Let $N\hookrightarrow\mathbb R^q$ be a compact isometric embedding with second fundamental form $A$ defined by $A_p(X,Y)=(D_XY)^\perp$. Let $(U,g)$ be a smooth Riemannian domain. A map $u\in W^{1,2}(U;N)$ is weakly harmonic, meaning stationary for the Dirichlet energy under all compactly supported projected ambient variations $u_t=\Pi(u+t\psi)$ with $\psi\in C_c^\infty(U;\mathbb R^q)$, if and only if, in the sense of distributions,

\begin{align*} \Delta_g u - A(u)(du,du)_g=0. \end{align*}