The Hessian(f, v) command computes the Hessian Matrix of the function f with respect to the variables in v. This is the Matrix with an (i,j)th entry of diff(f, v[i], v[j]).

If v is not provided, the differentiation variables are determined from the ambient coordinate system (see SetCoordinates), if possible.

If p is supplied, the computed Hessian matrix will be evaluated at the corresponding point.  The dimension of the point must equal the number of differentiation variables.

The det option specifies whether the determinant of the Hessian matrix is also returned.  If given as determinant = true, or just determinant, then an expression sequence containing the Hessian matrix and its determinant is returned.

If any options are given in opts, they will be passed on to the construction of the returned Matrix. For details on available options, see Matrix.

$\mathrm{with}\left(\mathrm{VectorCalculus}\right)\:$

$\mathrm{Hessian}\left(\cos \left(xy\right)\,\left[x\,y\right]\right)$

$\left[\begin{array}{cc}-y^2\cos \left(xy\right)& -\sin \left(xy\right)-yx\cos \left(xy\right)\\ -\sin \left(xy\right)-yx\cos \left(xy\right)& -x^2\cos \left(xy\right)\end{array}\right]$

$H≔\mathrm{Hessian}\left(\frac{1}{x^2+y^2+z^2}\,\left[x\,y\,z\right]\right)$

$H≔\left[\begin{array}{ccc}\frac{8x^2}{{\left(x^2+y^2+z^2\right)}^3}-\frac{2}{{\left(x^2+y^2+z^2\right)}^2}& \frac{8xy}{{\left(x^2+y^2+z^2\right)}^3}& \frac{8xz}{{\left(x^2+y^2+z^2\right)}^3}\\ \frac{8xy}{{\left(x^2+y^2+z^2\right)}^3}& \frac{8y^2}{{\left(x^2+y^2+z^2\right)}^3}-\frac{2}{{\left(x^2+y^2+z^2\right)}^2}& \frac{8yz}{{\left(x^2+y^2+z^2\right)}^3}\\ \frac{8xz}{{\left(x^2+y^2+z^2\right)}^3}& \frac{8yz}{{\left(x^2+y^2+z^2\right)}^3}& \frac{8z^2}{{\left(x^2+y^2+z^2\right)}^3}-\frac{2}{{\left(x^2+y^2+z^2\right)}^2}\end{array}\right]$

$\mathrm{SetCoordinates}\left(\mathrm{polar}\left[r,t\right]\right)\:$

$\mathrm{Hessian}\left(r\cos \left(t\right)\,\mathrm{shape}=\mathrm{symmetric}\,\mathrm{determinant}\right)$

$\left[\begin{array}{cc}0& -\sin \left(t\right)\\ -\sin \left(t\right)& -r\cos \left(t\right)\end{array}\right],-{\sin \left(t\right)}^2$

$\mathrm{Hessian}\left(r\cos \left(t\right)\,\left[r\,t\right]=\left[1\,\frac{\mathrm{\pi }}{3}\right]\right)$

$\left[\begin{array}{cc}0& -\frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2}& -\frac{1}{2}\end{array}\right]$

The determinant option was introduced in Maple 16.

For more information on Maple 16 changes, see Updates in Maple 16.