The gradient $\nabla u$ of $u\in W^{1,p}(M;N)$ is tangent to $N$ almost everywhere Let $M,N$ be (compact) Riemannian manifolds. $N$ is viewed as an embedded submanifold of $\Bbb R^K$. The Sobolev space $W^{1,p}(M;N)$ is defined as
$$
W^{1,p}(M;N):=\{ u\in W^{1,p}(M;\Bbb R^K)\ |\ u(x)\in N \quad\text{almost every   } x\in M\}.
$$

How do we show that $\nabla u$, the weak derivatives of $u$, belong to the tangent space $T_{u(x)}N$ almost every $x$?

I don't know if this is essential or not, but I am willing to assume that the structure/dimension of $M,N$ admits the approximation of every $u\in W^{1,p}(M;N)$ by smooth functions $u_n\in C^{\infty}_c(M;N)$.
 A: Here is another argument. Every Sobolev function in $\mathbb{R}^n$ has a representative which is absolutely continuous on almost all lines parallel to coordinate axes. This is Theorem 2 in Section 4.9.2 in  L. C. Evans, R. F.  Measure theory and fine properties of functions.  Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. Now $u\in W^{1,p}(M,\mathbb{R}^K)$, when represented in a local coordinate system maps line segments parallel to coordinate system to rectifiable curves in $N$ and hence the derivative of $u$ maps tangent vectors to vectors tangent to rectifiable curves in $N$ and hence tangent to $N$. This implies that $Du$ maps $T_xM$ to $T_{u(x)}N$ for almost all $x\in M$.
Regarding the last comment in the question, in general you cannot assume that $u\in W^{1,p}(M,N)$ can be approximated by smooth maps $C^\infty(M,N)$. For example $u:W^{1,p}(B^n,S^{n-1})$, $u(x)=x/|x|$ cannot be approximated by smooth maps $C^\infty(B^n,S^{n-1})$ when $n-1\leq p<n$ and one can easily modify this example to have a mapping $u\in W^{1,p}(S^{n},S^{n-1})$ that cannot be approximated by maps in $C^\infty(S^{n},S^{n-1})$ when $n-1\leq p<n$. This is a result due to Schoen and Uhlenbeck. For references and related results see for example this
 survey papepr.
A: We will assume $N$ is closed, while $M$ may have boundary. Also let $n = \mathrm{dim}(M).$
If $n \leq p < \infty$ we have $C^{\infty}(M;N)$ is dense in $W^{1,p}(M;N),$ so let $u_k$ be a sequence of smooth functions tending to $u \in W^{1,p}(M;N).$ Now let $x \in M$ and let $x_1,\dots,x_n$ be a choice of coordinates on $U \ni x.$ Then we get each $\partial_{x_i} u_k \rightarrow \partial_{x_i}u$ in $L^2(U;\mathbb R^{K+K})$ as $k \rightarrow \infty$ (here each $\partial_{x_i} u_k$ takes values in $TN \hookrightarrow \mathbb R^{K+K}$). Then passing to a subsequence we can assume each $\partial_{x_i}u_k \rightarrow \partial_{x_i}u$ almost everywhere on $U,$ so we get $\partial_{x_i}u(y) \in T_{u(y)}N$ for almost every $y \in U.$ Doing this around each point implies the result.
Note this argument only requires $u \in W^{1,p}_{\mathrm{loc}}(M,N)$ with $n \leq p,$ so in particular this also establishes the $p = \infty$ case.
