  Given a compact manifold $N^n$, an integer $k \in \mathbb{N}_*$ and an
exponent $1 \le p < \infty$, we prove that the class $C^\infty(\overline{Q}^m;
N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect
to the strong topology in the Sobolev space $W^{k, p}(Q^m; N^n)$ when the
homotopy group $\pi_{\lfloor kp \rfloor}(N^n)$ of order $\lfloor kp \rfloor$ is
trivial. We also prove the density of maps that are smooth except for a set of
dimension $m - \lfloor kp \rfloor - 1$, without any restriction on the homotopy
group of $N^n$
