Mathematical Tripos: IA Algebra & Geometry (Part I) Contents 0 Introduction i 0.1 Schedule . i 0.2 Lectures . ii 0.3 Printed Notes . ii 0.4 Example Sheets . iii 0.5 Acknowledgements. iii 0.6 Revision. iv 1 Complex Numbers 1 1.0 Why Study This? . 1 1.1 Introduction . 1 1.1.1 Real numbers . 1 1.1.2 The general solution of a quadratic equation . 1 1.1.3 Complex numbers . 1 1.2 Algebraic Manipulation of Complex Numbers . 2 1.3 Functions of Complex Numbers . 2 1.3.1 Complex conjugate . 2 1.3.2 Modulus . 3 1.4 The Argand Diagram . 3 1.5 Polar (Modulus/Argument) Representation . 5 1.5.1 Geometric interpretation of multiplication . 5 1.6 The Exponential Function . 6 1.6.1 The real exponential function . 6 1.6.2 The complex exponential function . 7 1.6.3 The complex trigonometric functions . 7 1.6.4 Relation to modulus/argument form . 8 This is a supervisor’s copy of the notes. It is not to be distributed to students. 1.6.5 Modulus/argument expression for 1 . 8 1.7 RootsofUnity .............................................. 8 1.8 De Moivre’s Theorem . 9 1.9 Logarithms and Complex Powers . 10 1.9.1 Complex powers . 10 1.10 Lines and Circles in the Complex Plane . 11 1.10.1 Lines . 11 1.10.2 Circles . 11 1.11 M¨obius Transformations . 11 1.11.1 Composition . 12 Mathematical Tripos: IA Algebra & Geometry (Part I) a c [email protected], Michaelmas 2006 1.11.2 Inverse . 12 1.11.3 Basic Maps . 13 1.11.4 The general M¨obiusmap . 14 2 Vector Algebra 15 2.0 Why Study This? . 15 2.1 Vectors . 15 2.1.1 Geometric representation . 15 2.2 Properties of Vectors . 16 2.2.1 Addition . 16 2.2.2 Multiplication by a scalar . 16 2.2.3 Example: the midpoints of the sides of any quadrilateral form a parallelogram . 17 2.3 Scalar Product . 18 2.3.1 Properties of the scalar product . 18 2.3.2 Projections . 18 2.3.3 Another property of the scalar product . 19 2.3.4 Example: the cosine rule . 19 2.4 Vector Product . 19 2.4.1 Properties of the vector product . 20 2.4.2 Vector area of a triangle/parallelogram . 21 2.5 Triple Products . 21 2.5.1 Properties of the scalar triple product . 21 2.6 Bases and Components . 22 2.6.1 Two dimensional space . 22 2.6.2 Three dimensional space . 23 2.6.3 Higher dimensional spaces . 24 2.7 Components . 24 2.7.1 The Cartesian or standard basis in 3D . 24 2.7.2 Direction cosines . 25 This is a supervisor’s copy of the notes. It is not to be distributed to students. 2.8 Vector Component Identities . 25 2.9 Polar Co-ordinates . 26 2.9.1 2D plane polar co-ordinates . 26 2.9.2 Cylindrical polar co-ordinates . 27 2.9.3 Spherical polar co-ordinates (cf. height, latitude and longitude) . 28 2.10 Suffix Notation . 30 2.10.1 Dyadic and suffix equivalents . 30 2.10.2 Summation convention . 31 2.10.3 Kronecker delta . 32 2.10.4 More on basis vectors . 32 2.10.5 The alternating tensor or Levi-Civita symbol . 33 Mathematical Tripos: IA Algebra & Geometry (Part I) b c [email protected], Michaelmas 2006 2.10.6 The vector product in suffix notation . 33 2.10.7 An identity . 34 2.10.8 Scalar triple product . 34 2.10.9 Vector triple product . 34 2.11 Vector Equations . 35 2.12 Lines and Planes . 35 2.12.1 Lines . 35 2.12.2 Planes . 36 2.13 Cones and Conic Sections . 37 2.14 Maps: Isometries and Inversions . 39 2.14.1 Isometries . ..