The substended angle of a circle

Kontilera

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

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

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!