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