Proving Vector Calculus: Cyclic Integral of (r dot ds)=0

[Suvadip]

How to prove that

cyclic integral of (r dot ds) =0

(symbols having usual meanings).

Please help me


2) It is always possible to find curl when vector function is known, but how to find the vector when its curl is known.

 

[matiasmorant]

if you are asking how to prove that
$$\oint \overset{ }{ \overset{\to }{r}}.\overset{ }{\overset{\to }{\text{ds}}}=0$$

where r is the position vector, that is r={x , y}; the we can prove it using green's theorem. "the integral of a vector field around a closed curve equal de integral of the rotational of the field over the surface enclosed"

and rot(r)=0, as you can check; so the integral we are considering equals the integral of 0 over the surface enclosed by the curve S; which is 0 of course, since the integrand is 0.