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Re: Defining a total derivative

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  • Subject: [mg127802] Re: Defining a total derivative
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Thu, 23 Aug 2012 02:55:03 -0400 (EDT)
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  • References: <k0nh82$avi$1@smc.vnet.net> <k11tv8$t7e$1@smc.vnet.net> <20120822091921.18F18684C@smc.vnet.net>

For the benefit of the O.P., in case of discomfort with using pure 
functions here=85 one can use ordinary explicit function definitions as well:

   g[fn_] := D[fn, x] + D[fn, y]
  
   g[x + y]

   f[x, y] := Sin[x y]

   g[g[f[x,y]]
   Nest[g, f[x,y], 2]

   gi[k_][fn_]:= Nest[g, fn, k]

   gi[3][f[x, y]]

On the other hand, if one wanted to get fancier, he could provide additional arguments that specify the names of the variables with respect to which the derivatives are taken. I leave this as an exercise.

  
On Aug 22, 2012, at 5:19 AM, Dr. Wolfgang Hintze <weh at snafu.de> wrote:

> On 22 Aug., 08:24, S16 <sowna... at gmail.com> wrote:
>> On Saturday, August 18, 2012 8:46:10 AM UTC+1, S16 wrote:
>>> Hi,
>>
>>> I am very new to Mathematica, so need a bit of help!
>>
>>> I want to define a function (called say G) that is defined as
>>
>>> G = =E2=88=82/=E2=88=82x - =
((=E2=88=82F/=E2=88=82x)/(=E2=88=82F/=E2=88=82=
>>
>>> y))*=E2=88=82/=E2=88=82y
>>
>>> Where F is some other function which will be defined.
>>
>>> So as you can see, G is a differenital operator. Want to define it so that I can just do G[ some function ] rather than repeatedly write out the whole thing.
>>
>>> Any help at all would be awesome!
>>
>>> -S16
>>
>> Sorry, my message came out formatted all wrong. I have actually managed to solve this issue- but have a different question.
>>
>> Say I have defined an operator G, which involves partial derivatives in x and y
>>
>> and I want to find expressions for G[G[ ]] , G[G[G[ ]]] - applying the operator multiple times. is there a way to define this on Mathematica (I want to put this in a package).
>>
>>
>
> Let's take an example.
>
> Define the operator g as
>
> In[7]:= g = D[#1, x] + D[#1, y] &
>
> Out[7]= D[#1, x] + D[#1, y] &
>
> Test it
>
> In[8]:= g[x + y]
>
> Out[8]= 2
>
> Chose a non trivial funcion
>
> In[20]:= f = Sin[x*y]
>
> Out[20]= Sin[x*y]
>
> Now iterate g and apply it immediately to f
>
> In[22]:= g[g[f]]
>
> Out[22]= 2*Cos[x*y] - x^2*Sin[x*y] - 2*x*y*Sin[x*y] - y^2*Sin[x*y]
>
> But this can be achieved more generally using Nest
>
> In[23]:= Nest[g, f, 2]
>
> Out[23]= 2*Cos[x*y] - x^2*Sin[x*y] - 2*x*y*Sin[x*y] - y^2*Sin[x*y]
>
> Now the step you wanted. Definiting the interation of g without
> applying it immediately.
>
> In[27]:= gi[k_] := Nest[g, #1, k] &
>
> Test it
>
> In[28]:= gi[2][f]
>
> Out[28]= 2*Cos[x*y] - x^2*Sin[x*y] - 2*x*y*Sin[x*y] - y^2*Sin[x*y]
>
> Now the third iteration
>
> In[29]:= gi[3][f]
>
> Out[29]= (-x^3)*Cos[x*y] - 3*x^2*y*Cos[x*y] - 3*x*y^2*Cos[x*y] -
> y^3*Cos[x*y] - 6*x*Sin[x*y] - 6*y*Sin[x*y]
>
> Best regards,
> Wolfgang

---
Murray Eisenberg                                     =
murray at math.umass.edu
Mathematics & Statistics Dept.      
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                  413 545-2859 (W)
710 North Pleasant Street                      fax   413 545-1801
Amherst, MA 01003-9305








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