This course is designed as a self-study program in differential calculus. The content is organized into “chapters” below.

chapter # | Topics |
---|---|

Preface | |

0 | The Spreadsheet |

1 | Philosophy, Numbers and Functions |

2 | The Exponential Function and Trigonometric Functions |

3 | Vectors, Dot Products, Matrix Multiplication and Distance |

4 | Area of a Parallelogram, Determinants, Volume and Hypervolume, the Vector Product |

5 | Vectors and Geometry in Two and Three Dimensions |

6 | Differentiable Functions, the Derivative and Differentials |

7 | Computation of Derivatives from their Definition |

8 | Calculation of Derivatives by Rule |

9 | Derivatives of Vector Fields and the Gradient in Polar Coordinates |

10 | Higher Derivatives, Taylor Series, Quadratic Approximations and Accuracy of Approximations |

11 | Quadratic Approximations in Several Dimensions |

12 | Applications of Differentiation: Direct Use of Linear Approximation |

13 | Solving Equations |

14 | Extrema |

15 | Curves |

16 | Some Important Examples and a Formulation in Physics |

17 | The Product Rule and Differentiating Vectors |

18 | Complex Numbers and Functions of Them |

19 | The Anti-derivative or Indefinite Integral |

20 | The Area under a Curve and its Many Generalizations |

21 | The Fundamental Theorem of Calculus in One Dimension |

22 | The Fundamental Theorem of Calculus in Higher Dimensions; Additive Measures, Stokes Theorem and the Divergence Theorem |

23 | Reducing a Line Integral to an Ordinary Integral and Related Reductions |

24 | Reducing a Surface Integral to a Multiple Integral and the Jacobian |

25 | Numerical Integration |

26 | Numerical Solution of Differential Equations |

27 | Doing Integrals |

28 | Introduction to Electric and Magnetic Fields |

29 | Magnetic Fields, Magnetic Induction and Electrodynamics |

30 | Series |

31 | Doing Area, Surface and Volume Integrals |

32 | Some Linear Algebra |

33 | Second Order Differential Equations |