Hessian form

Given a smooth manifoldMathworldPlanetmath M and f:M being in C2(M), if x is a critical pointMathworldPlanetmath of f, that is df=0 at x, then we can define a symmetric 2-form


where HT*2 and u and v are any vector fields that take the values ux and vx, respectively, at point x. Equality of the two defining expressions follows from the fact that x is a critical point of f, because then [u,v](f)=df([u,v])=0, where [u,v] denotes the Lie bracket of the two vector fields. The form H is called the Hessian form.

In local coordinates, the Hessian form is given by


Its components are those of the Hessian matrix in the same coordinates. The advantage of the above formulation is coordinate independence. However, the price is that the Hessian form is only defined at critical points. It does not define a tensor field as one would naïvely expect.

Using the Hessian form, it is possible to analyze the critical points of f (determine whether they are local minima, maxima, or saddle points) in a coordinate independent way.

Title Hessian form
Canonical name HessianForm
Date of creation 2013-03-22 15:00:03
Last modified on 2013-03-22 15:00:03
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 6
Author PrimeFan (13766)
Entry type Definition
Classification msc 26B12
Related topic HessianMatrix
Related topic RelationsBetweenHessianMatrixAndLocalExtrema
Related topic TestsForLocalExtremaForLagrangeMultiplierMethod