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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
to your first posting:

<< 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:

grad[f][x, y] // PushOnto[_Function] 

{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]
Through[grad[f][x, y]] // MatrixForm
gradList[{{f, f}, {f, f}}] //
   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.

Thank you in advance,

Sam

PS: Using Map[Apply[]] in the second case because Through doesn't seem 
to work with Lists of Lists. This is the subject of an earlier post...



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