let C be a smooth simple closed curve in the plane and n be a unit vector normal to C.
prove that the integral of n.ds over C is 0.
You cannot compute a line integral over a vector field.
Therefore the integral of n.ds over just makes no sense.
I think you might mean the following:
Define the vector-field by .
From here we see that and (sum of components of the normal vector).
Thus, by divergence theorem,