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5.54.3  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.
Find the hessian matrix of F(x,y,z)=2x2yxz3.
Input :

hessian(2*x^2*y-x*z^3 , [x,y,z])

Output :


To have the hessian matrix at the critical points, first input :


Output is the critical points :


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 :


and after simplification :


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