this is from "Sobolev Mappings between Metric Spaces" by P. Hajlasz (2.3):
The degree of a mapping $v: M \to N$ between two oriented $n$-dimensional compact manifolds without boundary can be de defined by the integral formula $$\deg v = \int_M \frac{\det Dv}{\text{vol } N}.$$ From the Hölder inequality it follows that the degree is continuous in the $W^{1,n}$ norm.
It is known that for $p = \dim M$ the class of smooth mappings build a dense subset of the class of Sobolev mappings (between manifolds). My question is how does one show the continuity? Especially the hint "Hölder inequality" leaves me clueless, since I need to show for $v_k \to v$ in $W^{1,n}$
$$\int_M \det D v_k - \det D v \underset{k \to \infty}{\longrightarrow} 0.$$
Any ideas?