Archive for December, 2014

290(5): Mathematical Background of Higher Order Infinitesimal Integrals

This is given in “Vector Analysis Problem Solver” chapter 16, the chapter on volume integrals. I use example 16.3 of VAPS to justify the correct expression for the intensity of black body radiation from the Planck distribution. This is Eq. (16) and has two terms missed entirely by the old theory. I worked out the integration mathematics with close logic by directly adapting the mathematics of volume integrals. Unless there is an error in this close logic the usual expression for the Stefan Boltzmann law is completely wrong. Rayleigh gave no justification in 1900 for his neglect of higher order infinitesimals, Jeans simply made a proportionality correction in 1905.

a290thpapernotes5.pdf

Happy New Year to Horst Eckardt and Family

Good to hear of the visit. This village of Craig Cefn Parc was started as a drift mining village, and I gave some steam coal to Steve Bannister to take back to Utah to present to the examination board alongside his Thesis on the first industrial revolution. Robert Cheshire found some coal mining artifacts near Hendy Drift Mine. It was an entirely Welsh speaking village with a vigorous and famous culture – the hymn singing and so on. My grandfather the Head Deacon (Prof Ddiacon) composed hymns in this same room. he was also the Capel Meister and choir leader, brass band leader and so on. For a living he was a coal miner (Autobiography Volume One above my coat of arms on www.aias.us). This house is to be preserved under the Newlands Family Trust as you know. This famous and indigenous culture unique to Wales has been completely destroyed in Craig Cefn Parc by closures of industry by remote non existent government, so I would say that things here can only get better, and I have posted many proposals on this blog. The older people loath what is happening. I founded the Newlands Family Trust to try to reverse the disastrous decline in values, language, sense of community and culture. I would say that investment is needed to counter the sharp rise in lawlessness and anti social conduct and to reinvigorate the culture, to protect the chapels and to make all the schools in Mawr teach in the Welsh language to all children from age about three onwards. Thi swoudl be simple to implement by any real government. I have created my own industry in this house, so if everyone did this, there would be no problem of unemployment and idle abusive wasters becoming vandals. The late Head Deacon, Cen Williams, greatly enjoyed listening to you playing the great Bach D Minor work on the organ of Elim, even though some stops didn’t work. One of the aims of the Trust would be to buy back his house (the one next door), put it back in to its pristine condition and to try to heavily protect the Mans, the Head Deacon’s house and the chapels. Maybe we can team up with the Llangiwg Community. It is not a good idea to throw our own culture in Wales into skips. They would not do that with the works of J. S. Bach, or Beethoven in Vienna. Some of the Bavarian churches like Vierzenheiligen are masterpeices of art, and they would not be thrown into skips or be lived in as houses – a ghastly desecration that should be outlawed.

In a message dated 30/12/2014 18:47:40 GMT Standard Time, writes:

I visited my mother near to Göttingen over Christmas, the village is even smaller than yours. Some peopled complained about too few investments into rural infrastructure by the community, may be similar as at your site.
I will try to find a mathematician as soon as I am back to work at 7th Jan.

Horst

Am 30.12.2014 um 18:39 schrieb EMyrone:

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
Sent: 30/12/2014 16:30:35 GMT Standard Time
Subj: Re: 290(4): The Correct Derivation of the Stefan Boltzmann Law

I 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 term

d 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.

Daily Report 29/12/14

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Discussion of Note 290(4)

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 Law

I 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 term

d 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.

The Role of B(3) in the Corrected Stefan Boltzmann Law

The question is whether B(3) adds a third state of polarization, and whether it is an oscillator. The B(3) field adds a classical energy density:

(E / V) (B(3)) = B(3) dot B(3)* / mu0

so adds an intensity:

I = (c / muo) B(3) dot B(3)*

to the usual intensity:

I = epso (E(1) dot E(1)* + E(2) dot E(2)*) + (B(1) dot B(1)* + B(2) dot B(2)*) / muo

so B(3) also changes the proportionality constant in the Stefan Boltzmann law if it is incorporated in this way. However B(3) does not have a phase so is not an oscillator as defined by Rayleigh. The only contribution of Jeans was to adjust a proportionality constant.

290(4): The Correct Derivation of the Stefan Boltzmann Law

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.

a290thpapernotes4.pdf

Daily Report 28/12/14

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