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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 <m3uatb$st7$1@speranza.aioe.org>,
>> Nasser M. Abbasi <nma@12000.org> wrote:
>>>On 11/11/2014 5:39 PM, none Rouben Rostamian wrote:
>>>
>>>> Thanks much, Joe, this is exactly what I was hoping for.
>>>> I had no idea about frontend() or the Physics package.
>>>>
>>>> This brings up a somewhat related question.  Doing
>>>>     frontend(int, [x(t)^2, x(t)]);
>>>> we get x(t)^3/3, which is fine.  The following, however,
>>>> issues an Error message:
>>>>      frontend(int, [x(t)^2, x(t)=a..b]);
>>>>
>>>> Is there a way to get that to work too?
>>>>
>>>> Rouben
>>>>
>>>
>>>I do not use frontend() either. But if all else fails, you
>>>can simply use calculus
>>>
>>>------------------------------
>>>restart;
>>>f := frontend(int,[x(t)^2, x(t)]);
>>>f := unapply(f,t);
>>>limit(f(t),t=b)-limit(f(t),t=a);
>>>-------------------------
>>>
>>>     (1/3)*x(b)^3-(1/3)*x(a)^3
>>
>> Hi Nasser, that's good but it's not exactly what
>> I had asked.  Your result integrates over t=a..b.
>> I wanted x(t)=a..b, but that's easy to fix:
>> f := frontend(int,[x(t)^2, x(t)]);
>> eval(f, x(t)=b) - eval(f, x(t)=a);
>>
>> This technique, however, does not work in a more complex case.
>> For instance, let's write x' and x'' for the first and second
>> derivatives of x.  We want to integrate x'' * x' * x^2 with
>> respect to x.  So we do:
>>
>>     frontend(int, [diff(x(t),t,t)*diff(x(t),t)*x(t)^2, x(t)]);
>>
>> and we get the expected 1/3 * x'' * x' * x^3.
>>
>> Now, how do we do the corresponding definite integral
>> where x(t)=a..b ?  The expected answer is
>> 1/3 * x''(t) * x'(t) * (b^3 - a^3).
>
> frontend(int, [diff(x(t),t,t)*diff(x(t),t)*x(t)^2, x(t)=a..b], [{`+`,`*`,`^`,`=`,`..`},{}]);  
>    1/3*diff(x(t),`$`(t,2))*diff(x(t),t)*(-a^3+b^3)

In this case, we could also do
  frontend(int, [diff(x(t),t,t)*diff(x(t),t)*x(t)^2, x(t)=a..b], [{Not(function)},{}]);
         1/3*diff(x(t),`$`(t,2))*diff(x(t),t)*(-a^3+b^3)


-- 
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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