Re: Defining a total derivative

*To*: mathgroup at smc.vnet.net*Subject*: [mg127788] Re: Defining a total derivative*From*: "Dr. Wolfgang Hintze" <weh at snafu.de>*Date*: Wed, 22 Aug 2012 05:19:21 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net*References*: <k0nh82$avi$1@smc.vnet.net> <k11tv8$t7e$1@smc.vnet.net>

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

**Follow-Ups**:**Re: Defining a total derivative***From:*Murray Eisenberg <murray@math.umass.edu>