Smooth manifolds -- The tangent space -- Regular values -- Vector bundles -- Constructions on vector bundles -- Integrability -- Local phenomena that go global.

"Preface In his inaugural lecture in 1854, Riemann introduced the concept of an "n fach ausgedehnten Grösse" - roughly something that has "n degrees of freedom" and which we now would call an n-dimensional manifold. Examples of manifolds are all around us and arise in many applications, but formulating the ideas in a satisfying way proved to be a challenge inspiring the creation of beautiful mathematics. As a matter of fact, much of the mathematical language of the 20th century was created with manifolds in mind. Modern texts often leave the readers with the feeling that they are getting the answer before they know there is a problem. Taking the historical approach to this didactic problem has several disadvantages. The pioneers were brilliant mathematicians, but still they struggled for decades getting the concepts right. We must accept that we are standing on the shoulders of giants. The only remedy I see is to give carefully chosen examples to guide the mind to ponder over the questions that you would actually end up wondering about even after spending a disproportionate amount of time; and in this way hopefully appreciating and internalizing the solutions when they are offered. These examples should be concrete. On the other end of the scale, also proofs should be considered as examples: they are examples of successful reasoning"--