       Re: partials of Sum[x[i]^2, {i,1,n}] (e.g.)

• To: mathgroup at smc.vnet.net
• Subject: [mg69554] Re: partials of Sum[x[i]^2, {i,1,n}] (e.g.)
• From: dimmechan at yahoo.com
• Date: Fri, 15 Sep 2006 06:45:56 -0400 (EDT)
• References: <eebfke\$4d8\$1@smc.vnet.net>

```***Mathematica as far as I know can work with limits in symbolic form.

***You can define the function you are interested as follows:

In:=
f[lst:{x_,y___}]:=Sum[(lst[[i]])^2,{i,1,Length[lst]}]

***where

In:=
?___
Out=
___ (three _ characters) or BlankNullSequence[ ] is a pattern object
that can \
stand for any sequence of zero or more Mathematica expressions. ___h or
\
BlankNullSequence[h] can stand for any sequence of expressions, all of
which \

***Then e.g.

In:=
D[f[{Cos[x],Sin[x],Tan[x],Exp[y]}],Sin[x]]
Out=
2 Sin[x]

In:=
D[f[{Cos[x],Sin[x],Tan[x],Exp[y]}],y]
Out=
2*E^(2*y)

***In fact you do not need here the limits. Indeed

In:=
f[lst:{x_,y___}]:=Apply[Plus,lst^2]

***where

In:=
?Apply
Out=
Apply[f, expr] or f @@ expr replaces the head of expr by f. Apply[f,
expr, \
levelspec] replaces heads in parts of expr specified by levelspec.
More...

***Then e.g.

In:=
D[f[{Cos[x],Sin[x],Tan[x]}],Tan[x]]
Out=
2*Tan[x]

Dimitris Anagnostou

Î?/Î? kj Î­Î³Ï?Î±Ï?Îµ:
> In symbolic manipulations, one often needs to leave some of the
> limits of an expression in symbolic form.  E.g. the n in:
>
>   f[x,...,x[n]] = Sum[x[i]^2, {i, 1, n}],
>
> (where I've used Mathematica notation loosely).
>
> Then one often finds derivations like
>
>   D[f[x,...,x[n]], x[k]] = 2 x[k], for all k in { 1,..., n }
>
> Is it possible to do something like this in Mathematica?
>
> More generally, can Mathematica fully understand expressions with
> symbolic limits?
>
> Basically, I have a slightly hairier expression that I want to take
> the partials of, set them all equal to zero to produce a system of
> n equations.  If that weren't enough, I'd like to solve this system
> of n equations using Mathematica.  This kind of manipulation is
> far more difficult, as far as symbolic math goes, than anything
> I've seen Mathematica do yet, because it requires Mathematica to
> understand the notion of an array with a "symbolic cardinality",
> but I thought I'd ask.
>
> Thanks!
>
> kj
> --
> NOTE: In my address everything before the first period is backwards;
> and the last period, and everything after it, should be discarded.

```

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