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Defining a total derivative
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S16
Aug 18, 2012, 3:46:10 AM8/18/12
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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
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
S16
Aug 22, 2012, 2:24:40 AM8/22/12
to
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).
Dr. Wolfgang Hintze
Aug 22, 2012, 5:20:55 AM8/22/12
to
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
Aug 23, 2012, 2:53:52 AM8/23/12
to
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:
---
Murray Eisenberg =
mur...@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
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 - =
>>> 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
>>
((=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).
>>
>>
>
>>
>>> 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).
>>
>>
>
Murray Eisenberg =
mur...@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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