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Contra-variant and Co-variant vectors

  1. Aug 17, 2013 #1
    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?
  2. jcsd
  3. Aug 17, 2013 #2


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    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 Ax' = ∂x'/∂x Ax = 1

    If the vector A is covariant, then Ax' = ∂x/∂x' Ax = 4
  4. Aug 17, 2013 #3
    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.)
  5. Aug 17, 2013 #4


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  6. Aug 17, 2013 #5


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    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.
  7. Aug 18, 2013 #6
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