Re: Derivative of a function with multiple variables

• To: mathgroup at smc.vnet.net
• Subject: [mg69502] Re: Derivative of a function with multiple variables
• From: dimmechan at yahoo.com
• Date: Thu, 14 Sep 2006 06:56:10 -0400 (EDT)
• References: <ee8grk\$itd\$1@smc.vnet.net>

```***See The Mathematica Book Section 3.5.1

***You can get what you want executing the command
D[f[x,y,z],{{x,y,z},2}]. Indeed

D[f[x,y,z],{{x,y,z},2}]
{{Derivative[2, 0, 0][f][x, y, z], Derivative[1, 1, 0][f][x, y, z],
Derivative[1, 0, 1][f][x, y, z]},
{Derivative[1, 1, 0][f][x, y, z], Derivative[0, 2, 0][f][x, y, z],
Derivative[0, 1, 1][f][x, y, z]},
{Derivative[1, 0, 1][f][x, y, z], Derivative[0, 1, 1][f][x, y, z],
Derivative[0, 0, 2][f][x, y, z]}}

**One can also get previous result using properly the built-in function
Outer

Information[Outer]
Outer[f, list1, list2, ... ] gives the generalized outer product of the
listi, forming all possible combinations of the
lowest-level elements in each of them, and feeding them as arguments
to f. Outer[f, list1, list2, ... , n] treats as separate
elements only sublists at level n in the listi. Outer[f, list1,
list2, ... , n1, n2, ... ] treats as separate elements only
sublists at level ni in the corresponding listi.
InputForm[Attributes[Outer] = {Protected}]

Outer[D,{f[x,y,z]},{x,y,z},{x,y,z}]
{{{Derivative[2, 0, 0][f][x, y, z], Derivative[1, 1, 0][f][x, y, z],
Derivative[1, 0, 1][f][x, y, z]},
{Derivative[1, 1, 0][f][x, y, z], Derivative[0, 2, 0][f][x, y, z],
Derivative[0, 1, 1][f][x, y, z]},
{Derivative[1, 0, 1][f][x, y, z], Derivative[0, 1, 1][f][x, y, z],
Derivative[0, 0, 2][f][x, y, z]}}}

***Cheers
Dimitris

> Hi all,
>
> I am sure this is easy to many of you
>
> Assume I have f = f[x,y,z]
> I would like to construct a simple function g[f[x,y,z]] that will
> generate the Hessian. i.e.
> g[f[x,y,z]]=
> {{Derivative[2,0,0][f][x,y,z], Derivative[1,1,0][f][x,y,z],
> Derivative[1,0,1][f][x,y,z]},
> {Derivative[1,1,0][f][x,y,z], Derivative[0,2,0][f][x,y,z],
> Derivative[0,1,1][f][x,y,z]},
> {Derivative[1,0,1][f][x,y,z], Derivative[0,1,1][f][x,y,z],
> Derivative[0,0,2][f][x,y,z]}}.
>
> Any help?
>
> --