12
votes
Accepted
Why are the symbols E, F, G, L, M, and N used for the coefficients of the fundamental forms?
See the paper by Gauss, Disquisitiones generales circa superficies curvas, from 1828. On page 24, we find:
$$
E\,dp^2 + 2F\,dp\,dq + G\,dq^2 .
$$
Since then everyone continued to use the letters $E,F,...
- 10.6k
11
votes
Accepted
Help translate from German a quote by Hermann Weyl in Space Time Matter
One must firmly protest these orgies of a formalism with which even technicians are getting harassed today.
(Literally: of the formalism, with which one is beginning to harass even technicians today.)...
- 7,809
10
votes
Non-Euclidean geometry: motivations to develop it at the times of Gauss?
The motivation of discoverers of non-Euclidean geometry (Gauss, Lobachevski and Bolyai) was their attempts to prove the Fifth postulate of Euclid (to deduce it from the other axioms, or to replace by ...
- 51.1k
9
votes
What is the history of staircase or 𝜋=4 paradox?
The "staircase paradox" (or "Pythagoras paradox") name appears to be recent, so it is hard to search for it. Wolfram calls it "diagonal paradox", but that may be conflating it with a different paradox ...
- 80k
9
votes
Accepted
Why is distance sometimes abbreviated S?
I think it most likely stands for spatium. E.g. Euler’s first book Mechanica (1736) uses $s$ throughout and first introduces it as follows (p. 13):
Theorema. (...) oportet determinare tempus, quo ...
- 7,809
8
votes
Accepted
When was a partition of unity discovered?
Partitions of unity were formally introduced by Dieudonne (C. R. 205 (1937) 593-595), and for some time they were even called "Dieudonne decompositions".
However is some special cases they were used ...
- 51.1k
7
votes
Accepted
Did Clifford introduce the "Clifford torus", and for what purpose?
The Clifford torus was introduced by Clifford in 1873, not as embedded into $\mathbb{R}^4$ or $\mathbb{C}^2$, but first projectively and then intrinsically, by identifying the opposite sides of a flat ...
- 80k
7
votes
What makes the musical isomorphism, musical?
The $\flat$ map in music lowers the pitch of a note (by one half) and the corresponding map lowers indices, the $\sharp$ map raises it, as well as the indices. And of course adding a flat after a ...
- 1,738
7
votes
Accepted
Did Riemann invent the Riemann curvature tensor?
A short answer: yes, he did.
Riemann's habilitation lecture was aimed at a broad non-mathematical audience,
so he did not use formulas in it, trying to explain everything in words. The curvature ...
- 51.1k
6
votes
Accepted
Who developed The Fundamental Theorem of Curves?
Existence claims as theorems became fashionable after Hilbert introduced the axiomatic method. Before that people more often talked about problems and constructions (following Euclid's, or rather ...
- 80k
6
votes
Accepted
Who first had the idea to study surfaces via rings of functions, as in algebraic geometry?
A canonical reference on this is Dieudonne's History of Algebraic Geometry. An abridged version Historical Development of Algebraic Geometry is freely available, see also Easton's slides.
Let me make ...
- 80k
5
votes
The history of different constructions of tangent spaces
I apologize for the limited information in this answer, but as the question has gone unanswered for 4 years, I plead that something is better than nothing.
Your fourth version appears in The ...
- 4,922
5
votes
Accepted
Who first wrote down $S^6$'s standard almost complex structure? And who first proved that it is not integrable?
According to this arxiv paper by Atiyah, existence and construction dates from 1947; non-integrability from 1951.
Here is Atiyah's history:
Ehresmann 1947: Introduced the notion of almost complex ...
- 7,237
5
votes
Does Gauss own two “Theorema”?
In his Disquisitiones generales circa superficies curvas (1827), §12, page 24, Gauss called egregium [sponte perducit ad egregium, i.e. spontaneously leads to excellent] the following Theorem:
Si ...
- 15.2k
4
votes
Who first had the idea to study surfaces via rings of functions, as in algebraic geometry?
The idea is usually attributed to Dedekind and Weber in Theorie der algebraischen Functionen einer Veränderlichen (1882): [1, 2, 3, 4, 5,...].
- 7,809
4
votes
Accepted
Nomizu's structural approach to differential geometry
The intended sense of "structure" comes from Bourbaki, who reformulated all of mathematics as a theory of structures founded on set theory in their multi-volume Elements of Mathematics (with ...
- 80k
4
votes
Accepted
First appearance of Hadamard's lemma on smooth functions
The reference to the book on wave propagation (Leçons sur la propagation des ondes et les équations de l'hydrodynamique. Cours du collège de France, Hermann, 1903)
available at https://archive.org/...
- 56
3
votes
Help translate from German a quote by Hermann Weyl in Space Time Matter
One has to object thoroughly to the orgies of formalism, with which recently even technicians have been bothered.
Man muß ... protestieren -> One has to object to (in the context of legislative ...
- 139
3
votes
Accepted
From where the so-named "elastica problem" is coming from?
A. G. Greenhill, The applications of elliptic functions, Macmillan, London & NY, 1892,
pp. 87-88.
If you read French, a much clearer and more comprehensive discussion is in
G.-H. Halphen, Traite ...
- 51.1k
3
votes
Complete list of publications of Rebecca Barlow
Math Sci Net lists 6 publications, dating from 1984-1999.
- 2,697
3
votes
Material models of Riemann surfaces
Kharkiv University (Ukraine) subscribed to all models made M. Schilling, who probably was a student of Klein, and who run a company making and selling these models. Currently they photograph them and ...
- 51.1k
3
votes
Accepted
What is the origin of French/Burmester's curves?
There seems to be little secondary literature on this so answering the OP questions fully would take some serious digging into the original sources. One promising secondary source that I was unable to ...
- 80k
3
votes
Accepted
How did Einstein arrive at his field equations without the Bianchi identities or variational formulation?
Pais does not "suggest" that Einstein wrote the equation in this form, he reproduces on p.256 what Einstein wrote on November 25, 1915 with reference to his presentation to the Prussian Academy: $R^{\...
- 80k
3
votes
Einstein already used the idea of time orientation when formulating General Relativity?
I believe the question started being studied in earnest in the 1950s, with formal definition and first results usually attributed to Markus (1955, p. 412) — by, e.g., Hawking in his famous 1966 prize ...
- 7,809
3
votes
Accepted
Who discovered that the electromagnetic tensor is the curvature of a connection?
I still have only a partial answer to the question. Looking at Trautman's lecture notes published in 1970 (https://doi.org/10.1016/0034-4877(70)90003-0), it is more than clear that he knew for sure ...
- 181
2
votes
Did Gauss formulate, or at least know of, the full essence of the Gauss-Bonnet Theorem?
Although I've already accepted one answer, I added this answer in order to clarify what is known about Gauss's work towards the general Gauss-Bonnet theorem and what is a matter of speculation; this ...
- 4,679
2
votes
Accepted
Summary of Gauss's work on geodesic lines on ellipsoid
I do not read German, but I can tell what I know about this. Gauss' worked on a generic ellipsoid (with three different axes) but this work was motivated by geodesy (rather than pure mathematics). So ...
- 51.1k
2
votes
Accepted
What makes the musical isomorphism, musical?
The linked MathOverflow page (from the Wiki page) says that Berger himself doesn't remember how the name came about. The ♭: V → V* and ♯: V* → V notation uses musical symbols, which to me suggests ...
- 2,578
2
votes
What is the etymology of the mathematical terms "sheaf, stalk, germ"?
Todd and Alexandre, I just want to offer a perspective to your discussion from a German language point of view. I wanted to put it as a comment, but it was getting too long so I am putting it as an ...
- 121
2
votes
What is the history of staircase or 𝜋=4 paradox?
This is a follow-up to some of my comments to the OP and to the answer that @Conifold gave. A few days ago I purchased a copy of the book that I had mentioned, Measure and the Integral by Lebesgue (...
- 3,582
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