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Re: Re: Derivative of Dot[]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg91156] Re: [mg91106] Re: [mg91055] Derivative of Dot[]
  • From: "Eitan Grinspun" <eitan at grinspun.com>
  • Date: Thu, 7 Aug 2008 04:43:16 -0400 (EDT)
  • References: <200808050759.DAA09628@smc.vnet.net>

On Wed, Aug 6, 2008 at 5:06 AM, Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
> I think the rule for D[X[t] . Y[t], t] is necessary, rather than
> "useful". Compare these two
>
>
>
> In[1]:= D[Dot[X[t] , Y[t]], t]
> Out[1]= X[t] . Derivative[1][Y][t] + Derivative[1][X][t] . Y[t]
>
> In[2]:= D[dot[X[t], Y[t]], t]
> Out[2]= Derivative[1][Y][t]*Derivative[0, 1][dot][X[t],
>        Y[t]] + Derivative[1][X][t]*Derivative[1, 0][dot][
>        X[t], Y[t]]
>
> If there was no built-in rule for D and Dot, then exactly the same
> thing that happens for dot would happen for Dot, which would be
> somewhat less convenient (you would need to define Derivative[1, 0]
> [Dot] and Derivative[0, 1][Dot] yourself).
> Other than that, I do not know of any obvious application. Note
> however, that Derivative is always first converted to D[ ], so any
> rules that are applied to Derivative and Dot are actually derived from
> rules for D and Dot. I don't see any direct use for them and actually
> I think they are basically an slightly unfortunate side effect.

If I understand correctly, what you are saying is that if Dot' had not
been defined, I would have to do the work of defining it. What I am
saying is that the way it is defined, it does not seem to be useful,
and I am assuming that this means that I am misunderstanding something
(i.e., that it must have a useful definition). As far as I can see,
Dot' = #1.1 + 1.#1& is useful iff 1 is interpreted as the identity
map, and not useful if 1 is a scalar (which mathematica refuses to
treat as the identity map). I could write rules/patterns to manipulate
this, but I was assuming that the developers had something in mind.

Eitan


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