## Vector CalculusVector calculus is the fundamental language of mathematical physics. It pro vides a way to describe physical quantities in three-dimensional space and the way in which these quantities vary. Many topics in the physical sciences can be analysed mathematically using the techniques of vector calculus. These top ics include fluid dynamics, solid mechanics and electromagnetism, all of which involve a description of vector and scalar quantities in three dimensions. This book assumes no previous knowledge of vectors. However, it is assumed that the reader has a knowledge of basic calculus, including differentiation, integration and partial differentiation. Some knowledge of linear algebra is also required, particularly the concepts of matrices and determinants. The book is designed to be self-contained, so that it is suitable for a pro gramme of individual study. Each of the eight chapters introduces a new topic, and to facilitate understanding of the material, frequent reference is made to physical applications. The physical nature of the subject is clarified with over sixty diagrams, which provide an important aid to the comprehension of the new concepts. Following the introduction of each new topic, worked examples are provided. It is essential that these are studied carefully, so that a full un derstanding is developed before moving ahead. Like much of mathematics, each section of the book is built on the foundations laid in the earlier sections and chapters. |

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### Contents

I | 1 |

II | 2 |

III | 3 |

IV | 4 |

V | 7 |

VI | 9 |

VII | 11 |

VIII | 14 |

XL | 76 |

XLI | 78 |

XLII | 83 |

XLIII | 85 |

XLIV | 87 |

XLV | 88 |

XLVI | 91 |

XLVII | 93 |

IX | 16 |

X | 17 |

XI | 21 |

XII | 22 |

XIII | 23 |

XIV | 25 |

XV | 26 |

XVI | 28 |

XVII | 30 |

XVIII | 31 |

XIX | 33 |

XX | 38 |

XXI | 39 |

XXII | 40 |

XXIII | 45 |

XXIV | 47 |

XXV | 48 |

XXVI | 51 |

XXVII | 52 |

XXVIII | 53 |

XXIX | 56 |

XXXI | 58 |

XXXII | 60 |

XXXIII | 61 |

XXXV | 65 |

XXXVI | 68 |

XXXVII | 70 |

XXXVIII | 72 |

XXXIX | 74 |

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### Common terms and phrases

a x b angle anti-symmetric Applying arbitrary b x c Cartesian coordinate system Chapter closed curve components conservative vector field consider cross product curvilinear coordinate defined definition density differential direction div and curl divergence theorem dot product electric field equal evaluated Example exp ax field F Find fluid flux follows formula free suffix function gives grad gradient Hence integral of F irrotational Laplace's equation Laplacian line integral magnitude mass matrix obeys obtained parameter partial derivatives perpendicular physical plane position vector rate of change result right-handed rotation scalar field scalar triple product scale factors second-rank tensor Section Show Similarly simplifies solution Stokes's theorem stress tensor suffix notation surface element surface integral symmetric tensor of rank unit normal unit vectors vector quantity velocity volume element volume integral wave written in terms zero