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There is a notation used in differential geometry and general relativity in which the partial derivative operators $\partial_\mu$ are used as the basis for the space of contravariant vectors, and their duals, notated $dx$, are the basis for covariant vectors. I've found a couple of sources that attribute this notation to Élie Cartan:

"for [covectors] a cute notational game due to Élie Cartan, is played" -- Mathematics for Physics II, Michael Stone, cns.gatech.edu/~predrag/courses/PHYS-6124-12/StGoChap11.pdf

"Élie Cartan proposed to use differential coordinates $dx^i$ as a convenient basis of 1-forms." -- Kosyakov, Classical Theory of Particles and Fields, quoted at https://physics.stackexchange.com/questions/144089/is-partial-derivative-a-vector-or-dual-vector

Cartan published his first paper in 1899 and lived until 1951. Can anyone give date, or a reference to the paper in which he first introduced this notation?

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  • $\begingroup$ Maybe Cartan's book Lecons sur la geometrie des espaces de Riemann, published in 1928, based on his lectures of the academic year 1925-1926. In 1936 this book was translated into Russian. $\endgroup$ Commented Jul 3, 2015 at 21:52
  • $\begingroup$ Cartan did not "introduce" what Stone is ascribing to him, no one did. Parts of the "game" developed in works of different people over time. Cartan himself pulled the strands together over the course 20-30 years. $\endgroup$
    – Conifold
    Commented Jul 21, 2015 at 1:15
  • $\begingroup$ Which notational trick is meant by this question? The notation $dx^i$ and $\partial_\mu$ is much older than Cartan. (The first one going back to Leibniz the second one to probably Legendre.) $\endgroup$ Commented Apr 2, 2019 at 18:10

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Some sluething revealed to me that the first source was really Cartan's La Géométrie des espaces de Riemann, published in 1925 three years before the Lecons sur la geometrie des espaces de Riemann. Looking at the list of Cartan's works, this is the first work on the Riemannian differential geometry. The link for the first source is in the original French; About 60-80% of the text is understandable with only knowledge of English (from my experience, for I do not know French) because the French words for mathematical terms are very similar. For the basis of covariant vectors being $dx$, here are the relevant sections (three sections, close together but with paragraphs in between, images have been posted in French): enter image description here


enter image description here


enter image description here


This is the source for the statement ""Élie Cartan proposed to use differential coordinates dxi as a convenient basis of 1-forms," though his terminology is outdated in this context (no mention of basis or of differential forms, even though both are being discussed). I believe that in the second source linked (the one given by Mauro ALLEGRANZA) the notation is more modern (being more 'official' than this one) but I haven't checked.

This is where the cute notational game is played. Although the partial derivate operators aren't explicitly shown, because the inner product of the contravariant and covariant bases equals the metric tensor (second image), by definition the contravariant base would be equivalent to the partial derivative operator. I believe that the more modern notation is in the second source linked, but as I have before stated, I haven't checked.


As an addition to this answer, wythagoras translated the French text to English. Note that he had to reform some of the sentences or split them in two sentences.

Citation 1.

An entire system of curvlinear coordinates $u^1$, $u^2$, ... $u^n$ in $n$-dimensional Euclidean space correspond to a determined form of linear elements of the space $$ds^2=\sum_{i,j} g_{ij} du^i du^j$$ where the coefficients $g_{ij}$ are certain functions in $u^1$, $u^2$, ... $u^n$.
The two fundamental problems that arise are the following:

The linear element of the Euclidean space is a priori supposed to give rise to a point such that the curvlinear coordinates can be compared to the Cartesian coordinates.

Find the conditions under which the functions $g_{ij}$ are able to represent the $ds^2$ from the Euclidean space.

Citation 2.

Imagine for each point $M$ the curvlinear coordinates $u^1$, $u^2$, ... $u^n$ a system of cartesian coordinates (R) with the origin on that point. Now consider the point on infinitesimal distance of $M$, namely $M'$, and compare it to $du^1$, $du^2$, ... $du^n$. The coordinate axes are therefore tangents to the line coordinates and the scalar products of the vectors of the coordinates $e_1, e_2, e_n$ is the quantity $g_{ij}$.

Citation 3.

The conditions for integrability of equation (4) provide the other equations. First notice that the first member of equation (4), which we call $Dx^i$, represent the components of motion of a mobile point, of which the coordinates, comparing to (R), would be $x^i$. Likewise the components of the geometric variation of a vector $X^i$ would be: $$DX^i = dX^i + \sum_{k,r} \Gamma^i_{k,r}X^k du^r$$

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  • $\begingroup$ Thanks for your efforts, but this isn't what I was attempting to ask about in the question. I was specifically asking about the notational tricks described in the question. $\endgroup$
    – user466
    Commented Jul 4, 2015 at 18:56
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Differentials and their linear combinations already appear in full glory in Cartan's first paper on Pfaffian systems Sur certaines expressions différentielles et le problème de Pfaff (1899), where he works out some of their algebra. But he was not the first to use them, the paper mentions Pfaff, Natani, Clebsch, Lie, Frobenius and Darboux, and in L’intégration des systèmes d’équations aux différentielles totales (1901) he references Biermann's 1885 paper Ueber n simultané Differenlialgleichungen der Form $\sum X_\mu dx_\mu$. "It was Pfaff who pioneered the study of exterior differential systems by his formulation of the Pfaff problem in [Pfaff 1814-15]. The exterior derivative of a pfaffian form, called the bilinear covariant, was introduced by Frobenius in 1877 and efficiently used by Darboux in [Darboux 1882]", see Exterior Differential Systems.

Did they think of $dx_\mu$ as "basis of $1$-forms"? Certainly not Pfaff, linear algebra as we know it did not emerge until Sylvester and Cayley later into the century, but it does not mean that he could not play the "notational game" without the terminology. Cartan's innovation, which came in his 1922 book Lecons sur les Invariants Integraux, was extending the exterior differential, and the game, to higher order forms, but he inherited manipulation of $d$'s from much earlier. In fact, at a formal level some of this game goes back to Leibniz, who would write differential equations in differentials to separate variables, etc. As for $\partial$'s the game, if not the notation, goes back to at least moving trihedrons in Darboux's Leçons sur la Théorie Générale des Surfaces, although it was Cartan who greatly advanced the method and generalized it to higher dimensions.

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  • $\begingroup$ Thanks for taking the time to write this up. There's a lot of interesting material here. However, this doesn't answer the question. $\endgroup$
    – user466
    Commented Jul 12, 2015 at 1:59
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It is not quite clear, even from the more explicit post on physics stack exchange, which of the two games you are interested in. They are different. One is the idea of the various bundles involved, e.g., that a 1-form is not (as the earlier mathematicians thought of it) "an expression of the form pdX + qdY+ rdZ" (plus, they well knew how the expresion transforms under changes of variable), but rather a section of a vector bundle. The next generation (Ehresmann, Chevalley, Lichnerowicz, H. Cartan, Bourbaki in general) learned the essence of fibre bundles from Cartan's work, and developed their better way of expressing it from their attempts to clarify his very difficult to read papers. The other is the exterior calculus, i.e., the union of the Grassmann calculus with Riemannian geometry, all the rules for calculating with tensors, d, curvature 2-form with values in a Lie algebra, etc. Of this, Cartan's only truly original contribution was the definition of the exterior derivative, d, for forms of higher degree. And its notation. But d for higher degree forms does not seem to be what Kosyakov was really referring to. He seems to be referring to Cartan's systematisation of ideas that do go back to Hamilton, Grassmann, Stokes, Frobenius, Pfaff, Riemann, and Ricci. Cartan's sytematisation and notation was better, but the ideas were there earlier.
In particular, $dx_i$ as a notation was early, but as a section of the cotangent bundle, that is due to Cartan (or perhaps Ricci). And I do not see that Cartan used $\partial_x$ early on.

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