This is a problem many of the grad students have probably encountered, it's in Chapter 0 of Riemannian Geometry by Do Carmo. Do Carmo proved that the tangent bundle of a differentiable manifold is itself a differentiable manifold by constructing a differentiable structure on TM, where M is a differentiable manifold. So I wanted to take the differentiable structure that Do Carmo gives in the book, assume that two parametrizations overlap and show that the differential at some point in the overlap has positive determinant. My real question then is, what exactly IS the differential of the overlap map? Thanks guys.