hessian takes two arguments : an
expression F of n real variables and a vector of these variable names.

hessian returns the hessian matrix of F, that is the matrix of the
derivatives of order 2.

Example

Find the hessian matrix of F(x,y,z)=2x^{2}y−xz^{3}.

Input :

hessian(2*x

`^`

2*y-x*z`^`

3 , [x,y,z])Output :

[[4*y,4*x,-(3*z

`^`

2)],[2*2*x,0,0],[-(3*z`^`

2),0,x*3*2*z]]To have the hessian matrix at the critical points, first input :

solve(derive(2*x

`^`

2*y-x*z`^`

3,[x,y,z]),[x,y,z])Output is the critical points :

[[0,y,0]]

Then, to have the hessian matrix at this points, input :

subst([[4*y,4*x,-(3*z

`^`

2)],[2*2*x,0,0], [-(3*z`^`

2),0,6*x*z]],[x,y,z],[0,y,0])Output :

[[4*y,4*0,-(3*0

`^`

2)],[4*0,0,0],[-(3*0`^`

2),0,6*0*0]]and after simplification :

[[4*y,0,0],[0,0,0],[0,0,0]]