This result states extrinsic weak harmonic map equation: given (U,g) be a smooth Riemannian domain and let N R q be a compact isometric embedding with second fundamental form A. A map u W 1,2 (U;N) is weakly harmonic if and only if, in.... It is useful in harmonic map theory and elliptic regularity, where variational identities, curvature estimates, and compactness arguments control geometric objects.