       Re: More universal way of writing gradient

• To: mathgroup at smc.vnet.net
• Subject: [mg112933] Re: More universal way of writing gradient
• From: "David Park" <djmpark at comcast.net>
• Date: Wed, 6 Oct 2010 03:16:53 -0400 (EDT)

```The Presentations package has a command PushOnto that allows more selective
application of arguments to patterns anywhere in an expression. So in answer

<< Presentations`

{{Sin}, {Cos}}[x] // PushOnto[Sin | Cos]

{{Sin[x]}, {Cos[x]}}

In answer to your present question, first note that since Version 7
Mathematica has a nice input form for Function in terms of a Function arrow,
entered as esc fn esc or \[Function]. So we can write:

f := {x, y} \[Function] Sin[x] Exp[y]
grad := u \[Function] {Derivative[1, 0][u], Derivative[0, 1][u]}

Then we can use your grad function as follows, but we could also use Through
here as well as PushOnto:

{E^y Cos[x], E^y Sin[x]}

I don't really understand your gradList example, so here I just assumed that
you wanted the gradient of four different functions in a matrix.

f1 := {x, y} \[Function] Sin[x] Exp[y]
f2 := {x, y} \[Function] Cos[x] Exp[y]
f3 := {x, y} \[Function] Sin[y] Exp[x]
f4 := {x, y} \[Function] Cos[y] Exp[x]

Map[grad, {{f1, f2}, {f3, f4}}, {2}][x, y] // PushOnto[_Function]

{{{E^y Cos[x], E^y Sin[x]}, {-E^y Sin[x], E^y Cos[x]}}, {{E^x Sin[y],
E^x Cos[y]}, {E^x Cos[y], -E^x Sin[y]}}}

David Park
djmpark at comcast.net
http://home.comcast.net/~djmpark/

From: Sam Takoy [mailto:sam.takoy at yahoo.com]

Hi,

My question is not related to the gradient at all, but rather strictly
the grammar of Mathematica. Gradient is just an example.

My question is: what's the elegant way to write the following function
so that it applies to single functions as well as ("rectangular") lists
of functions?

grad[u_] := {Derivative[1, 0][u], Derivative[0, 1][u]}
gradList[u_] := {Map[Derivative[1, 0], u, {2}],
Map[Derivative[0, 1], u, {Length[Dimensions[u]]}]}

f[x_, y_] := Sin[x] Exp[y]
Map[Apply[#, {x, y}] &, #, {Length[Dimensions[#]]}] & // MatrixForm

I'm sure I could wrap grad and gradList into a function with an If, but
I'm sure there is a more natural way.