# Is $\Gamma^i_{jk}$ the Christoffel symbol or the Christoffel symbols?

For years, I have been perplexed that the expression $\Gamma^i_{jk}$ is often referred to in the plural as "the Christoffel symbols", although sometimes it is referred to in the singular as "the Christoffel symbol". I have found this even in the 1926 book "The absolute differential calculus", which is a translation of Levi-Civita's 1925 Italian book "Lezioni di calcolo differenziale assoluto". Since Levi-Civita approved the translation, I imagine he approved the use of "Christoffel's symbol of the first kind" and "Christoffel's symbol of the second kind" which appear in there, pages 109–110, although he changed to the plural on page 169.

Another early writer, Hermann Weyl, in his 1918 "Raum, Zeit, Materie", page 91, referred to "Christoffelsche Dreiindizes-Symbole", in the plural.

In the original 1869 paper by Christoffel, "Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades", the word "symbol" was not really used, and the words which are used don't give a clear indication either way of the singular or the plural.

Modern authors seem to have a slight preference for the plural, although it doesn't sound quite right, and there is quite a lot of inconsistency.

Now here are my theories.
Theory 1: Because initially there was the symbol of the first kind and symbol of the second kind, the plural was adopted to refer to the two kinds of symbols. I.e. there were two symbols. Nowadays we mostly only use the second kind. But the use of the plural has stuck, erroneously, due to everyone just following everybody else like sheep. In other words, it's a misunderstanding, which everyone copies because the original reason for the plural is lost in the mists of time.

Theory 2: Since the array of coefficients is not a tensor, it could not be called "the Christoffel tensor". It could have been called "the Christoffel array", but almost no one does that. So since there is no "container" term for the full array, people use the plural "symbols" to mean the elements of the array. In other words, each element of the array is one symbol.

The problem with both of these theories is that so many authors alternate between singular and plural, with no obvious reason.

So here is my question:
Is there a correct terminology, with a definitive ruling on which usage is correct?

• In index notation, there tends to be an ambiguity as to whether a symbol with indices stands for a single object or a collection of objects. According to modern conventions, for example, $v^i$ stands for the whole object, because the Latin i is understood to be an abstract index. Before abstract index notation, this would have been understood as a notation for a component, but would also have been used sometimes to mean the whole object anyway. The ambiguity, when it exists, seems harmless to me, and probably the same ambiguity applies to the Christoffel symbol(s).
– user466
Oct 1, 2015 at 0:56
• There is the same issue with the Euler-Lagrange equation(s) which can be written in components. I am also not sure what the source of correctness would be for terminology in general, other than prevailing usage. Lexicographers occasionally suggest etymological rules, e.g. not mixing words of different origin, but "combinatorics" say is universally accepted despite “the dubious honor of being composed of elements of three languages" jstor.org/stable/10.4169/… Christoffel symbol(s) do not seem to be misleading or confusing either. Oct 1, 2015 at 1:23
• @Conifold: So some named objects, and also named equations (which hadn't occurred to me), were referred to in the plural a hundred years ago because each component was thought of as an individual object, whereas the modern way of thinking is that the whole collection is a single object. I guess that's the accumulated effect of set theory on modern thinking. Now we spontaneously think of collections of objects as objects themselves. My conclusion, then, would be that I am justified to use the singular, to indicate a modern way of thinking! (After all, many language don't even have plural!!) Oct 1, 2015 at 4:03
• A "definitive ruling" in mathematical terminology? Don't hold your breath. Oct 2, 2015 at 13:27