*Difficulty:*★★★☆☆ undergraduate

I have written on the topic of “covariant” and “contravariant” vectors (and higher-rank tensors) previously, and have been intending to write an update for a number of years. It must be noted some authors recommend avoiding these terms completely, including Schutz 2009 §3.3:

Most of these names are old-fashioned; ‘vectors’ and ‘dual vectors’ or ‘one-forms’ are the modern names.

Let’s return to Schutz’ reason soon. I have followed this naming practice myself, except I prefer to say “covector” or “1-form”, rather than “dual vector” which can be clumsy. (What would you call the vector which is the (metric) dual to a given 1-form: the “dual of a dual vector”, or a “dual-dual-vector”!?) People also talk about “up[stairs] indices” and “down[stairs] indices”, which seems alright.

But if you want to be cheeky, you might say a vector is **co**variant, while a 1-form is **contra**variant — the exact opposite of usual terminology! I remember a maths graduate student at some online school stating this. Similar sentiments are expressed by Spivak 1999 vol. 1 §4:

Nowadays such situations are always distinguished by calling the things which go in the same direction “covariant” and the things which go in the opposite direction “contravariant”. Classical terminology used these same words, and it just happens to have reversed this: a vector field is called a

contravariant vector field, while a section ofTis called a^{*}Mcovariant vector field. And no one has had the gall or authority to reverse terminology so sanctified by years of usage. So it’s very easy to remember which kind of vector field is covariant, and which is contravariant — it’s just the opposite of what it logically ought to be.

While I love material which challenges my conceptual understanding, and Spivak’s humorous prose is fun; trying to be too “clever” with terms can hamper clear communication. Back to Schutz, who clarifies:

The reason that ‘co’ and ‘contra’ have been abandoned is that they mix up two very different things: the transformation of a basis is the expression of

newvectors in terms ofoldones; the transformation of components is the expression of thesameobject in terms of the new basis.

If you take the *components* of a fixed vector in a given basis, they transform contravariantly when the basis changes. But if you consider the vector as a whole — a single geometric object — and ask how a basis vector (specifically) is mapped to a new basis vector, the change is “covariant”. (Recall, as Schutz explains: “The property of transforming *with* basis vectors gives rise to the *co* in ‘covariant vector’ and its shorter form ‘covector’.”) In general, if you fix a set of components, by which I mean fixing an ordered set of numbers like (0,1,0,½) say, and then change the basis vectors these numbers refer to, then the change of a vector (as a whole entity) is “covariant”, so-called. For 1-forms, the converse of these statements apply. Some diagrams would make this paragraph clearer, but I leave this as an exercise, sorry.

However, it seems to me the most accurate description is that vectors *don’t change at all*, when you change a basis! Picture a vector as an arrow in space, then the arrow does not move. In this sense vectors are neither contravariant nor covariant, but *invariant*! (We could also say *generally covariant*, since they are geometric entities independent of any coordinate system. This is the usual modern meaning of the word “covariant”, but it’s a bit different to the covariant–contravariant distinction, so for clarity I avoid this language here.) In conclusion, one of the clearest descriptions is to simply say: vector, or 1-form / covector. Or, given historical usage, it is especially clear to say a vector’s “*components* transform contravariantly”. See the Table.

object | clarification | vector | 1-form |
---|---|---|---|

components | same (co)vector, but components in a new basis | contravariant | covariant |

(co)vector | same (co)vector, treated as a whole | invariant | invariant |

basis (co)vector | transform to different (co)vector | covariant | contravariant |

*Addendum:* I recently learned a dual basis need not be made up of 1-forms, as in the usual formulation in differential geometry, but of vectors instead! Recall the defining relation between a coordinate basis and cobasis: , or for an arbitrary frame. In particular, each cobasis element is orthogonal to the “other” 3 vectors. But we can take the duals , which are vectors but obey the same orthogonality relations, via the metric scalar product: . (By relating to the standard approach I may have made things look complicated, but this should be visualised as simply finding new vectors orthogonal to existing vectors.) (On a separate note, “dual” here means as an individual vector, not dual as a basis.) It seems this vector approach to a dual basis was the original one. In 1820 an Italian mathematician Giorgini distinguished between *projezione oblique* (parallel projections) and *projezioni ortogonali* (orthogonal projections) of line segments, now termed contravariant and covariant. According to one historian (Caparrini 2003 ), this was “one of the first clear-cut distinctions between the two types of projections in analytic geometry.” However priority goes to Hachette 1809 . Today in geometric algebra, also known as Clifford algebra, a dual basis is also defined as vectors not 1-forms, via (Doran & Lasenby 2003 §4.3).