# Making sense out of covariance and contravariance

I just read about co- and contravariant vectors and I am not sure that I got it right: If we imagine that we have a n-dimensional manifold $M$ then a tangent space is spanned by the vectors $\partial_1,...,\partial_n.$ These guys transform from one coordinate system to another by

$$\frac{\partial}{\partial x^i} = \frac{\partial y^j}{\partial x^i} \frac{\partial}{\partial y^j}.$$ This transform is according to wikipedia called the covariant transform. Now, it is worth noticing that normally the covectors are the elements in the dual space. The basis vectors of the dual space are given by $dx_1,...,dx_n.$ They transform differently as

$$dx^i = \frac{\partial x^i}{\partial y^j} dy^j.$$

Despite, although we transform covectors this transform is called contravariant. So somehow it seems as if the kind of transform does not fit to the kind of vector we are considering here and I don't see why this happens.

If you have any questions, please leave me a comment.

• Related: physics.stackexchange.com/q/79013/2451 and links therein. – Qmechanic Apr 28 '15 at 20:20
• Are you getting confused by the prefix "co-". In "covariant" it's used in the sense of "varying with". In "covector" it's used in the sense of "dual to, opposite". In other words, "co-" has two opposing meanings here and so I can see it's a bit confusing to find that covectors don't transform covariantly. – Dan Piponi Apr 28 '15 at 21:26
• @DanPiponi that's a neat explanation. But doesn't the prefix "co-" in "covector" have the same etymology as the "co-" in "covariant"? I would have guessed that both come from latin "with" "together", which presupposes two things, as in "dual". – Michael Apr 4 '19 at 12:28
• I wasn't completely correct. Covectors do transform covariantly and vectors transform contravariantly. Covectors are covariant because the matrix to transform their components to a new basis is the same matrix used to construct the new basis. (Contravariant) vectors use the inverse. Summary here: en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors But I'm still pretty sure both senses of co- are in use here. See the brief mention of the categorical "coconcept" in the wikipedia article where it mentions the terminological conflict - the source of my error in fact. – Dan Piponi Apr 4 '19 at 14:12

## 2 Answers

Basically, vectors are called contravariant because their components transform oppositely to the basis vectors: if our change of coordinates is such that

$$\frac{\partial}{\partial x^i} = \frac{\partial y^j}{\partial x^i} \frac{\partial}{\partial y^j}$$

then if we have a vector $\mathbf{V}$, its components $V^i_x$ in the $x$ coordinates are related to its components $V^i_y$ by

$$V^i_x = \frac{\partial x^i}{\partial y^j} V^j_y.$$

By the same logic, 1-forms are called covectors or covariant vectors because their components transform like the basis vectors, while the basis covectors transform like the components of vectors.

In modern mathematical terminology, a functor is called covariant when it preserves the direction of the morphisms, contravariant if it reverses it. For a given differentiable map between manifolds (of which a special case would be open sets within the same manifold), the derivative is a map between the associated tangent bundles. This defines a covariant functor. Pullback of differential forms (covector fields) is a map between the covector bundles in the opposite direction, and defines a contravariant functor. In other words, the association of the bundle of vector fields to a manifold is a covariant functor, of the bundle of 1-forms is a contravariant functor. A pretty confusing (apparent) discrepancy in terminology.

Spivak, in his comprehensive introduction to differential geometry vol 1, says about it (page 113)

Classical terminology used these same words [covariant and contravariant], and it just happens to have reversed this: a vector field is called a contravariant vector field, while a section of $T^\ast M$ is called a covariant 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 contravariant - it's just the opposite of what it logically ought to be.