What are Contravariant and Covariant Components of a Vector? Part 2

The contravariant and covariant componets of a vector is central concept of differential geometry. Here we discuss why they are useful and necessary to extend the concept of the length of a vector to a large class of coordinate systems. An example coordinate system having oblique axes is analyzed and explicit formulas for the contravariant and covariant components are found. This provides a foundation for understanding tensors and the metric tensor in higher dimensions.

Category:

Education

Tags:

• Differential
• geometry
• Contravariant
• Covariant
• componenets
• vectors
• tensors
• metric
• higher
• dimensions

[AllOtherNamesTaken2]

Why is one set of coordinates contravariant and the other one covariant?

[Mathview]

@AllOtherNamesTaken2 Good question. The terms are somewhat arbitrary. Most current references don't bother explaining the term origin. i.e. Contra - variant to what? Co - variant with what else? The two types of vectors are defined via a coordinate transformation rules, or algebraically via the dual space concept. When changing coordinate systems the components of a covariant vector change by the factor (d x_old / d x_new). One could say it varies "with" x_old, so co-variant. More later.

[StudyAcademic]

I DON'T GET THAT FORMULA FOR VECTOR C in v and u coordinate system. Why did you use tan? how did you find upper componets?

[Mathview]

@StudyAcademic The formula can be found by doing trigonometry on the figure that shows the two coordinate systems. It's a little tricky, but you can get the tangent function and the rest from simple trig. on the drawing.

[Mathview]

BTW The notation that is more standard usage has the matrix of g with two lower indices as the metric tensor, and g with two upper indices as the inverse matrix. The calculations in the video are self consistent, though. I guess I should re-do this video sometime and fix-up this detail. Also, a good exercise is to derive this from scratch yourself on a blank sheet of paper. The trig calculation is all you need to do to get these formulas.

[scarface14441]

i know this is dumb but witch line is b im not sure

[Mathview]

@scarface14441 Ah..good Q. We have two coordinate systems in the plane. The oblique coordinates with non-orthogonal axes, and the regular old Cartesian coordinates, with orthogonal axes. In the Cartesian system the coordinates of the vector C are (a,b). So Cx = a, Cy = b. If we instead use the oblique coordinates, we can construct the covariant and contravariant components of C, as described in the video, which permits us to write the length of C, sqrt(a*a+b*b) as Csub1*Csup1 + Csub2*Csup2.

[MrSlayer4ever]

Very useful videos.I was trying to understand how tensors work but it is difficult for me to study from books because there are some things that I don't understand like covariant and contravariant components of a vector.Finally I know what they are.I'm moving on to your videos about tensor algebra hoping that they will be as helpful as these were.Thanks.

[yn30s]

Wow, that makes a lot of sense. I've been studying Reimannian metrics in detail so I actually know exactly what you're talking about. Of course Lee could have just indicated this relationship directly but I think he likes to torture you (which I must admit often does lead to a deeper appreciation even if it takes longer). Thanks so much for the communication, you have helped me to understand this a lot.

[Mathview]

@yn30s Great comments! I agree, simple is better. So ... In any Euclidean Space, like the tangent space TMp at point p, we can define a new inner product ( x | y ) as (x | G | y ) where G is a symmetric positive definite matrix ( a well known Linear Algebra result.) The length of a vector X with this inner product is then (X|G|X) and we can consider the vectors (X| and G|X) as separate vectors, the co and contra vectors, and the matrix representation of G as the metric tensor.