How to compute the gradient and Hessian of a vector field symbolically with sympy.

[Isaque Soares]

Is there a way to compute the gradient and hessian matrices of a vector field like u = (x) i + (2yz) j + (3xy) k.

[Faisal Riyaz]

Hi Isaque,

The vector module has a gradient function but it works only for scalar fields. There is an implementation given

in this issue. Unfortunately, it has not been merged.

The hessian function in the matrices module also works for scalars only. You will have to iterate over each component

separately.

Thanks

Faisal

On Sat, Jul 4, 2020 at 4:54 AM Isaque Soares <isaque...@gmail.com> wrote:

Is there a way to compute the gradient and hessian matrices of a vector field like u = (x) i + (2yz) j + (3xy) k.

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[Kasper Peeters]

Is there a way to compute the gradient and hessian matrices of a vector field like u = (x) i + (2yz) j + (3xy) k.

I know you didn't ask for this, but just in case it is of any help, here's how you do it in Cadabra (https://cadabra.science).

Cadabra uses Sympy under the hood, so it's at least a partial Sympy answer.

Set-up with:

   {x,y,z}::Coordinate;
   {i,j,k,l}::Indices(values={x,y,z});

   \partial{#}::PartialDerivative;

   rl:= { u_{x} = x, u_{y} = 2 y z, u_{z} = 3 x y };

The gradient can then be computed using:

   grad:= g_{i j} = \partial_{i}{ u_{j} };

   evaluate(grad, rl, rhsonly=True);

and the Hessian with:

   H:= H_{i j k} = \partial_{i j}{ u_{k} };

   evaluate(H, rl, rhsonly=True);

Hope this helps.

Kasper

[Oscar Benjamin]

Also Faisal just opened a pull request implementing this:

https://github.com/sympy/sympy/pull/19703

Does that look like what you expected? Feel free to comment on the
pull request if you have any suggestions.

Oscar

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[Nicolas Guarin]

You could also use a package that I wrote for that purpose

https://continuum-mechanics.readthedocs.io/en/latest/

If you are interested in Cartesian coordinates only you could just do the following

u = Matrix([x, 2*y*z, 3*x*y])
J = u.jacobian([x, y, z])

H = Array([J[:, k].jacobian([x, y, z]) for k in range(3)], (3, 3, 3))

to get

[[0  0  0]  [0  0  0]  [0  0  0]]
[[       ]  [       ]  [       ]]
[[0  0  0]  [0  0  2]  [0  2  0]]
[[       ]  [       ]  [       ]]
[[0  3  0]  [3  0  0]  [0  0  0]]