Answer to Question #55. Are there pictorial examples that distinguish covariant and contravariant vectors?

Answer to question # 55 "Are there pictorial examples that distinguish covariant and contravariant vectors ?" D. Neuenschwander, Am. American J. of Physics in print Neuenschwander 1 asked how to visualize the distinction between co-and contravariant vectors. Most of all textbooks introduce this distinction on an abstract level, the only exception I know is Stephani 2 , and below I will show how I present it in my lectures "Introduction to diierential geometry" at Potsdam university. If no metric exists at all, then covariant vectors and contravariant vectors are diierent types of objects. If a metric exists, then there is a canonical isomorphism between them; so we introduce vectors, and after xing a coordinate system, we speak about their covariant and their contravariant components. In the following, we will deal with the second case only, because it is more easy to visualize: The chalkboard has a canonical metric which makes it a at two-dimensional Riemannian manifold. Neuenschwander 1 wrote that the mentioned distinction is necessary when dealing with curved spaces. This is not wrong, but it is a little bit misleading , and I prefer to say: ".. . is necessary when dealing with a non-rectangular coordinate system." Example: We x a point (the "origin" O) in the Eu-clidean plane, then there is a one-to-one correspondence between points and 1