The winding number n for a map from S2→S2 is given by eq.\,(92.35), where φ^aφ^a=1. We will prove that n is invariant under an infinitesimal deformation of φ^. Since any smooth deformation can be made by compounding infinitesimal ones, this will prove that n is invariant under any smooth deformation.
a) Consider an infinitesimal deformation of φ^, φ^→φ^+δφ^. Show that φ^⋅δφ^=0 and that φ^⋅∂iφ^=0.
b) Use these results to show that εabcδφ^a∂iφ^b∂jφ^c=0.
c) Use this to show that δn=0.