I am doing self-reading on some degree theory using “Topological Degree Theory and Applications” by *Donal O’Regan, Yeol Je Cho, and Yu-Qing Chen*; I am totally new to this field and not quite familiar with the notion. We first define the degree of a $C^1$ map as follows:

where the $J_f := det Df$. I am stuck on this basic theorem:

I do not see how a direct computation works in this case. Could anyone help me with this? Thanks!