Questions on Covariance and Contravariance of Vectors

[geordief]

from https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors

"Examples of vectors with contravariant components include the position of an object relative to an observer"Can anyone help me with this at all ?(maybe just a guide as to how to follow it up)Also at the top of that Wiki page (first diagram),are those two orthogonal coordinate systems (the Tangent and the Dual ) aligned wrt each other as a direct function of the curve of the surface where they coincide? (there is a surface there isn't there?)

 

[pervect]

from https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors

"Examples of vectors with contravariant components include the position of an object relative to an observer"Can anyone help me with this at all ?(maybe just a guide as to how to follow it up)

That works well enough on the plane, though I wouldn't recommend it as an example to be used in more complicated geometries. In particular, it won't quite work as an example of a vector in General relativity.

I don't understand what you needs clarified, though, I can say I'd recommend a more abstract definition of what a vector is as I wrote in post 3 in the thread https://www.physicsforums.com/threads/vectors-and-dual-vectors.968660/#post-6151336. But I'm not sure if you'd find that helpful, I don't have a good feeling as to your background or why you are interested in the topic.

Also at the top of that Wiki page (first diagram),are those two orthogonal coordinate systems (the Tangent and the Dual ) aligned wrt each other as a direct function of the curve of the surface where they coincide? (there is a surface there isn't there?)

The diagram isn't all that clear, but I believe that the yellow vectors are drawn along the line of constant coordinates (the coordinates are labelled x1 and x2). The blue lines are drawn perpendicular to the yellow lines, I believe.

I would again not do things the same way as wiki did.

 

[stevendaryl]

from https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors

"Examples of vectors with contravariant components include the position of an object relative to an observer"Can anyone help me with this at all ?(maybe just a guide as to how to follow it up)Also at the top of that Wiki page (first diagram),are those two orthogonal coordinate systems (the Tangent and the Dual ) aligned wrt each other as a direct function of the curve of the surface where they coincide? (there is a surface there isn't there?)

I think it's best not to worry at first about things being covariant or contravariant, but instead to get a clear idea of what a tangent vector and a dual vector (one-form, or covector) are.

There are two mathematical objects that are the most fundamental in studying manifold (space or spacetime).

1. A parametrized path. This is a curve through the space, which can be mathematically represented as a continuous function $\mathcal{P}(s)$ from real numbers to points in space. The parameter $s$ is basically any real-valued quantity that increases smoothly along the path. For instance, if your space is 3D space, then a location as a function of time would be an example of a parametrized path. In that case $s$ would be time. But a curve drawn on a flat piece of paper is also a parametrized path, where the parameter is the distance traveled along the curve.
2. A scalar field. This is a function $\phi(\mathcal{P})$ on the space that gives a scalar value (which could be a real number, or a complex number, or maybe something even more exotic, like quaternions, maybe) at each point in the space. So an example is the altitude (height above or below sea level) as a function of position in the 2D space that is the surface of the Earth. The requirement is that the value changes smoothly as you move from point to point.

Note: You don't need coordinates to make sense of these objects. A curve on a piece of paper is the same, regardless of whether you set up a coordinate system on the paper, and regardless of whether you use cartesian or polar coordinates. Altitude makes sense regardless of whether you describe your location using latitude and longitude.

Now, if you have a parametrized path $\mathcal{P}(s)$ and you also have a scalar field $\phi(\mathcal{P})$, then you can combine them to get a function from reals to reals (or reals to complex numbers): $F(s) = \phi(\mathcal{P}(s))$.

So where do vectors come in? Well, simply speaking, a tangent vector is a linear approximation to a parametrized path, and a one-form (or covector or dual vector) is a linear approximation to a scalar field. A tangent vector to a path $\mathcal{P}(s)$ is sometimes suggestively denoted as $\frac{D\mathcal{P}}{Ds}$, and within a coordinate system has the components $(\frac{D\mathcal{P}}{Ds})^j = \frac{dx^j}{ds}$, where $x^j$ is the value of coordinate number $j$ (viewed as a function of $s$ as one travels along the path $\mathcal{P}(s)$).

The one-form $d\phi$, which is a linear approximation to the scalar field $\psi$ has within a coordinate system the components $(d\phi)_j = \frac{\partial \phi}{\partial x^j}$.

The fact that tangent vectors transform differently than one-forms or dual vectors follows from their definitions as linear approximations to paths or scalar functions.

 

[plob]

A tangent vector to a path P(s)P(s)\mathcal{P}(s) is sometimes suggestively denoted as DPDsDPDs\frac{D\mathcal{P}}{Ds}, and within a coordinate system has the components (DPDs)j=dxjds(DPDs)j=dxjds(\frac{D\mathcal{P}}{Ds})^j = \frac{dx^j}{ds}, where xjxjx^j is the value of coordinate number jjj (viewed as a function of sss as one travels along the path P(s)P(s)\mathcal{P}(s)).

Hi
as described this sounds like a unit tangential vector. So I'm wondering if that is true. would it ever not be equal to one?

 

[plob]

Sorry about the formatting problem in my quote

 

[Orodruin]

Hi
as described this sounds like a unit tangential vector. So I'm wondering if that is true. would it ever not be equal to one?

Without a metric there is no such thing as a unit vector and the construction you quoted does not require a metric.

 

[plob]

Without a metric there is no such thing as a unit vector and the construction you quoted does not require a metric.

Hi stevendaryl can you explain to me what he means?

 

[Orodruin]

Hi stevendaryl can you explain to me what he means?

A "unit" vector means a vector of length one. If your vector space does not have an appropriate inner product defined on it (i.e., a metric), then the length of a vector (which is defined as the square root of the inner product with itself) does not have any meaning. The construction that @stevendaryl showed you is a general construction with no requirement that a metric should exist and therefore you cannot generally speak of unit vectors with that construction unless your space has an inner product.

As an example that should show you that it will not necessarily be a unit vector even in a space with an inner product, consider the curve $\vec x(s) = s(\vec e_1 + \vec e_2 + \vec e_3)$ in three-dimensional space. Its tangent vector in those coordinates would be $\dot{\vec x} = \vec e_1 + \vec e_2 +\vec e_3$, which has length $\sqrt{3}$.

 

[stevendaryl]

Hi stevendaryl can you explain to me what he means?

Length is not defined for every manifold. The example that I like to point to is thermodynamics, where the state of a quantity of a gas can be characterized by a point on the space $V,T$ where $V$ is the volume and $T$ is the temperature. If the state is changing with time, then there will be an associated tangent vector $v$ with components $v^V = \frac{dV}{dt}$ and $v^T = \frac{dT}{dt}$. The units of $v^V$ are volume per unit time and the units of $v^T$ are degrees per unit time. How would you combine those components to get a "length" of that tangent vector? What you would expect from experience with Cartesian space is that:

$|v| = \sqrt{(v^V)^2 + (v^T)^2}$

but that expression makes no sense (for unit reasons, if for no other).

 

[plob]

For instance, if your space is 3D space, then a location as a function of time would be an example of a parametrized path. In that case sss would be time.

Hi I guess one of my problems is that it is easy for me to hear a 'for instance' like that one example and then over generalize.

ie, hope that THAT is as tough as things get. But I guess spatial reasoning can't help with certain instances

I guess I've hijacked the OP.

 

[robphy]

As @Orodruin says, length requires the specification of a metric.
And, as @stevendaryl says, not every manifold has a metric.

Here's another point to drive the point home.
With vectors, you always add (according to the parallelogram rule) and scalar multiply.

Consider a vector from $(0,0)$ to $(1,1)$, call it $\overrightarrow{(1,1)}$.
What is its length?

Without a metric, it's not defined.
(The best you can do is compare vectors along that line. One is a scalar multiple of the other.
So, $\overrightarrow{(2,2)}$ is twice as long as $\overrightarrow{(1,1)}$.)

With a metric, the length associated with metric depends on the choice of metric.

• With the Euclidean metric (with coordinate labels $x,y$),
  the length is the square-root of $\Delta x^2+\Delta y^2$... so $\sqrt{2}$.
• With the Minkowski metric for special relativity [this subforum] (with coordinate labels $t,y$),
  the length is the square-root of $\Delta t^2-\Delta y^2$... so $\sqrt{0}$.
• With the Galilean metric for Galilean relativity (with coordinate labels $t,y$),
  the length is the square-root of $\Delta t^2$... so $\sqrt{1}$.

In Minkowski and Galilean spacetime, the "length" is the time read on that observer's wristwatch [for timelike worldlines].