Math 240: Calculus III
                  Edvard Fagerholm
                 edvardf@math.upenn.edu
                   June 20, 2012
                        Contents
                        1 About                                                                            3
                            1.1   Content of these notes . . . . . . . . . . . . . . . . . . . . . .       3
                            1.2   About notation . . . . . . . . . . . . . . . . . . . . . . . . . .       4
                        2 Vector Spaces                                                                    5
                            2.1   Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    5
                            2.2   Definition of a vector space . . . . . . . . . . . . . . . . . . . .      6
                            2.3   Span of vectors . . . . . . . . . . . . . . . . . . . . . . . . . .      7
                            2.4   Linear independence of vectors . . . . . . . . . . . . . . . . . .      10
                            2.5   Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     12
                            2.6   Generalizing our Definition of a Vector (optional) . . . . . . .         15
                        3 Matrices                                                                       18
                            3.1   Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .    18
                            3.2   Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . .      19
                            3.3   The transpose of a matrix . . . . . . . . . . . . . . . . . . . .       24
                            3.4   Some important types of matrices . . . . . . . . . . . . . . . .        25
                            3.5   Systems of linear equations and elementary matrices . . . . . .         27
                            3.6   Gaussian elimination . . . . . . . . . . . . . . . . . . . . . . .      31
                            3.7   Rank of a matrix . . . . . . . . . . . . . . . . . . . . . . . . .      35
                            3.8   Rank and systems of linear equations . . . . . . . . . . . . . .        37
                            3.9   Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . .      40
                            3.10 Properties of the determinant . . . . . . . . . . . . . . . . . .        42
                            3.11 Some other formulas for the determinant . . . . . . . . . . . .          47
                            3.12 Matrix inverse . . . . . . . . . . . . . . . . . . . . . . . . . . .     50
                            3.13 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . .       55
                            3.14 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . .      63
                        4 Higher-Order ODEs                                                              68
                            4.1   Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . .    68
                            4.2   Homogeneous equations . . . . . . . . . . . . . . . . . . . . .         70
                            4.3   Nonhomogeneous equations . . . . . . . . . . . . . . . . . . .          72
                            4.4   Homogeneous linear equations with constant coefficient . . . . 74
                            4.5   Undetermined Coefficients . . . . . . . . . . . . . . . . . . . .         77
                            4.6   Variation of parameters (optional) . . . . . . . . . . . . . . . .      80
                            4.7   Cauchy-Euler equations       . . . . . . . . . . . . . . . . . . . . .  81
                                                                  1
                             4.8   Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . .     83
                                   4.8.1   Free undamped motion . . . . . . . . . . . . . . . . . .        83
                                   4.8.2   Free damped motion . . . . . . . . . . . . . . . . . . .        84
                                   4.8.3   Driven motion . . . . . . . . . . . . . . . . . . . . . . .     86
                         5 Systems of linear ODEs                                                          88
                             5.1   Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . .    88
                             5.2   Homogeneous linear systems . . . . . . . . . . . . . . . . . . .        92
                             5.3   Homogeneous linear systems – complex eigenvalues . . . . . . 95
                             5.4   Solutions by diagonalization . . . . . . . . . . . . . . . . . . .      98
                         6 Series solutions to ODEs                                                        99
                             6.1   Solutions around ordinary points       . . . . . . . . . . . . . . . .  99
                             6.2   Solutions around singular points . . . . . . . . . . . . . . . . . 101
                         7 Vector Calculus                                                               102
                             7.1   Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
                             7.2   Independence of Path . . . . . . . . . . . . . . . . . . . . . . . 105
                             7.3   Multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . 108
                             7.4   Green’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 111
                             7.5   Change of variable formula for multiple integrals . . . . . . . . 116
                             7.6   Surface integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 120
                             7.7   Divergence theorem . . . . . . . . . . . . . . . . . . . . . . . . 125
                             7.8   Stokes theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 127
                                                                   2
                 1   About
                 1.1   Content of these notes
                 These notes will cover all the material that will be covered in class. What is
                 mostly missing is going to be pictures and examples. The course text, Zill,
                 Cullen, Advanced Engineering Mathematics 3rd Ed, will be very useful for
                 more thorough and harder (read: longer) examples than what I’m willing to
                 spend my time on typing up. You’ll also find more problems to complement
                 the homework that I will assign, since the more problems you do the better.
                    All the theory that’s taught in class will be in these notes and some more.
                 During lecture I might sometimes skip parts of some proofs that you’ll find
                 in these notes or instead just present the general idea by doing e.g. a special
                 case of the general case. This will usually happen when:
                   1. The full proof will not add anything useful to your understanding of
                      the topic.
                   2. Thefull proof is not a computational technique that will turn out useful
                      when solving practical problems. This is an applied class after all.
                    Embedded in each section you’ll find some examples. After almost every
                 definition there will be something I would call a trivial example that should
                 help you check that you are understanding what the definition is saying.
                 You should make sure you understand them before moving on, since not
                 understanding them is a sign that you’ve misunderstood something.
                    Each section also ends with a list of all the most basic computational
                 problems related to that topic. These are things you will be expected to
                 perform in your sleep, so make sure you understand them. Almost any
                 problem you will encounter in this class will reduce to solving a sequence of
                 these problems, so they will be the ”bricks” of most applications that you’ll
                 encounter.
                                              3