Hessian matrix and gradient vector of functions

Questions
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1

Hi all, I am writing to inquire about the Hessian matrix and the gradient vector of some functions, this because I have left several exercises functions of two variables and three variables, to find the Hessian matrix and the gradient vector, there is some way to make these calculations in Mathematica?

2

http://reference.wolfram.com/mathematica/ref/Grad.html

(search for Hessian on the following page)
http://reference.wolfram.com/mathematica/tutorial/Differentiation.html

3

Thanks for helping David, I consulted the links that you have shown me and I found everything I needed. They are very useful for the job I’m doing,. Thanks again

4

You can do this using D[]. Given a function with (say) arguments x and y you can write:

D[f[x,y], {{x,y},2}]

You can use Grad[] twice, of course, but D[] will get you there directly.

Some background info here:

On that page, they present an implementation of a Hessian operator:

HessianH[f_, x_] := D[f, {x,2}]

You’d call it like this:

HessianH[f[x,y], {x,y}]

If you don’t want to have to repeat the argument list, you could do something like:

H[f_, x_]:= HessianH[Apply[f, x], x]

So now you can say:

H[f, {x,y}]

As of v12, you can access a pre-defined version of this function as follows:

ResourceFunction["HessianMatrix"][f[x,y], {x,y}]

which emits a matrix in the normal way. More info here:

https://resources.wolframcloud.com/FunctionRepository/resources/HessianMatrix