# What is a covector, covariant tensor, etc?

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.

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Dick
Homework Helper
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
Staff Emeritus
Gold Member
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.

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