complex integration"; see for example (Not getting $-\frac{\pi}{4}$ for my integral. Help with algebra).