Hessian in abstract notation

Question

For functions $u$ and $f$, is $$\mathrm{Hess} f(\nabla u,\nabla u)=(\nabla_i\nabla_j f)(\nabla^i u)(\nabla^j u),$$ or is it $(\nabla_i\nabla_j f(\nabla^j u))(\nabla^i u)$?

Answer 1

Let's write $\operatorname{Hess}(f) = \nabla^2 f$ and $\nabla u = (\nabla^i u)\partial_i$. Then: $$(\nabla^2f)(\nabla u, \nabla u) = (\nabla^2f)((\nabla^i u)\partial_i, (\nabla^j u)\partial_j) = (\nabla^i u)(\nabla^j u) \cdot (\nabla^2 f)_{ij} = (\nabla^i u)(\nabla^j u) \nabla_{i} \nabla_{j} f$$

Your second expression is therefore incorrect (assuming we interpret $(\nabla_i\nabla_j f(\nabla^j u))$ to mean $(\nabla^2 g)_{ij}$ where $g =f \nabla^j u$, which is the only reasonable interpretation where it doesn't coincide with your first expression).