- Chapter 1: The Geometry of Euclidean Space
- Section 1.1: Vectors in Two- and Three-Dimensional Space
- Section 1.2: The Inner Product, Length, and Distance
- Section 1.3: Matrices, Determinants, and the Cross Product
- Section 1.4: Cylindrical and Spherical Coordinates
- Section 1.5:
*n*-Dimensional Euclidean Space

- Chapter 2: Differentiation
- Section 2.1: The Geometry of Real-Valued Functions
- Section 2.2: Limits and Continuity
- Section 2.3: Differentiation
- Section 2.4: Introduction to Paths and Curves
- Section 2.5: Properties of the Derivative
- Section 2.6: Gradients and Directional Derivatives

- Chapter 3: Higher-Order Derivatives: Maxima and Minima
- Section 3.1: Iterated Partial Derivatives
- Section 3.2: Taylor's Theorem
- Section 3.3: Extrema of Real-Valued Functions
- Section 3.4: Constrained Extrema and Lagrange Multipliers
- Section 3.5: The Implicit Function Theorem (Optional)

- Chapter 4: Vector-Valued Functions
- Section 4.1: Acceleration and Newton's Second Law
- Section 4.2: Arc Length
- Section 4.3: Vector Fields
- Section 4.4: Divergence and Curl

- Chapter 5: Double and Triple Integrals
- Section 5.1: Introduction
- Section 5.2: The Double Integral Over a Rectangle
- Section 5.3: The Double Integral Over More General Regions
- Section 5.4: Changing the Order of Integration
- Section 5.5: The Triple Integral

- Chapter 6: The Change of Variables Formula and Applications of Integration
- Section 6.1: The Geometry of Maps from R2 to R2
- Section 6.2: The Change of Variables Theorem
- Section 6.3: Applications
- Section 6.4: Improper Integrals (Optional)

- Chapter 7: Integrals Over Paths and Surfaces
- Section 7.1: The Path Integral
- Section 7.2: Line Integrals
- Section 7.3: Parametrized Surfaces
- Section 7.4: Area of a Surface
- Section 7.5: Integrals of Scalar Functions Over Surfaces
- Section 7.6: Surface Integrals of Vector Fields
- Section 7.7: Applications of Differential Geometry, Physics, and Forms of Life

- Chapter 8: The Integral Theorems of Vector Analysis
- Section 8.1: Green's Theorem
- Section 8.2: Stokes' Theorem
- Section 8.3: Conservative Fields
- Section 8.4: Gauss' Theorem
- Section 8.5: Differential Forms

Section 1.4 | Section 1.5 | Chapter 1 Review | ||
---|---|---|---|---|

Exercise 1 | Exercise 1 | Exercise 1 | Exercise 26 | Exercise 51 |

Exercise 2 | Exercise 2 | Exercise 2 | Exercise 27 | Exercise 52 |

Exercise 3 | Exercise 3 | Exercise 3 | Exercise 28 | Exercise 53 |

Exercise 4 | Exercise 4 | Exercise 4 | Exercise 29 | Exercise 54 |

Exercise 5 | Exercise 5 | Exercise 5 | Exercise 30 | Exercise 55 |

Exercise 6 | Exercise 6 | Exercise 6 | Exercise 31 | Exercise 56 |

Exercise 7 | Exercise 7 | Exercise 7 | Exercise 32 | Exercise 57 |

Exercise 8 | Exercise 8 | Exercise 8 | Exercise 33 | |

Exercise 9 | Exercise 9 | Exercise 9 | Exercise 34 | |

Exercise 10 | Exercise 10 | Exercise 10 | Exercise 35 | |

Exercise 11 | Exercise 11 | Exercise 11 | Exercise 36 | |

Exercise 12 | Exercise 12 | Exercise 12 | Exercise 37 | |

Exercise 13 | Exercise 13 | Exercise 13 | Exercise 38 | |

Exercise 14 | Exercise 14 | Exercise 14 | Exercise 39 | |

Exercise 15 | Exercise 15 | Exercise 15 | Exercise 40 | |

Exercise 16 | Exercise 16 | Exercise 16 | Exercise 41 | |

Exercise 17 | Exercise 17 | Exercise 17 | Exercise 42 | |

Exercise 18 | Exercise 18 | Exercise 18 | Exercise 43 | |

Exercise 19 | Exercise 19 | Exercise 19 | Exercise 44 | |

Exercise 20 | Exercise 20 | Exercise 20 | Exercise 45 | |

Exercise 21 | Exercise 21 | Exercise 21 | Exercise 46 | |

Exercise 22 | Exercise 22 | Exercise 22 | Exercise 47 | |

Exercise 23 | Exercise 23 | Exercise 48 | ||

Exercise 24 | Exercise 24 | Exercise 49 | ||

Exercise 25 | Exercise 50 |