where and and are any vector fields that take the values and , respectively, at point . Equality of the two defining expressions follows from the fact that is a critical point of , because then , where denotes the Lie bracket of the two vector fields. The form 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.
|Date of creation||2013-03-22 15:00:03|
|Last modified on||2013-03-22 15:00:03|
|Last modified by||PrimeFan (13766)|