# What is a covector, covariant tensor, etc?

• ForMyThunder

#### 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.

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!

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

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.

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.

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.

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.