I want to obtain the topological charge or winding number of the map $$ f_n(\mathbf{r})=(\sin \theta \cos (n \varphi), \sin \theta \sin (n \varphi), \cos \theta) $$ and my lecture notes say that it is given by $$ Q=\frac{1}{8 \pi} \epsilon_{i j k} \int d^2 S_k f_n \cdot\left(\partial_i f_n \wedge \partial_j f_n \right) . $$ However, I am really confused about what the indexes $ijk$ mean: are they the polar angular variables? Why has the surface area index k?
I would really appreciate any clarification!
