1 Complex Numbers
 1.1 Basic Properties
  1.1.1 Introduction
  1.1.2 Basic Definitions
 1.2 Polar Form of Complex Numbers
  1.2.1 Vector Representation
  1.2.2 Argument and Modulus
  1.2.3 Manipulations in Vector/Polar Form
  1.2.4 Complex exponential
  1.2.5 De Moivre’s Theorem
 1.3 The Exponential Function
  1.3.1 A Power Series for {e}^{x}
  1.3.2 Power Series for \mathop{sin}\nolimits (x) and \mathop{cos}\nolimits (x)
 1.4 Euler’s Formula
  1.4.1 The Unit Circle
  1.4.2 Roots, n–th roots of unity
 1.5 Trigonometric and Hyperbolic Functions
  1.5.1 Definitions
  1.5.2 Inverse hyperbolic functions
  1.5.3 Derivatives
  1.5.4 Hyperbolic Identities
2 Second Order Linear Differential Equations
 2.1 Ordinary 2nd Order Linear Differential Equations
  2.1.1 Origin of ODEs
  2.1.2 Definitions
  2.1.3 How to Solve Them
 2.2 ODE with constant coefficients I
  2.2.1 Undamped oscillator
  2.2.2 Initial Value Problem
  2.2.3 Exponential
  2.2.4 Summary
 2.3 2nd order ODE with constant coefficients II
  2.3.1 Positive discriminant
  2.3.2 negative discriminant
  2.3.3 The Marginal Case
  2.3.4 Summary
 2.4 Inhomogeneous Equations
  2.4.1 Solution to inhomogeneous equations
 2.5 * Green’s function approach
  2.5.1 * Initial Conditions for the Homogeneous Case
  2.5.2 * The Inhomogeneous Case: Effect of the External Force
3 Functions of more than one variable
 3.1 Functions of several variables
  3.1.1 Symmetries
 3.2 Partial Derivatives
  3.2.1 Reminder: Derivative of a function of one variable
  3.2.2 Derivatives for functions of two variables
  3.2.3 Higher Derivatives, Notation
  3.2.4 Minima, Maxima and Saddle points
 3.3 Curves on Surfaces
  3.3.1 Parametric curves in the x–y–plane
  3.3.2 Parametric urves on Surfaces
  3.3.3 Change of height along a Curve
 3.4 The Gradient
  3.4.1 Definition of the Gradient
  3.4.2 Examples
  3.4.3 Gradient and Differential; Geometrical Meaning
4 Series and Limits
 4.1 Finite and Infinite Series
  4.1.1 Finite Series of Natural Numbers (Gauß)
  4.1.2 Finite Geometric Progression
  4.1.3 Binomial
  4.1.4 Infinite Series
 4.2 Taylor–Series
  4.2.1 The Exponential Function
  4.2.2 Power Series for \mathop{sin}\nolimits (x) and \mathop{cos}\nolimits (x)
  4.2.3 General Case
  4.2.4 Example: The Exponential Function \mathop{exp}\nolimits (x)
 4.3 Taylor–Expansion of Functions
  4.3.1 Convergence: Expansion of f(x) =\mathop{ ln}\nolimits (1 + x)
  4.3.2 Alternative way to generate a Taylor Series
  4.3.3 Taylor expansion of f(x) around an arbitrary x = a
 4.4 Further Examples for Series and Limits
  4.4.1 Newtonian Limit of Relativistic Energy
  4.4.2 Limits
5 Two–by–Two Matrices
 5.1 Two-by-Two Matrices: Introduction
  5.1.1 Linear Equations of Two Unknowns
  5.1.2 Two–by–Two Matrices: Definition
  5.1.3 Linear Mappings and Matrix Operatings
 5.2 Two–by–Two Matrices: Linear Mappings
  5.2.1 Specific Linear Mappings 1: the Unit Matrix
  5.2.2 Specific Linear Mappings 2: Stretching and Shrinking
  5.2.3 Specific Linear Mappings 3: Projections
  5.2.4 Specific Linear Mappings 4: Rotations
  5.2.5 Specific Linear Mappings 5: Reflections
 5.3 Two–by–Two Matrices: Index Notation and Multiplication
  5.3.1 Basis Vectors and Index Notation
  5.3.2 Multiplication of a Matrix with a Scalar
  5.3.3 Matrix Multiplication: Definition
  5.3.4 Matrix Multiplication: Index Notation
 5.4 Inverse of a Matrix
  5.4.1 Motivation
  5.4.2 Definition and Theorem
 5.5 Eigenvalues and eigenvectors
  5.5.1 A physics example