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Re: A Use for Interpretation
*To*: mathgroup at smc.vnet.net
*Subject*: [mg85754] Re: A Use for Interpretation
*From*: David Bailey <dave at Remove_Thisdbailey.co.uk>
*Date*: Thu, 21 Feb 2008 17:57:20 -0500 (EST)
*References*: <200802190700.CAA27893@smc.vnet.net> <fphcts$8ij$1@smc.vnet.net>
Murray Eisenberg wrote:
> Very nice! Some other places where this might be employed are in laws
> of exponents or laws of logs. Or applying trig identities. Or Laplace
> transform.
>
> Ideally, this could be used not just by the teacher, but by the student
> who would demonstrate that she knows the rules by "teaching them to the
> computer". Yet there's a "catch" to that: the Mathematica
> sophistication needed, what with ReplaceRepeated, RuleDelayed, HoldForm,
> ReleaseHold, and FreeQ. Not to mention the "strange" definition of the
> function, like "integrate" here, which must be defined in terms of itself.
>
> David Park wrote:
>> Since Version 6 appeared and I first encountered Interpretation I found it
>> rather difficult to understand what use it might have. The examples in Help
>> seem almost bizarre.
>>
>> Here is one use where it works rather well. Teachers might often be
>> frustrated in demonstrating the behavior of basic function such as D,
>> Integrate or Limit because Mathematica automatically evaluates using its
>> built-in rules. With Interpretation we can define a pseudo routine that
>> displays as the real routine and then apply our own basic transformation
>> rules. Here is an example with a linear breakout of Integrate.
>>
>> integrate[integrand_, var_] :=
>> Interpretation[HoldForm[Integrate[integrand, var]],
>> integrate[integrand, var]]
>>
>> We then define rules that breakout sums and constant factors. The rules even
>> display in a nice form.
>>
>> rule1 = integrate[a_ + b_, x_] -> integrate[a, x] + integrate[b, x]
>> rule2 = integrate[a_?(FreeQ[#, x] &) b_, x_] -> a integrate[b, x]
>>
>> Then we can demonstrate how these rules work, performing the actual
>> integration in the last step.
>>
>> integrate[5 a Sin[x] + y Cos[x] + x^2, x]
>> % //. rule1
>> % //. rule2
>> % /. Interpretation[a_, b_] :> ReleaseHold[a]
>>
>
I think there is a better way to do this. I use
Inert[Integrate]Sin[x],{x,0,Pi}] and use a MakeBoxes rule to display the
result as a grey-shaded integral. The advantage of this method is that
it is possible to edit the contents of the inert integral and re-input
the result.
David Bailey
http://www.dbaileyconsultancy.co.uk
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