A differentiable manifold, defined with the help of collections of charts, comes with basic notions of calculus before the introduction of the metric. We start with the definitions of vectors as directional derivatives and 1-forms via the natural dual pairing and build up general tensors from these two. Partial derivatives for functions extend to the Lie derivative, while a special subclass of tensors known as differential forms admits the so-called exterior derivative. We develop calculus based on these most basic structures, ending with the Stokes theorem. This sets the stage for the Riemannian geometry, given in two alternate forms in Chapters 4 and 5.