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
$$
W^{1,p}(M;N):=\{ u\in W^{1,p}(M;\Bbb R^K)\ |\ u(x)\in N \quad\text{almost every } x\in M\}.
$$



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> How do we show that $\nabla u$, the weak derivatives of $u$, belong to the tangent space $T\_{u(x)}N$ almost every $x$?
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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 $u\_n\in C^{\infty}\_c(M;N)$.



The gradient $\nabla u$ of $u\in W^{1,p}(M;N)$ is tangent to $N$ almost everywhere
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.


