MA231Vector Analysis
                 Stefan Adams
          2010, revised version from 2007, update 02.12.2010
                       Contents
                       1 Gradients and Directional Derivatives                                            1
                                                                  m      n
                       2 Visualisation of functions f: R → R                                              4
                           2.1   Scalar fields, n = 1 . . . . . . . . . . . . . . . . . . . . . . . .       4
                           2.2   Vector fields, n > 1 . . . . . . . . . . . . . . . . . . . . . . . .       8
                           2.3   Curves and Surfaces      . . . . . . . . . . . . . . . . . . . . . . .  10
                       3 Line integrals                                                                  16
                           3.1   Integrating scalar fields . . . . . . . . . . . . . . . . . . . . . .    16
                           3.2   Integrating vector fields     . . . . . . . . . . . . . . . . . . . . .  17
                       4 Gradient Vector Fields                                                          20
                           4.1   FTCfor gradient vector fields . . . . . . . . . . . . . . . . . .        21
                           4.2   Finding a potential . . . . . . . . . . . . . . . . . . . . . . . .     26
                           4.3   Radial vector fields . . . . . . . . . . . . . . . . . . . . . . . .     30
                       5 Surface Integrals                                                               33
                           5.1   Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    33
                           5.2   Integral   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  36
                           5.3   Kissing problem . . . . . . . . . . . . . . . . . . . . . . . . . .     40
                       6 Divergence of Vector Fields                                                     45
                           6.1   Flux across a surface . . . . . . . . . . . . . . . . . . . . . . .     45
                           6.2   Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    48
                       7 Gauss’s Divergence Theorem                                                      52
                       8 Integration by Parts                                                            59
                       9 Green’s theorem and curls in R2                                                 61
                           9.1   Green’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . .     61
                           9.2   Application . . . . . . . . . . . . . . . . . . . . . . . . . . . .     66
                       10 Stokes’s theorem                                                               72
                           10.1 Stokes’s theorem      . . . . . . . . . . . . . . . . . . . . . . . . .  72
                           10.2 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . .      78
                       11 Complex Derivatives and M¨obius transformations                                80
                       12 Complex Power Series                                                           89
                     13 Holomorphic functions                                                   99
                     14 Complex integration                                                    103
                     15 Cauchy’s theorem                                                       107
                     16 Cauchy’s formulae                                                      117
                         16.1 Cauchy’s formulae    . . . . . . . . . . . . . . . . . . . . . . . . 117
                         16.2 Taylor’s and Liouville’s theorem . . . . . . . . . . . . . . . . . 120
                     17 Real Integrals                                                         122
                     18 Power series for Holomorphic functions                                 126
                         18.1 Power series representation . . . . . . . . . . . . . . . . . . . . 126
                         18.2 Power series representation - further results  . . . . . . . . . . 128
                     19 Laurent series and Cauchy’s Residue Formula                            130
                         19.1 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
                         19.2 Laurent series . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
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                 Preface
                 Health Warning: These notes give the skeleton of the course and are not a
                 substitute for attending lectures. They are meant to make note-taking easier
                 so that you can concentrate on the lectures. An important part in vector
                 analysis are figures and pictures. These will not be contained in these notes.
                 For any figure which appears on the blackboard in my lectures I leave some
                 empty space with a reference number which coincides with the number I am
                 using in the lectures. You can fill the diagrams and figures by your own.
                    These notes grew out of hand written notes from Jochen Voß who gave
                 this course 2005 and 2006. I thank him very much for letting me using his
                 notes.
                 Any remarks and suggestions for improvements would help to create better
                 notes for the next year.
                 Stefan Adams
                 Motivation
                 What is Vector Analysis?
                 In analysis differentiation and integration were mostly considered in one di-
                 mension. Vector analysis generalises this to curves, surfaces and volumes in
                 Rn,n ∈ N. As an example consider the “normal” way to calculate a one
                 dimensional integral: You may find a primitive of a function f and use the
                 fundamental theorem of calculus, i.e. for f = F′ we get
                                     Z bf(x)dx = F(b)−F(a).
                                      a
                 The value of the integral can be determined by looking at the boundary
                 points of the interval [a,b]. Does this also work in higher dimensions? The
                 answer is given by Gauss’s divergence theorem.
                 Notation
                 One of the main problems in vector analysis is that there are many books
                 with all possible different notations. During the whole course I outline alter-
                 native notations in use. It is one of the objectives to acquaint you with the
                 different notations and symbols. Note that most of the material originated
                 from physics and hence many books are using notations and symbols known
                 by people in physics.
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