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MathGroup Archive 2007

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Re: Some questions about vector

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74560] Re: Some questions about vector
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Mon, 26 Mar 2007 02:06:24 -0500 (EST)
  • Organization: The Open University, Milton Keynes, UK
  • References: <eu54qn$l30$1@smc.vnet.net>

Qi Zhang wrote:
> input:
> 
> Clear[f];
> f[v] = Sum[(i*v[i]), {i, 1, N}]

It would be better not to use capitalize symbols: the symbol N is 
already defined as a system built-in function (Numerical approximation 
of an expression).

> D[f[v], v[i]]
> 
> the output is
> 
> \!\(1\/2\ N\ \((1 + N)\)\)

Which is correct from Mathematica point of view. The differential 
operator D has higher precedence than Sum. Therefore, the expression 
D[f[v], v[i]] is first interpreted as, "take the first derivative of 
something (something being the unevaluated sum: at this stage we could 
have a constant, that would change nothing) which depends on the 
expression v[i] with respect to the expression v[i]." Since Mathematica 
does not know anything about the symbolic expression v[i] (except that 
it is not a number), we have D[v[i], v[i]] == 1. So we are left with 
Sum[i, {i, 1, N]. The evaluation process continues, and, since 
Mathematica knows something about this sum (built-in mathematical 
knowledge), it returns the well known formula 1/2*n*(1+n) for the sum of 
the first n integers.

> However what I expect is i
> 

Here, I do not follow you for i is not a constant coefficient. On what 
basis did you conclude that the correct result must be i?

> What should I do in order to get the the answer I expected?
> 
> Thanks a lot.
> 

In addition, the following threads are worth reading:

. "Derivative of Sum" at 
http://forums.wolfram.com/mathgroup/archive/2004/Apr/msg00591.html

. "partial derivative of a sum" at
http://forums.wolfram.com/mathgroup/archive/2005/Aug/msg00547.html

Regards,
Jean-Marc


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