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Re: Differentiating with respect to an expression

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Joe Riel <joer@san.rr.com> writes:

> rouben@shadow.(none) (Rouben Rostamian) writes:
>
>> In article <87sihpl61w.fsf@san.rr.com>, Joe Riel  <joer@san.rr.com> wrote:
>>>rouben@shadow.(none) (Rouben Rostamian) writes:
>>>
>>>> The following issue comes up quite often in the context of
>>>> analytical mechanics.  I have a clunky solution for it.
>>>> I wonder if there is a clever way.
>>>>
>>>> Let's say we have L = x^2 + x'^2 * x''^2, where x is a function of t,
>>>> and in the usual mathematical notation, x' and x'' are the first and
>>>> second derivatives of x.
>>>>
>>>> We want to compute the derivative of L with respect to x'.  The
>>>> answer should be 2*x'*x''^2.  Here is the way I do it in Maple:
>>>>
>>>> restart;
>>>> L := x(t)^2 + diff(x(t),t)^2 * diff(x(t),t,t)^2;
>>>> subs(diff(x(t),t,t)=Z2, diff(x(t),t)=Z1, %);
>>>> diff(%, Z1);
>>>> subs(Z1=diff(x(t),t), Z2=diff(x(t),t,t), %);
>>>>
>>>> The L shown above is simple enough so that we don't need a
>>>> CAS to compute the derivative.  The L in a real example will
>>>> be the result of a long chain of calculations, will depend on
>>>> several functions and their derivatives, and will take up a
>>>> couple of screenfuls.
>>>>
>>>> If there is a clever way to compute that derivative, I would
>>>> like to know.
>>>
>>>A low-level way to do this is with frontend:
>>>
>>>frontend(diff, [L,diff(x(t),t)]);
>>>       2*diff(x(t),t)*diff(diff(x(t),t),t)^2
>>>
>>>An alternative method is to use the Physics package:
>>>
>>>with(Physics):
>>>
>>>diff(L, diff(x(t),t));
>>>       2*diff(x(t),t)*diff(diff(x(t),t),t)^2
>>>
>>># alternatively, with Physics
>>>
>>>diff(L, D(x)(t));
>>>     2*D(x)(t)*`@@`(D,2)(x)(t)^2
>>>
>>
>> Thanks much, Joe, this is exactly what I was hoping for.
>> I had no idea about frontend() or the Physics package.
>>
>> This brings up a somewhat related question.  Doing
>>    frontend(int, [x(t)^2, x(t)]);
>> we get x(t)^3/3, which is fine.  The following, however,
>> issues an Error message:
>>     frontend(int, [x(t)^2, x(t)=a..b]);
>>
>> Is there a way to get that to work too?
>>
>> Rouben
>
> The optional third argument of frontend can be used for that.
> See the help page.  It consists of a list of two sets.
> The first set is significant here, it contains the types 
> that frontend does *not* freeze.  By default the `+`, `*`, and `^` 
> types are not frozen.  In this case, you also do *not* want
> to freeze `=` and `..`.  So the following works:
>
>   frontend(int, [x(t)^2, x(t)=a..b], [{`+`,`^`,`*`,`..`,`=`},{}]);
>        1/3*b^3-1/3*a^3
>
> Frequently one can use the Not type to specify what you do
> want frozen.  For example,
>
>   frontend(int, [x(t)^2, x(t)=a..b], [{Not(identical(x(t)))},{}]);
>        1/3*b^3-1/3*a^3

As mentioned, frontend is a rather low-level tool.
It has its uses, but is somewhat awkward to use.
A useful trick, when you're having trouble formulating
the appropriate set of types that will not be frozen,
is to use 'print' for the operator to see what has been frozen:


  frontend(print, [x(t)^2, x(t)=a..b]);                        
               O^2, O

  # unfreeze defaults, and equations
  frontend(print, [x(t)^2, x(t)=a..b], [{`+`,`*`,`^`,`=`},{}]);
              O^2, O = O

  # unfreeze defaults, equations, and ranges
  frontend(print, [x(t)^2, x(t)=a..b], [{`+`,`*`,`^`,`=`,`..`},{}]);
              O^2, O = a .. b

That last one is what we want to pass to int, so replacing
the 'print' with 'int' gives the desired result.

-- 
Joe Riel

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Thread

Differentiating with respect to an expression rouben@shadow.(none) (Rouben Rostamian) - 2014-11-11 22:10 +0000
  Re: Differentiating with respect to an expression Joe Riel <joer@san.rr.com> - 2014-11-11 14:35 -0800
    Re: Differentiating with respect to an expression rouben@shadow.(none) (Rouben Rostamian) - 2014-11-11 23:39 +0000
      Re: Differentiating with respect to an expression "Nasser M. Abbasi" <nma@12000.org> - 2014-11-11 18:50 -0600
        Re: Differentiating with respect to an expression "Nasser M. Abbasi" <nma@12000.org> - 2014-11-11 19:44 -0600
        Re: Differentiating with respect to an expression rouben@shadow.(none) (Rouben Rostamian) - 2014-11-12 01:55 +0000
          Re: Differentiating with respect to an expression Joe Riel <joer@san.rr.com> - 2014-11-11 18:00 -0800
            Re: Differentiating with respect to an expression Joe Riel <joer@san.rr.com> - 2014-11-11 18:03 -0800
              Re: Differentiating with respect to an expression rouben@shadow.(none) (Rouben Rostamian) - 2014-11-12 02:44 +0000
                Re: Differentiating with respect to an expression Joe Riel <joer@san.rr.com> - 2014-11-11 19:16 -0800
                Re: Differentiating with respect to an expression rouben@shadow.(none) (Rouben Rostamian) - 2014-11-12 04:53 +0000
      Re: Differentiating with respect to an expression Joe Riel <joer@san.rr.com> - 2014-11-11 17:45 -0800
        Re: Differentiating with respect to an expression Joe Riel <joer@san.rr.com> - 2014-11-11 17:52 -0800
        Re: Differentiating with respect to an expression rouben@shadow.(none) (Rouben Rostamian) - 2014-11-12 02:03 +0000

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