derivatives of indexed variables

[George Jefferson]

What am I missing?

total derivative of unspecified y:

In[1]:= Dt[y,x]

Out[1]= Dt[y, x]

But if we take the derivaitve of an undefined indexed variable
it is assumed to be a constant..

In[2]:= Dt[a[1],x]

Out[2]= 0

note however that the indexed symbol could easily represent a non-constant.

In[3]:= a[1]=y; Dt[a[1],x]

Out[3]= Dt[y, x]

clearly the trouble is that there is no distinction between an indexed
symbol and a function evaluated at a fixed point..
Is there a workaround?

[Scott Herod]

(Original article follows)

Mathematica does not know the difference between indexed functions
and functions of an integer. Apparently when you try to compute the
derivative of something like a[1] you get

Dt[a[1],x] = Dt[1,x]*Derivative[1][a][1]

This is useful if you want to make constants without having to worry about
derivatives acting correctly. Unfortunately it makes it hard to create
indexed functions.

One workaround that I have used is to move back and forth between two forms
of the function a[i]. I create a sequence of functions a1, a2, etc
and convert between them using;

Cat[dumx_,dumy_] := ToExpression[StringJoin[ToString[dumx],ToString[dumy]]];
subI = {a[i_] :> Cat[a,i]};

and

subII = Table[Cat[a,i] -> a[i], {i,numberofa}]

=================================
In[3]:= a[4] /. subI

Out[3]= a4

In[4]:= numberofa = 10

Out[4]= 10

In[6]:= a4 /. subII

Out[6]= a[4]
=================================

Of course this is pretty crude and requires that you know how many indexed
functions you are going to have.

A second method is to make "a" an array of Function objects.

For example:
===================================
In[9]:= Clear[a]

In[10]:= a = {Sin, Cos, (1 + #)^2 &};

In[11]:= a[[1]][t]

Out[11]= Sin[t]

In[12]:= a[[3]][x]

2
Out[12]= (1 + x)
==================================

Then you can differentiate elements of the vector "a"

=================================

In[14]:= Dt[a[[3]],x]

2
Out[14]= Dt[(1 + #1) , x] &

In[15]:= %[x^2]

2
Out[15]= 4 x (1 + x )
=================================

Scott Herod
Applied Mathematics
University of Colorado, Boulder
she...@newton.colorado.edu