Editing Hessian matrix (section)

=== Vector-valued functions ===

If <math>f</math> is instead a [[vector field]] <math>\mathbf{f} : \R^n \to \R^m,</math> that is,
<math display=block>\mathbf f(\mathbf x) = \left(f_1(\mathbf x), f_2(\mathbf x), \ldots, f_m(\mathbf x)\right),</math>
then the collection of second partial derivatives is not a <math>n \times n</math> matrix, but rather a third-order [[tensor]]. This can be thought of as an array of <math>m</math> Hessian matrices, one for each component of <math>\mathbf{f}</math>:
<math display=block>\mathbf H(\mathbf f) = \left(\mathbf H(f_1), \mathbf H(f_2), \ldots, \mathbf H(f_m)\right).</math>
This tensor degenerates to the usual Hessian matrix when <math>m = 1.</math>