George Keeling
Gold Member
 33
 4
1. The problem statement, all variables and given/known data
I am studying co and contra variant vectors and I found the video at youtube.com/watch?v=8vBfTyBPu4 very useful. It discusses the slanted coordinate system above where the X, Y axes are at an angle of α. One can get the components of v either by dropping perpendiculars to the axes (v_{i}) or by dropping a line parallel to the other axis (v^{i}). These give correct results for the norm v^{i}v_{i} and the dot product v^{i} w_{i}. (I have not shown w). So the v^{i} are called contravariant and the v_{i} are called covariant. According to the video Dirac thought this was a great example.
But both v_{i} and v^{i} contravary with a change of scale of the basis vectors. This contradicts some definitions of contravariant and covariant components, e.g. this one on Wikipedia. These definitions say that covariant components covary with a change of scale.
Is there a simple resolution to this apparent contradiction?
2. Relevant equations
Norm and dot product were calculated by expressing v_{i} and v^{i} in Cartesian coordinates. Quite a lot of equations! Along the way we found the metric for the slanted coordinate system.
3. The attempt at a solution
We can demonstrate the problem by drawing basis vectors on the diagram and then another set double their size.
I am studying co and contra variant vectors and I found the video at youtube.com/watch?v=8vBfTyBPu4 very useful. It discusses the slanted coordinate system above where the X, Y axes are at an angle of α. One can get the components of v either by dropping perpendiculars to the axes (v_{i}) or by dropping a line parallel to the other axis (v^{i}). These give correct results for the norm v^{i}v_{i} and the dot product v^{i} w_{i}. (I have not shown w). So the v^{i} are called contravariant and the v_{i} are called covariant. According to the video Dirac thought this was a great example.
But both v_{i} and v^{i} contravary with a change of scale of the basis vectors. This contradicts some definitions of contravariant and covariant components, e.g. this one on Wikipedia. These definitions say that covariant components covary with a change of scale.
Is there a simple resolution to this apparent contradiction?
2. Relevant equations
Norm and dot product were calculated by expressing v_{i} and v^{i} in Cartesian coordinates. Quite a lot of equations! Along the way we found the metric for the slanted coordinate system.
3. The attempt at a solution
We can demonstrate the problem by drawing basis vectors on the diagram and then another set double their size.
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