### The Maxwell's equations, the Beatles and the differential geometry

The following video is a song about Maxwell's equations. Lyrics are written by David Olson with the basis of Let it be by Beatles.
Enjoy!
Interestingly, Maxwell's equations have been drastically reduced into a language of di fferential geometry. These four sets of equations which perfectly describe the theory of electromagnetism have been reduced to a set of two equations which lay the foundations of most new theories in the physical world today.
The most revolutionary quantum leap in the history of theoretical physics is the birth of general relativity and quantum eld theory (the standard model of elementary particle). These theories describe nature better than any physicist ever had at hand, although they have not been uni ed into a coherent picture of the world. One of the main ingredients of these theories is di erential geometry. Euclidean geometry was abandoned in favour of di erential geometry and classical eld theories had to be quantized.
Maxwell's equations in the language of di erential geometry lead to a generalization to these new theories, and these equations are a special case of Yang-Mills equations (beyond the scope of this essay), which is also gauge invariant and describe not only electromagnetism but also the strong and weak nuclear forces. This essay is nothing but the tip of the iceberg.
(from Maxwell's Equations in Terms of Differential Forms (pdf) by Solomon Akaraka Owerre)
Read also: The poem of the Maxwell's equations in pdf written by Lynda Williams.

### Pythagoras by Henry Swinburne

Pythagoras spent the last years of his life at Metapontum. After his decease, the house he had dwelt in was converted into a temple of Ceres, and reforted to with the greatest veneration by the Metapontines, who were truly sensible of the advantages they had derived from his instructions(1)
This philosopher was one of the most exalted characters of antiquity; one of the few sages who did not confine their views to private and partial objects, but made their learning of use to nations at large, whom they instructed, enlightened and directed in the paths of moral virtue and real glory. Many ridiculous stories are related of his opinions and doctrines, which give us the idea of a visionary or impostor; but we should be cautious how we admit implicity anecdotes respecling the great men of distant ages, when we find them clash with what is allowed to have been their general line of conduct. Perhaps Pythagoras found it necessary, in order to captivate the veneration and confidence of a credulous superstitious people, that he should propagate strange and marvellous sigments, and thereby allure them to listen attentively to the lessons, and obey the injunctions of a lawgiver. He was the legislator, the reformer of Magna Graecia. To him and his disciples the little states that composed it owe a celebrity which they were not entitled to from extent of dominion or conquests. Their ruin may be attributed to the neglect of his precepts; or, indeed, in some shape to the very great successes attending his institutions, which rasfed those republics to such an uncommon pitch of prosperity, as intoxicated and finally corrupted their citizens.
(...)
Pythagoras, after his long peregrinations in search of knowledge, fixed his refidence in this place [Croton], which some authors think his native one, at least that of his parents, supposing him to have been born in the isle of Samos, and not at some town of that name in Italy. This incomparable sage spent the latter part of his life in training up disciples to the rigid exercise of sublime and moral virtue, and instructing the Crotoniates in the true arts of government, such as alone can insure happiness, glory, and independence.

(1) Some authors write that he died, and that the temple was dedicated at Croton.

(from Travels in the two Sicilies by Henry Swinburne - 1783-1790)

### Note on the theory of hypergeometric functions

In the few last days I have search for more links about Mary Frances Winston, and I find the paper written by Mary when she was at Göttingen. So I try to translate(1) it using Google. I hope that this version is enough clear.

Riemann defined in its basic treatise the related $P$-functions first of all such as, whose exponents differ by integers, while the exponents are subject only to the condition that their sum is equal to 1: $\alpha' + \alpha'' + \beta' + \beta'' + \gamma' + \gamma'' = 1 \qquad (1)$ and that none of the differences $\pm (\alpha' - \alpha''), \; \pm (\beta' - \beta''), \; \pm (\gamma' - \gamma'')$ should be an integer. Meanwhile, the analytical definition provided herein is not the essence of the relationship. Rather all of the following developments about Riemann's related funcrions rest thereon, if only related functions should have the same monodromy. Now, Prof. Klein in his lectures about the hypergeometric function in the winter 1893-94, essentially noted that the latter is generally a result of the aforementioned analytical definition and that a special case exists, which is an exception. It is those exponents' system in which there exists the relation: $\pm (\alpha' - \alpha'') \pm (\beta' - \beta'') \pm (\gamma' - \gamma'') = 2k + 1 \qquad (2)$ where $k$ is an arbitrary integer. Here are the P-functions: $P \begin{pmatrix} 0 & \infty & 1 & \\ \alpha' + a' & \beta' + b' & \gamma' + c' & x \\ \alpha'' + a'' & \beta''+ b'' & \gamma'' + c'' & \end{pmatrix} \qquad (3)$ (where $a'$, $a''$, $b'$, $b''$, $c'$, $c''$ are integers of zero-sum) divide into two separate bands such that only the P-functions of the individual band are related to each other, i.e. have the same monodromy.
I was busy trying to find the analytical features of these two bands, and so the Riemann's initial analytical definition of the complete set, that fits all cases included in the Riemann treatise. My result is this:
It finds a relation instead of (2), so the left side of (1) breaks down in two integer triplets $\alpha' + \beta' + \gamma' \qquad \text{and} \qquad \alpha'' + \beta'' + \gamma''$ On of such triplet (due to (1)) is necessarily positive, the other null or negative. We assume the definite expression for the triplet $(\alpha' + \beta' + \gamma')$ to be positive. The function (3) is given by the $P$-function if and only if the corresponding triplet $\alpha' + a' + \beta' + b' + \gamma' + c'$ is also positive.
The proof is very simple. It is sufficient to establish the singular points $0$, $\infty$, $1$ corresponding fundamental branches of the P-function in the form of hypergeometric series and then to make the comparison. For example, for $x = 0$we have: $P^{(\alpha')} = x^{\alpha'} \cdot (1-x)^{\gamma'} \cdot F (\alpha' + \beta' + \gamma', \alpha' + \beta'' + \gamma', 1 + \alpha' - \alpha'', x)$ $P^{(\alpha'')} = x^{\alpha''} \cdot (1-x)^{\gamma''} \cdot F (\alpha'' + \beta' + \gamma'', \alpha'' + \beta'' + \gamma'', 1 + \alpha'' - \alpha', x)$ and here we see immediately that the one or other of the $F$-series breaks that occur (and therefore represents a rational integral function of $x$), according as $(\alpha' + \beta' + \gamma') \qquad \text{or} \qquad (\alpha'' + \beta'' + \gamma'')$ It is a null or negative integer. This is the essence. Here I don't discuss further details.
Winston, F.M. (1895). Eine Bemerkung zur Theorie der hypergeometrischen Function, Mathematische Annalen, 46 (1) 160. DOI: 10.1007/BF02096208 (Göttinger Digitalisierungszentrum | Academic Search)
(1) It seems that an english version of the paper exists, but I cannot find it.

### Mary Frances Winston

Mary Frances Winston was the first women mathematicians from United States who obtained the PhD in mathematics in Europe, at Göttingen, where she went in order to work with Felix Klein in the end of the 1893. She was not the only woman at Göttingen in that period: Grace Chisholm, English mathematician, the first woman admitted to Göttingen, and Margaret Maltby, a physicist from U.S.A.
At about this time, early in the 1890s, there had been discussion in Germany concerning admission of women to the universities. While the Prussian Minister of Culture was not unsympathetic to the idea, the overseer of the University at Göttingen was firmly against it. In spite of that, it was decided that foreign women should be admitted to study mathematics. Felix Klein, the mathematician responsible for bringing Chisholm and Winston to Göttingen, explained later that "Mathematics had here rendered a pioneering service to the other disciplines. With it matters are, indeed, most straightforward. In mathematics, deception as to whether real understanding is present or not, is least possible."
The story of Chisholm, Maltby and Winston has a great importance in the path towards equality of rights between men and women (not only in science), so I decided to extract from How many women mathematicians can you name? (pdf) by Judy Green the paragraphs about Mary Frances Winston:
In the summer of 1893 Klein came to the United States with mathematical models to be displayed at the Columbian Exposition in Chicago and to speak at the International Mathematical Congress held in conjunction with the Exposition. In Chicago Klein met Mary Winston, a graduate student at the University of Chicago whose undergraduate degree was from the University of Wisconsin. After teaching for two years in Milwaukee she studied with Charlotte Scott at Bryn Mawr before coming to the University of Chicago in its inaugural year, 1892. Klein agreed to sponsor her admission to the university but could not provide her with financial support.
Although Winston applied for a European fellowship from the Association for Collegiate Alumnae, she did not receive it and was able to go to Germany only because of the generosity of a woman mathematician, Christine Ladd-Franklin, who personally provided her with a \$500 stipend. Mary Winston arrived in Göttingen in the fall of 1893 and waited for Klein to clear the way for her admission to the university. A few weeks after her arrival, Winston wrote her family that the people in Göttingen were very skeptical as to her chances for admission; they were wrong.
Two years after coming to Germany, Winston published a short paper in a German mathematical journal. The authors of a 1934 book about mathematics in nineteenth century America note that this particular journal contains fifteen articles published by Americans between 1893 and 1897. They then list the authors of fourteen of these articles, omitting only the name Mary Winston. Winston's paper was based on a talk she had given in the mathematics seminar at Göttingen within months of her arrival in Germany. That talk was the first such given by a woman and she wrote her family that the presentation "went off reasonably well... I do not think that anyone will draw the conclusion from it that women cannot learn Mathematics."
Upon her return to the United States in 1896, Mary Winston took a job teaching high school in Missouri. The following year she received her Ph.D. from Göttingen and became Professor of Mathematics at Kansas State Agricultural College, now Kansas State University. Three years later she resigned and married Henry Byron Newson, a mathematician at the University of Kansas. Henry Byron and Mary Winston Newson had three children born in 1901, 1903, and 1909. Mary Winston was widowed in 1910 when her youngest child was just three months old. She moved in with her parents, who were then living in Lawrence. She returned to teaching, but not to mathematical research, a few years later at Washburn College in Topeka, Kansas. Her son reported that she took that job because Topeka was within commuting distance of Lawrence and her parents could care for the children during the week. Newson remained at Washburn until 1921; she spent the rest of her career at Eureka College in Illinois, retiring in 1942.