I've been reading on differential topology, and all the examples they give me are very abstract . They speak of arbitrary charts (V,ψ), and (U,φ), which map pieces of the m-manifold onto ℝm, which is fine, I understand the concepts, but how does one describe the functions for real, certainly you don't just say that there is some function named ψ, that maps V to ℝm, you need to include the actual function in some cases. So what exactly are these functions, can someone give me an actual tangible example, preferably on something simple like a 1-manifold? P.S. I am familiar with the stereographic projections for the unit sphere, but I don't know how to reproduce something of similar effect on a different manifold.