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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.