I am teaching some problem classes for a first course on general relativity and would like to be able to give examples of contravariant and covariant vectors, as an introduction to the theory of tensors.
A standard example for a high school student of a vector field might be the velocity of a particle or of a river or of the wind or something. The problem being that, arising as derivatives, these are all manifestly covariant objects, which we would now like to convey to students as being actually dual to a "true" vector field.
Does anyone have either a good example of a naturally arising, contravariant object or of a good way to illustrate to students the concept of a contravariant object and the distinction with covariant objects?
I don't agree on the covariant character of the velocity field. The velocity field is of the same kind of the displacement field, i.e. it is (naturally) a contravariant vector (the temporal derivative doesn't matter). A natural first example of covariant vector is the "area covector". Usually it is confused (or simply associated) with a vector: in three dimensions it is equal to the cross product of the (vector) sides of the associated parallelogram. The area is a covariant vector: when it acts on a vector (e.g. a velocity) it gives a scalar (e.g. the flux of a physical quantity). In the same way, when the area covector acts on a displacement vector, it gives the volume scalar (the volume of the parallelepiped defined by the surface and the displacement vector)