I have solved the vector identity closed integral (n^)dS=0
by using n^ as the unit vector in spherical polar system and dS as the corresponding appropriate area element.It is easy.But I think if it will do.Are there any other method to do it?

\int_S \hat{n}.dS = 0

This?

I think it depends on how strinct your math teacher is. I think a specific case is fine, but mathematicians get picky about things like this.