Hessian matrix : hessian

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) = 2x²y - xz³.
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]]