This result states euler-lagrange equation for harmonic maps: given (M,g) be a compact smooth Riemannian manifold without boundary, let (N,h) be a smooth Riemannian manifold, and let u:M N be a smooth map. Define the Dirichlet energy.... It is useful in harmonic map theory and elliptic regularity, where variational identities, curvature estimates, and compactness arguments control geometric objects.