What is a covector, covariant tensor, etc?

[ForMyThunder]

Sorry, but some of you may think this is a stupid question. (I'm only 16 years old.)

I have just now gotten into the field of tensors and topology, after studying vector calculus and differential equations and I have two questions:

a) What exactly is a covector?

b) What is the difference between a contravariant and covariant tensor, and any mixture of the two, in a geometrical sense? (I already know the transformation equations but I find it hard to deduce a relative view of these.)

Any help will be especially appreciated as I cannot find any source that will explain this to me. Thank you.

 

[Dick]

I don't think words will help you much. But a vector, is, uh, a vector. It points in a direction. A covector is a thing like a gradient. They transform oppositely under coordinate transformations, they are duals. If you know the tranformation rules then you know the difference. See, I told you words wouldn't help much. The interesting thing is that the product of a vector and a covector is the rate of change along a direction which is an INVARIANT. Because the opposite transformation rules cancel out. Welcome to the Forums!

 

[Hurkyl]

There is an analogy:
vectors : curves :: covectors : functions

 

[TMM]

Hey, sounds like I'm in the same exact position you are. (16 and just covering tensor notation.) I think of covariance and contravariance in terms of how scale factors work out and all that. When I first read about it I didn't gain a good intuition of them at all until I started doing lots of problems and began to notice the properties the literature talks about emerging in the math I was doing.

 

[ForMyThunder]

Thanks. I have heard of tensors being something like transformations but I got confused by thinking how a transformation could be transformed by those equations. I will keep reading up on this. Thank you.

 

[HallsofIvy]

The crucial point about tensors is that they transform "homogeneously" when you change coordinate systems. That is, the components of the tensor in coordinate system x' are just a sum of products of the components of the tensor in coordinate system x times numbers depending on the two coordinate systems. In particular, if the components of a tensor are 0 in one coordinate sytem, they are 0 in any coordinate system. If we write an equation as A= B, where A and B are tensors, that is the same as A- B= 0. If that is true in one coordinate system, it is true in any. Since physical laws are independent of human-created coordinate system, tensor equations are the natural way to write them.

 

[ForMyThunder]

Thanks! I just figured out that I had this book that explained dyads on the last chapter but I have only read half of it. I will finish it now that I know what it is talking about.