TPID 4
#26
(06/18/2014, 12:59 PM)sheldonison Wrote:
(06/18/2014, 12:23 PM)tommy1729 Wrote: Then how do you explain the violation of the chain law ?

regards

tommy1729

The flaw in your proof is that k is an integer, but you state k as a real number. "sexp ' (w+k) = exp^[k] ' (sexp(w)) * sexp ' (w) = 0". For k as a fraction, the chain rule does not apply.

What ? I never saw any exceptions to the chain rule for analytic functions ?

sexp(w-1) chain law works. sexp(w+1) chain law works.
In fact the idea that the derivative of sexp relates to a product is the result of the chain law.

So you say :
1) sexp(w+k) is analytic in w,k,w+k and sexp(w) is analytic in w.
2) sexp(w+k) = exp^[k](sexp(w))
3) exp^[k] is also analytic.
4) Yet the derivative of exp^[k](sexp(w)) IS NOT exp^[k] ' (sexp(w)) * sexp ' (w) despite that all functions involved are analytic and the conditions for the chain law are fullfilled ?

Even for the logarithm the chain rule applies even though it has a singularity :

D ln(f(z)) = f ' (z)/f(z) ... = chain law !

The chain law applies to all analytic functions and even many more.

There are many proofs of the chain law and I never read " an exception " ??

Could I be so so wrong in Calc 101 ??

I can imagine going to 0 superexponentially fast but reaching it seems impossible.

Maybe the functional equation is no longer satisfied in the regions or variants of sexp that you use ? That could explain it.

I would like to note that if f ' (h) as h approaches 0 could have as limit 0 BUT that is not the derivative f ' (0).
So in other words , do not confuse the derivative and infinitesimal with the ball from analysis.
As a concrete example : a very mild singularity that is very flat.
There is something known as the Cauchy-Riemann equations.

Maybe Im making a fool of myself if im terribly wrong but calculus 101 and analysis 101 make a strong case imho.

In math one cannot simply say : Ok you have proved this but my numeric example is counterproof.

The proofs of the chain rule are the most beautiful short proofs.

regards

tommy1729


Messages In This Thread
TPID 4 - by tommy1729 - 08/23/2012, 04:26 PM
RE: TPID 4 - by tommy1729 - 08/24/2012, 03:12 PM
RE: TPID 4 - by tommy1729 - 03/28/2014, 12:04 AM
RE: TPID 4 - by sheldonison - 06/15/2014, 06:22 PM
RE: TPID 4 - by tommy1729 - 04/26/2014, 12:24 PM
RE: TPID 4 - by sheldonison - 04/27/2014, 04:37 AM
RE: TPID 4 - by tommy1729 - 04/27/2014, 01:40 PM
RE: TPID 4 - by tommy1729 - 06/15/2014, 06:35 PM
RE: TPID 4 - by sheldonison - 06/15/2014, 06:42 PM
RE: TPID 4 - by tommy1729 - 06/15/2014, 07:09 PM
RE: TPID 4 - by sheldonison - 06/15/2014, 07:35 PM
RE: TPID 4 - by tommy1729 - 06/15/2014, 08:10 PM
RE: TPID 4 - by mike3 - 06/17/2014, 09:30 AM
RE: TPID 4 - by tommy1729 - 06/17/2014, 12:21 PM
RE: TPID 4 - by sheldonison - 06/17/2014, 06:16 PM
RE: TPID 4 - by mike3 - 06/17/2014, 09:48 PM
RE: TPID 4 - by sheldonison - 06/17/2014, 11:43 PM
RE: TPID 4 - by tommy1729 - 06/18/2014, 12:23 PM
RE: TPID 4 - by sheldonison - 06/18/2014, 12:59 PM
RE: TPID 4 - by tommy1729 - 06/18/2014, 10:21 PM
RE: TPID 4 - by sheldonison - 06/18/2014, 10:41 PM
RE: TPID 4 - by tommy1729 - 06/18/2014, 11:15 PM
RE: TPID 4 - by tommy1729 - 06/17/2014, 10:46 PM
RE: TPID 4 - by tommy1729 - 06/16/2014, 09:21 PM
RE: TPID 4 - by tommy1729 - 06/16/2014, 10:45 PM
RE: TPID 4 - by tommy1729 - 06/16/2014, 10:49 PM
RE: TPID 4 - by tommy1729 - 06/16/2014, 10:57 PM
RE: TPID 4 - by tommy1729 - 06/17/2014, 10:48 PM
RE: TPID 4 - by tommy1729 - 06/18/2014, 10:38 PM
RE: TPID 4 - by tommy1729 - 07/07/2014, 11:56 PM
RE: TPID 4 - by tommy1729 - 06/18/2022, 10:40 PM

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