Covariant and Contravariant: What Are the Differences in Differential Geometry?

[ObsessiveMathsFreak]

Thread Necromancy!

I'd like to bump this old thread to reopen discussion on the idea of covariance and contravariance.

To begin on the wrong foot, I think that covariance and contravariance are in fact red herrings in the study of tensors, but before I go any further, I'd like to explain myself through equations. A word of warning first. I won't be using superscripts and subscripts to denote contravariant or covariant components. In fact, I'm going to avoid using the terms contravariant and covariant at all, for reasons which I hope will become clearer later

I'm going to talk about rank 1 tensors first, i.e. vectors. I'll distinguish the two types of vector by use of lower case for the first type and uppercase for the second. So the basis vectors, and components for the first type will be denoted by \mathbf{e}_u^i and a_u^i, and for the second type by \mathbf{E}_u^i and A_u^i. Here the subscript "u" denotes the coordinate system, and the superscript i is denoting the index.

Any vector \mathbf{w} can be represented by either coordinate system. Using the Einstein summation convention;

\mathbf{w} = a_u^i \mathbf{e}_u^i = A_u^i \mathbf{E}_u^i.

OK so far these two vector types seem to simply be two different basis for the vector space. Now we distinguish them by how their representations change under a change of coordinates.

If we make a change of coordinates from the system "u" to the system "v", We must change both the basis vectors and the components to obtain the representations of \mathbf{w} in the new coordinate system.

The first type of vector transforms in the following way.

\mathbf{e}_v^i = \frac{\partial u^j}{\partial v^i}\mathbf{e}_u^j
a_v^i = \frac{\partial v^i}{\partial u^j}a_u^j

And the second type of vector transform in, in some sense, an opposite way.
\mathbf{E}_v^i=\frac{\partial v^i}{\partial u^j} \mathbf{E}_u^j
A_v^i = \frac{\partial u^j}{\partial v^i} A_u^j

As you can see, whichever transformation you decribe as covariant or contravariant, for each type of vector, its components transform one way and its basis vectors transform the other way. This is I think a big part of the confusion between the two terms, and I think, the primary reason for their inappropriateness as decriptive terms. Depending on your point of view, one type is contravariant and the other covariant, but this depends on whether you are speaking from a component viewpoint or a basis vector viewpoint.

More later.

 

[ObsessiveMathsFreak]

At this time, I'd like to drop the terms contravariant and covariant, and instead simply concentrate on these two types of transforming vectors. I would like, if I may, to denote the lower case type as a tangent vector, and, if I may be so bold, denote the upper case type as a cotangent vector. I'm not entirely sure about the appropriateness of the nomenclature here, but I will press on.

So represented by tangent vectors
\mathbf{w} = a_u^i \mathbf{e}_u^i.
And by cotangent vectors
\mathbf{w} = A_u^i \mathbf{E}_u^i.

So, if we make a change of cooridinate system from the system \mathbf{u}=(u^1,u^2,\ldots,\u^n) to the system \mathbf{v}=(v^1,v^2,\ldots,\v^n), both the basis vectors and components of the tangent and cotangent vectors change.

Lets denote the Jacobian matrix for the transformation from u to v by J with
J_{ij} = \frac{\partial v^i}{\partial u^j}
And the inverse Jacobian matrix J^{-1} is then given by
J_{ij} = \frac{\partial u^i}{\partial v^j}

To keep things from becoming totally abstract (this may already be a lost cause), I'd like to deal specifically with 3d vectors in 3d space. The following arguments do extend to higher, and lower, dimensions.

First, consider the tangent basis vectors\mathbf{e}_u^1 , \mathbf{e}_u^2, \mathbf{e}_u^3. These are vectors in their own right, which have a representation in carteasian coordinates. For the sake of argument, let's represent the carteasian coordinates basis vectors in carteasian space as column vectors. For example
\mathbf{e}_u^1=\left[\begin{array}{c}a\\b\\c\end{array}\right], or something.

Using this column vector representation, we can create a 3x3 matrix out of the tangent basis vectors.

e_u = \left[\begin{array}{ccc}<br /> \mathbf{e}_u^1 &amp; \mathbf{e}_u^2 &amp; \mathbf{e}_u^3 \end{array}\right]

Using this shorthand, we can express the tangent basis vector transformation rules as


\left[\begin{array}{ccc}<br /> \mathbf{e}_v^1 &amp; \mathbf{e}_v^2 &amp; \mathbf{e}_v^3 \end{array}\right] = \left[\begin{array}{ccc}<br /> \mathbf{e}_u^1 &amp; \mathbf{e}_u^2 &amp; \mathbf{e}_u^3 \end{array}\right] J^{-1}

Or if you like

e_v = e_u J^{-1}

Also, let's express the components of the tangent vector a_u^1\ ,\ a_u^2\ ,\ a_u^3 as a column vector
\mathbf{a_u}=\left[\begin{array}{c}a_u^1\\a_u^2\\a_u^3\end{array}\right]

Using this notation, the transformation rule for the tangent vector components can be expressed as
\left[\begin{array}{c}a_v^1\\a_v^2\\a_v^3\end{array}\right]= J \left[\begin{array}{c}a_u^1\\a_u^2\\a_u^3\end{array}\right]

Or if you like

\mathbf{a_v} = J \mathbf{a_u}

Next, I'd like to do something similar for the cotangent vectors, except this time I'll do things with row vectors, instead of column vectors.

So let's represent the carteasian coordinates of cotangent basis vectors as a row vector, For example:
\mathbf{E}_u^1=\left[\begin{array}{ccc} a &amp; b &amp; c \end{array}\right], or something.

Again, form a matrix, but this time the basis vectors are rows, not columns.

E_u = \left[\begin{array}{c}\mathbf{E}_u^1\\\mathbf{E}_u^2\\\mathbf{E}_u^3\end{array}\right]

So in this way we can represent the transformation rule for cotangent basis vectors as
\left[\begin{array}{c}\mathbf{E}_v^1\\\mathbf{E}_v^2\\\mathbf{E}_v^3\end{array}\right]=J \left[\begin{array}{c}\mathbf{E}_u^1\\\mathbf{E}_u^2\\\mathbf{E}_u^3\end{array}\right]

Or if you like

E_v = J E_u

And finally, let's express the components of the cotangent vector A_u^1\ ,\ A_u^2\ ,\ A_u^3 as a row vector
\mathbf{A_u} = \left[\begin{array}{c}A_u^1 \\ A_u^2 \\ A_u^3 \end{array}\right]

And so the transformation rule for cotangent components can be expressed as
\left[\begin{array}{c}A_v^1 \\ A_v^2 \\ A_v^3 \end{array}\right] = \left[\begin{array}{c}A_u^1 \\ A_u^2 \\ A_u^3 \end{array}\right] J^{-1}
Or if you like
\mathbf{A_v} = \mathbf{A_v} J^{-1}

So summing up, we have the tangent vector components as columns, and the cotangent vector components as rows, with the tangent and cotangent basis matrices, and the four transformations as follows,

\left[\begin{array}{ccc}<br /> \mathbf{e}_v^1 &amp; \mathbf{e}_v^2 &amp; \mathbf{e}_v^3 \end{array}\right] = \left[\begin{array}{ccc}<br /> \mathbf{e}_u^1 &amp; \mathbf{e}_u^2 &amp; \mathbf{e}_u^3 \end{array}\right] J^{-1}

\left[\begin{array}{c}a_v^1\\a_v^2\\a_v^3\end{array}\right]= J \left[\begin{array}{c}a_u^1\\a_u^2\\a_u^3\end{array}\right]

\left[\begin{array}{c}\mathbf{E}_v^1\\\mathbf{E}_v^2\\\mathbf{E}_v^3\end{array}\right]=J \left[\begin{array}{c}\mathbf{E}_u^1\\\mathbf{E}_u^2\\\mathbf{E}_u^3\end{array}\right]

\left[\begin{array}{c}A_v^1 \\ A_v^2 \\ A_v^3 \end{array}\right] = \left[\begin{array}{c}A_u^1 \\ A_u^2 \\ A_u^3 \end{array}\right] J^{-1}

More later, weh I get to my main point.

 

[ObsessiveMathsFreak]

Just looking at those four transformations again, in shorthand.

e_v = e_u J^{-1}
\mathbf{a_v} = J \mathbf{a_u}

E_v = J E_u
\mathbf{A_v} = \mathbf{A_v} J^{-1}

Now, I don't know about you, But I'm very tempted to refer to the transformation involving the jacobian as covariant, and those involving the inverse jacobian as contravariant. Perhaps I'm backwards here, but my point is that these transformations are opposite. With good reason too, as now for a tangent vector

\mathbf{w} = e_v \mathbf{a_v} = e_u J^{-1}J \mathbf{a_u} = e_u \mathbf{a_u}
and for a cotangent vector
\mathbf{w} = \mathbf{A_v} E_v = \mathbf{A_v} J^{-1}J E_u = \mathbf{A_u} E_u

So you can see the reason for when if the components transform either covariantly or contravariantly, the basis vectors must transform oppositely, so that those jacobians cancel.

My main point is, I think that reffering to vectors as being either tangent or cotangent is a lot more appropriate as referring to them as being contravariant or covariant. Contravariant and covariant depend on whether you are talking about the basis vectors, or the components. Tangent and cotangent refer to the vector as a whole object. I'm not entirely sure how this idea would extend to higher ranked tensors, but for vectors at least, I find thinking this way to be far less confusing than thinking of contra and covariance.

The question is obviously raised on what exactly the tangent and cotangent vectors are. Well, my understanding of them comes from the basis vectors. The tangent basis vectors are just the regular tangent space basis for a manifold. i.e.

\mathbf{e}_u^i \equiv \frac{\partial \mathbf{x}}{\partial u^i} \equiv \frac{\partial }{\partial u^i}
Where \mathbf{x}=\mathbf{x}(u^1,u^2,\ldots,u^n) is a point in cartesian space.

The cotangent basis vectors, to my understanding, are in fact the gradients of the coordinates in cartesian space.

\mathbf{E}_u^i \equiv \nabla u^i \equiv grad(u^i)
Where u^i=u^i(\mathbf{x}) is the inverse function of the coordinate mapping into cartesian space..

In other words, \mathbf{E}_u^i is normal to the level surfaces of u^i in cartesiann space. I think. I wish I had a good diagram here.

What are people's thoughts on all this?

 

[Chris Hillman]

At this time, I'd like to drop the terms contravariant and covariant, and instead simply concentrate on these two types of transforming vectors. I would like, if I may, to denote the lower case type as a tangent vector, and, if I may be so bold, denote the upper case type as a cotangent vector. I'm not entirely sure about the appropriateness of thttps://www.physicsforums.com/editpost.php?do=editpost&p=1205732he nomenclature here, but I will press on.

The standard terms you seek would be a vector field and its dual covector field.

A covector is a (simple) one-form, i.e. a real-valued function on vector fields. Given one vector field on a smooth manifold, you can extend this to a basis of vector fields on the manifold, and then the dual one-form takes value unity on the original vector field and zero on all the others.

To be precise, you are discussing a coordinate vector field \partial_w and its dual covector dw, which is the exterior derivative of the coordinate w (that is, a monotonic function on our manifold). Since the exterior derivative of a function is dual to a vector field called the gradient of the function in ordinary vector calculus, we can say this: a coordinate is just a monotonic function; it is associated with a unique vector field (the gradient) and a unique covector field (the dual of the gradient). Just as you said!

To make a coordinate system on some open neighbhood N of a d-manifold, you choose d coordinates on N, such that the corresponding gradients are never parallel. The coordinate vector fields are then also called a holonomic basis, meaning that their Lie brackets always vanish. Then, the Riemann curvature tensor is obtained immediately as the difference between iterated covariant derivatives performed in either order. Also, in this case, given an arbitrary vector field \vec{X}, by applying the various coordinate covector fields we can pick off the "components" of \vec{X} with respect to the given coordinate basis. If we have a frame field (an orthonormal set of vector fields) expressed in terms of our coordinate basis, we can then convert these to the components wrt the frame field. These are sometimes called the "physical components" since in specific scenarios they correspond, in principle, to measurable quantities.

You might be interested in some very recent posts by myself in other threads (past week) in which I mentioned the Coll-Morales classification of coordinate charts on Lorentzian manifolds, and a very recent post by myself briefly discussing various types of derivatives which appear in differential geometry.

 

[mathwonk]

obsessmath, your post, although correct in detail, isquite consistent with the usual terminology. i.e. coordinates of vectors ARE dual vectors. since they are scalar valued linear functions on vectors.

that is why the coordinates transform contravariantly, and the basis vectors themselves transform covariantly. i.e. a choice of (covariant) basis vectors at each point also determines a dual choice of (contravariant) basic covectors at each point, namely the linear coordinate functions for that basis.

 

[mathwonk]

you have put your finger on the fundamental confusion for many people, i.e. they confuse vectors with coordinates of vectors. the coordinates are functions on vectors, not vectors themselves.

it is hard to appreciate this when we have been told, misleadingly, all our life that a vector is an n - tuple of numbers. you seem to be doing this yourself however.

think about it physically, swing a rock around a string and let go, in one second its two positions determine an arrow of velocity, not a sequence of numbers. the numbers are assigned to the arrow as a vehicle of measurement, hence are functions on the arrow.

i.e. the coordinates are dual vectors, or covectors.

 

[mathwonk]

here is one waY TO SE THAT COORDINATES ArE NOT a natural property of vectors. choose just one vector at a point. what are its coordinates? they are not determiend. you need to choose a whole basis of vectors before you get coordinates for even one vector.

a coordinate system determines a basis fo vectors at each point nd also a bsis of coordinate functions, but a coordinate system is something different from a vector. i.e. coordinates are not just another way to view vectors.let me say you are very intelligent to notice this duality. but it is still confusing.

 

[mathwonk]

look at this example: suppose we are given a single vector at each point, i.e. a vector field. then we do not have a basis, hence no coordinates, and cannot take the coordinate point of view, without artificially introducing coordinates.

we wish to understand thigns as intrinsically as possible, hence must use exclusively the covariant point of view here.

simialrly if we have a single one form given naturally, we have only one covector at each point, hence again no dual basis, hence must intrinsically study it by the contravariant point of view.

coordinates are artficial and obscure the diference betwen covariance and contravariance. the intrinsic properties are the phenomena with real physical meaning, hence should be kept distinct for best understanding.

 

[ObsessiveMathsFreak]

Let me see if I understand. You are saying that a coordinate system, say in 3d, defines not only basis tangent vectors \mathbf{e}_1 ,\mathbf{e}_2,\mathbf{e}_3, but basis cotangent vectors or one-forms \acute{E}_1,\acute{E}_2,\acute{E}_3, and that the representation of a vector in that coordinate system is
\mathbf{v} = \acute{E}_1(\mathbf{v})\mathbf{e}_1+\acute{E}_2(\mathbf{v})\mathbf{e}_2+\acute{E}_3(\mathbf{v})\mathbf{e}_3

Now, the basis one forms transform with the jacobian under a change of coordinates , \mathbf{a_v} = J \mathbf{a_u}, and the basis tangent vectors transform with the inverse Jacobian , e_v = e_u J^{-1}.

So the basis one forms are covariant, and the basis tangent vectors are contravariant? Is that the right way around? And if so, then all one forms in the cotangent space are covariant and all vectors in the tangent space are contravariant? Is it correct to say that?

I understand that a vector is an object quite independant of its coordinates, or basis, or what have you. Obviously it is something that exists independantly of coordinate systems, or even our ability to create coordinate systems.

But then, what is all this talk about contravariant vectors and covariant vectors? Under a change of coordinates, the vector itself does not change at all. Only its representation changes. So the vector itself is neither contravariant nor covariant? Only its representations are? Or, are "normal" vectors always considered to be tangent vectors that transform contravariantly?

Would it be correct to say that what are being called covariant vectors are actually one forms, and what are being called contravariant vectors are actually (tangent) vectors? If so, there would be no such thing as a covariant vector, there would only be covariant forms?

A lot of questions here, but I think the fog is clearing.

 

[mathwonk]

clearing fog is what we live for. more power to you my friend,

 

[gvk]

Let me see if I understand. You are saying that a coordinate system, say in 3d, defines not only

basis tangent vectors \mathbf{e}_1 ,\mathbf{e}_2,\mathbf{e}_3, but basis cotangent vectors or one-forms

\acute{E}_1,\acute{E}_2,\acute{E}_3,... .

Actually, the right way to think is the following:
the vector is an invariant object which does not depend on any coordinate system at all!
This point of view was formulated above
"COORDINATES ArE NOT a natural property of vectors"

Vectors (and tensors) exist without any coordinate system. The coordinate systems are only needed to describe them in a proper manner.

The final physical results do not depend on the choise of coords.
If space has a metric, which, by the way, also does not depend on the coordinate systems, there is no difference between covarient and
contravarient (or tangent and cotangent) vectors-tensors.
It is the same vector(tensor).
In fact, you confirm this using the same notation /omega for covariant and contravariant vectors. Right?

 

[ObsessiveMathsFreak]

OK, I think I've got the representation of a vector part now.

If you consider a vector w as a tangent to some curve s(u,v) in, let's say a 2d plane, parameterised by coordinates u and v, then
\mathbf{w} = \frac{d \mathbf{s}}{dt} = \frac{\partial \mathbf{s}}{\partial u} \frac{du}{dt} + \frac{\partial \mathbf{s}}{\partial v} \frac{dv}{dt}

But \frac{du}{dt} and \frac{dv}{dt} can themselves be represented by (excuse the lapse into cartesians for a moment)

\frac{du}{dt} = \frac{\partial u}{\partial x} \frac{dx}{dt} + \frac{\partial u}{\partial y} \frac{dy}{dt} = \nabla u \cdot (\frac{dx}{dt},\frac{dy}{dt})

\frac{dv}{dt} = \frac{\partial v}{\partial x} \frac{dx}{dt} + \frac{\partial v}{\partial y} \frac{dy}{dt} = \nabla v \cdot (\frac{dx}{dt},\frac{dy}{dt})

But since (\frac{dx}{dt},\frac{dy}{dt})=\frac{d \mathbf{s}}{dt}=\mathbf{w}, what we can actually say is that

\frac{du}{dt}=\nabla u \cdot \mathbf{w}
\frac{dv}{dt}=\nabla v \cdot \mathbf{w}

But, if we want to avoid using the dot product and cartesian gradients, instead of using \nabla u \ \ \nabla v, we can define the one forms d\acute{u}\ \ \ d\acute{v} so that \nabla u \cdot \mathbf{w} = d\acute{u}(\mathbf{w}), etc.

These one forms have representations like d\acute{u} = \frac{\partial u}{\partial x} d\acute{x} + \frac{\partial u}{\partial y} d\acute{y}, in the caresian coordinate system, but of course, like vectors, one-forms have no preferred coordinate system! We can just visualise them as a field of gradient lines! So we just refer to the objects d\acute{u}\ \ \ d\acute{v}, which operate on vectors to give a number and so the representation of the vector becomes.

\mathbf{w}= d\acute{u}(\mathbf{w})\frac{\partial \mathbf{s}}{\partial u} + d\acute{v}(\mathbf{w})\frac{\partial \mathbf{s}}{\partial v}

And letting the partial derivatives of s be our basis vectors we would have.

\mathbf{w}= d\acute{u}(\mathbf{w})\mathbf{e}_u + d\acute{v}(\mathbf{w})\mathbf{e}_v

So, this concept of the coordinates being one forms seems clear to me now.

However I'm still not solid on what all this talk of covariant and contravariant vectors is. Is this a misnomer? Should we really be speaking of contravariant vectors and covariant one-forms?

 

[gvk]

However I'm still not solid on what all this talk of covariant and contravariant vectors is. Is this a misnomer? Should we really be speaking of contravariant vectors and covariant one-forms?

No, it is not misnomer. Covariant and contravariant are different if you don't have a metric in your space. Go through another thread in this forum:

https://www.physicsforums.com/showthread.php?p=413246#post413246

and may be it helps.