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 $\frac{{\partial }^2}{\partial v_i\partial v_j}f$.

If the point p is supplied, the computed Hessian will be evaluated at that point. The dimension of the point must equal the number of variables in v.

If the right side of det is true, an expression sequence containing the Hessian Matrix and its determinant, in that order, is returned. If the right side of det is false, the Hessian Matrix is returned. If this parameter is the word determinant, it is interpreted as determinant=true. If the det parameter is not specified, it defaults to determinant=false.

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.

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

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

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

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

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

$\mathrm{eval}\left(H\,\left[x=0.3\,y=0.7\,z=0.1\right]\right)$

$\left[\begin{array}{ccc}\mathrm{-2.239761612}& 8.179998928& 1.168571275\\ 8.179998928& 13.34118873& 2.726666310\\ 1.168571275& 2.726666310& \mathrm{-5.355951680}\end{array}\right]$

$\mathrm{Hessian}\left(x^2+y^2\,\left[x\,y\right]\,\mathrm{shape}=\mathrm{symmetric}\right)$

$\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$

$H,d≔\mathrm{Hessian}\left(x^2+y\exp \left(x\right)\,\left[x\,y\right]=\left[1\,-1\right]\,\mathrm{determinant}\right)$

$H,d≔\left[\begin{array}{cc}2-\ⅇ& \ⅇ\\ \ⅇ& 0\end{array}\right],-{\left(\ⅇ\right)}^2$

The p parameter was introduced in Maple 16.

The determinant option was introduced in Maple 16.

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