Welcome back, I trust you had a nice Christmas. I think that both methods can be tried, the (d omega)^n to dx and the multiple integral method, where (d omega) squared = d omega d omega and (d omega) cubed = d omega d omega d omega. This is analogous to dV = dxdydz . If you know an unbiased mathematician by all means invite him to look at the mathematics.
To: EMyrone@aol.com
Sent: 30/12/2014 16:30:35 GMT Standard Time
Subj: Re: 290(4): The Correct Derivation of the Stefan Boltzmann LawI am back now from my holiday.
I went through the notes for paper 290 today. Nearly unbelievable that the Stefan Boltzmann law contains terms that have been overlooked so far. I am not sure how powers of the differential integration interval d omega have to be treated. One argument would be that differential calculus is linear in the infinitesimal limit (see definition of the derivative), therefore higher orders of d omega can be neglected.
Another approach would be to make a variable substitution to obtain a linear termd omega^n –> dx
Then omega and E have to be transformed to functions of x to abtain an integrand like
omega(x) E(x) dx.
This is the standard way of handling this to my opinion. To be honest, I am not very convinced that powers of d omega can be handled as n-fold integrals. But I do not know a comparable case in physics. Perhaps a mathematician could help here.
Horst
Am 30.12.2014 um 11:08 schrieb EMyrone:
This is given in Eq. (2), which must be evaluated numerically. However, it is easily shown by hand in the high temperature or low frequency limit (5) that the obsolete derivation is completely wrong. Therefore the law could not have been tested experimentally with great precision. Incorrect mathematics cannot be tested experimentally with great precision. The error made by Lord Rayleigh and Sir James Jeans was to assume that the higher order infinitesimals (d omega) squared and (d omega) cubed can be neglected in the calculation of (omega + d omega)) cubed – omega cubed. As shown in this note they cannot be neglected at all because one is double integrated and the second is triple integrated.