## Differential Forms and ConnectionsThis book introduces the tools of modern differential geometry--exterior calculus, manifolds, vector bundles, connections--and covers both classical surface theory, the modern theory of connections, and curvature. Also included is a chapter on applications to theoretical physics. The author uses the powerful and concise calculus of differential forms throughout. Through the use of numerous concrete examples, the author develops computational skills in the familiar Euclidean context before exposing the reader to the more abstract setting of manifolds. The only prerequisites are multivariate calculus and linear algebra; no knowledge of topology is assumed. Nearly 200 exercises make the book ideal for both classroom use and self-study for advanced undergraduate and beginning graduate students in mathematics, physics, and engineering. |

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Excellent book, clear and precise with a careful build-up of material throughout the book, has questions and workings out for you to do for a solid mathematical notational understanding.

### Contents

II | ix |

IV | 3 |

V | 5 |

VII | 6 |

VIII | 10 |

IX | 11 |

X | 15 |

XI | 18 |

LXXIII | 114 |

LXXIV | 115 |

LXXV | 117 |

LXXVI | 118 |

LXXVII | 120 |

LXXVIII | 123 |

LXXX | 128 |

LXXXII | 130 |

XII | 19 |

XIII | 21 |

XIV | 22 |

XVI | 26 |

XVII | 29 |

XIX | 31 |

XX | 33 |

XXI | 37 |

XXII | 39 |

XXIII | 41 |

XXIV | 45 |

XXV | 47 |

XXVI | 48 |

XXVIII | 51 |

XXX | 53 |

XXXII | 58 |

XXXIII | 59 |

XXXIV | 62 |

XXXVI | 67 |

XXXVII | 69 |

XXXVIII | 70 |

XXXIX | 73 |

XL | 74 |

XLIII | 76 |

XLV | 77 |

XLVII | 79 |

XLIX | 83 |

L | 85 |

LII | 86 |

LIII | 89 |

LV | 92 |

LVII | 93 |

LIX | 95 |

LXI | 96 |

LXIV | 98 |

LXV | 100 |

LXVI | 102 |

LXVII | 103 |

LXIX | 105 |

LXX | 108 |

LXXI | 109 |

LXXII | 112 |

LXXXIII | 132 |

LXXXV | 137 |

LXXXVI | 139 |

LXXXVII | 142 |

LXXXVIII | 145 |

LXXXIX | 146 |

XC | 149 |

XCI | 150 |

XCII | 151 |

XCIII | 154 |

XCIV | 158 |

XCVI | 159 |

XCVII | 161 |

XCVIII | 162 |

C | 165 |

CII | 167 |

CIII | 170 |

CIV | 172 |

CV | 176 |

CVII | 179 |

CVIII | 181 |

CIX | 182 |

CX | 185 |

CXI | 187 |

CXII | 192 |

CXIII | 195 |

CXIV | 200 |

CXV | 204 |

CXVII | 210 |

CXVIII | 214 |

CXIX | 217 |

CXX | 220 |

CXXI | 221 |

CXXIII | 223 |

CXXV | 229 |

CXXVI | 231 |

CXXVII | 236 |

CXXVIII | 240 |

CXXIX | 242 |

CXXX | 245 |

CXXXI | 247 |

### Common terms and phrases

adapted moving orthonormal algebra atlas calculate called canonical volume form Chapter chart connection forms construction coordinate system cotangent bundle covariant exterior derivative defined definition denoted diffeomorphism differentiable structure differential forms differential manifold dimension dx a dy dy a dz equations equivalence class Euclidean space example Exercise expressed exterior differentiation exterior power exterior product fiber follows formula frame field gauge potential geometric given Hint identity immersion integral isomorphic Lie derivative matrix moving orthonormal frame n-dimensional submanifold n-form notation one-to-one open set open subset orthonormal basis orthonormal coframe field orthonormal frame orthonormal frame field parametrized surface Proof Prove pullback quaternion quaternion line bundle Riemannian metric Section smooth function smooth map submanifold submanifold-with-boundary submersion Suppose tangent bundle tangent space tangent vector transition functions trivialization vector bundle morphism vector calculus vector field vector space verify zero