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Defining a total derivative
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More options Aug 18 2012, 3:46 am
Newsgroups: comp.soft-sys.math.mathematica
From: S16 <sowna...@gmail.com>
Date: Sat, 18 Aug 2012 07:46:10 +0000 (UTC)
Local: Sat, Aug 18 2012 3:46 am
Subject: Defining a total derivative
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

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More options Aug 22 2012, 2:24 am
Newsgroups: comp.soft-sys.math.mathematica
From: S16 <sowna...@gmail.com>
Date: Wed, 22 Aug 2012 06:24:40 +0000 (UTC)
Local: Wed, Aug 22 2012 2:24 am
Subject: Re: Defining a total derivative

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).

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More options Aug 22 2012, 5:20 am
Newsgroups: comp.soft-sys.math.mathematica
From: "Dr. Wolfgang Hintze" <w...@snafu.de>
Date: Wed, 22 Aug 2012 09:20:55 +0000 (UTC)
Local: Wed, Aug 22 2012 5:20 am
Subject: Re: Defining a total derivative
On 22 Aug., 08:24, S16 <sowna...@gmail.com> wrote:

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

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More options Aug 23 2012, 2:53 am
Newsgroups: comp.soft-sys.math.mathematica
From: Murray Eisenberg <mur...@math.umass.edu>
Date: Thu, 23 Aug 2012 06:53:52 +0000 (UTC)
Local: Thurs, Aug 23 2012 2:53 am
Subject: Re: Defining a total derivative
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 <w...@snafu.de> wrote:

> On 22 Aug., 08:24, S16 <sowna...@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=

---
Murray Eisenberg                                     =
mur...@math.umass.edu
Mathematics & Statistics Dept.
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