Let $(U,g)$ be a smooth Riemannian domain and let $N\hookrightarrow\mathbb R^q$ be a compact isometric embedding with second fundamental form $A$. A map $u\in W^{1,2}(U;N)$ is weakly harmonic if and only if, in the sense of distributions,
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\begin{align*}
\Delta_g u + A(u)(du,du)=0.
\end{align*}