Contra-variant and Co-variant vectors

[exmarine]

This should be a simple question for you guys. I am trying to construct trivial examples of contra-variant and co-variant vectors. Suppose I have a 2D Cartesian system with equal units out the x and y axes, and a vector A with components (2,1). Suppose my prime 2D Cartesian system is parallel, but the units along the x’ axis are twice as long as those in the un-primed (and y’) axes. I think my A’ vector has components (1,1). Is that correct? Is that then a contra-variant vector? What is an example of a co-variant vector?
Thanks.

 

[Bill_K]

the units along the x’ axis are twice as long as those in the un-primed (and y’) axes.

If the units are twice as long, then the value of the x' coordinate is half as great: x' = x/2. Then

∂x'/∂x = 1/2 and ∂x/∂x' = 2

If the vector A is contravariant, then A^x' = ∂x'/∂x A^x = 1

If the vector A is covariant, then A_x' = ∂x/∂x' A_x = 4

 

[exmarine]

Ok thanks.

Then follow-on question: What could a covariant vector ever be used for? What could it represent? (I am reading GRT textbooks, so I'll eventually run into it I guess.)

 

[WannabeNewton]

You're using a covector whenever you use $\nabla f$. There is a precise mathematical definition of a covector and of a regular vector (contravariant vector). See for example: https://www.physicsforums.com/showthread.php?t=689904

 

[Nugatory]

bv

Then follow-on question: What could a covariant vector ever be used for? What could it represent? (I am reading GRT textbooks, so I'll eventually run into it I guess.)

Trivial example:

If you transform from a coordinate system in which distance is measured in meters to one in which it is is measured in kilometers ($x'=\frac{x}{1000}$) the $x$ coordinate of a contravariant upper-index vector such as velocity will be smaller by a factor of 1000; If an object's velocity was 1000 m/sec in the old coordinate system it will be 1 km/sec in the new one.

A covariant quantity would be something like altitude change per meter, which transforms the other way. If the altitude changes by 1 cm per meter of horizontal distance you cover, it will change by 1000 cm per kilometer after the transformation. From this, you might correctly conclude that the gradient is a example of a useful covector.

It's worth the exercise of writing down the metric tensor for two-dimensional cartesian space (it's just the 2x2 identity matrix) and then applying the tensor coordinate transformation rule for this trivial coordinate transform, just to see how $g_{ij}$ differs from $g^{ij}$ after the transform.

 

[Daverz]

The wiki page is pretty good:

http://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors