Editing Hessian matrix (section)

=== Generalizations to Riemannian manifolds ===

Let <math>(M,g)</math> be a [[Riemannian manifold]] and <math>\nabla</math> its [[Levi-Civita connection]]. Let <math>f : M \to \R</math> be a smooth function. Define the Hessian tensor by
<math display=block>\operatorname{Hess}(f) \in \Gamma\left(T^*M \otimes T^*M\right) \quad \text{ by } \quad \operatorname{Hess}(f) := \nabla \nabla f = \nabla df,</math>
where this takes advantage of the fact that the first covariant derivative of a function is the same as its ordinary differential. Choosing local coordinates <math>\left\{x^i\right\}</math> gives a local expression for the Hessian as
<math display=block>\operatorname{Hess}(f)=\nabla_i\, \partial_j f \ dx^i \!\otimes\! dx^j = \left(\frac{\partial^2 f}{\partial x^i \partial x^j} - \Gamma_{ij}^k \frac{\partial f}{\partial x^k}\right) dx^i \otimes dx^j</math>
where <math>\Gamma^k_{ij}</math> are the [[Christoffel symbols]] of the connection. Other equivalent forms for the Hessian are given by
<math display=block>\operatorname{Hess}(f)(X, Y) = \langle \nabla_X \operatorname{grad} f,Y \rangle \quad \text{ and } \quad \operatorname{Hess}(f)(X,Y) = X(Yf)-df(\nabla_XY).</math>