This result states extrinsic euler-lagrange equation for weak harmonic maps: given N R q be a compact isometric embedding with second fundamental form A defined by A p(X,Y)=(D XY) . Let (U,g) be a smooth Riemannian domain. A map u W 1,2 (U;N) is weakly.... It is useful in harmonic map theory and elliptic regularity, where variational identities, curvature estimates, and compactness arguments control geometric objects.