The substended angle of a circle

[Kontilera]

The subtended angle of a circle

Hello! I came across the following formula for the change in the subtended angle (of a circle), when we move our obersvation point $$\vec{x}$$ by $$\delta \vec{x},$$

$$\delta \Omega = \oint \hat{x} \cdot \frac{(\delta \vec{x} \times d\vec{l})}{|\vec{x}|^2}.$$

The integration path is the circumference of the circle.

Do you know any litterature or easy derivation that can prove this fact?

Thanks for your help, really appreciated!

 

[pat_connell]

You can find a proof of this formula in the book "Geometric Mechanics: Toward a Unification of Classical Physics" by Chicone andMarsden (2006). The proof is based on the use of Stokes' theorem.