Introduction to Analysis in Several Variables
                               (Advanced Calculus)
                                   Michael Taylor
                   Math. Dept., UNC
                   E-mail address: met@math.unc.edu
              2010 Mathematics Subject Classification. 26B05, 26B10, 26B12, 26B15, 26B20
             Key words and phrases. real numbers, complex numbers, Euclidean space, metric
              spaces, compact spaces, Cauchy sequences, continuous function, power series,
               derivative, mean value theorem, Riemann integral, fundamental theorem of
               calculus, arclength, exponential function, logarithm, trigonometric functions,
               Euler’s formula, multiple integrals, surfaces, surface area, differential forms,
             Stokes theorem, degree, Riemannian manifold, metric tensor, geodesics, curvature,
                       Gauss-Bonnet theorem, Fourier analysis
                            Contents
                            Preface                                                                    xi
                            Some basic notation                                                        xv
                            Chapter 1.  Background                                                      1
                              1.1.  One variable calculus                                               2
                              Exercises                                                                13
                              1.2.  Euclidean spaces                                                   17
                              Exercises                                                                22
                              1.3.  Vector spaces and linear transformations                           22
                              Exercises                                                                30
                              1.4.  Determinants                                                       31
                              Exercises                                                                36
                            Chapter 2.  Multivariable differential calculus                             41
                              2.1.  The derivative                                                     41
                              Exercises                                                                54
                              2.2.  Inverse function and implicit function theorem                     58
                              Exercises                                                                67
                              2.3.  Systems of differential equations and vector fields                  70
                              Exercises                                                                82
                            Chapter 3.  Multivariable integral calculus and calculus on surfaces       89
                              3.1.  The Riemann integral in n variables                                90
                              Exercises                                                               115
                              3.2.  Surfaces and surface integrals                                    119
                              Exercises                                                               138
                                                                                                       vii