## Geometrical Methods of Mathematical PhysicsIn recent years the methods of modern differential geometry have become of considerable importance in theoretical physics and have found application in relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This textbook provides an introduction to these methods - in particular Lie derivatives, Lie groups and differential forms - and covers their extensive applications to theoretical physics. The reader is assumed to have some familiarity with advanced calculus, linear algebra and a little elementary operator theory. The advanced physics undergraduate should therefore find the presentation quite accessible. This account will prove valuable for those with backgrounds in physics and applied mathematics who desire an introduction to the subject. Having studied the book, the reader will be able to comprehend research papers that use this mathematics and follow more advanced pure-mathematical expositions. |

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### Contents

I | 1 |

III | 5 |

IV | 9 |

V | 11 |

VI | 13 |

VII | 16 |

VIII | 20 |

IX | 23 |

LXXII | 130 |

LXXIII | 131 |

LXXIV | 132 |

LXXV | 134 |

LXXVII | 135 |

LXXVIII | 136 |

LXXIX | 137 |

LXXX | 138 |

XI | 26 |

XII | 28 |

XIII | 29 |

XIV | 30 |

XVI | 31 |

XVII | 34 |

XVIII | 35 |

XIX | 37 |

XX | 38 |

XXI | 42 |

XXII | 43 |

XXIV | 47 |

XXV | 49 |

XXVI | 50 |

XXVII | 51 |

XXVIII | 52 |

XXIX | 55 |

XXX | 56 |

XXXI | 57 |

XXXII | 58 |

XXXIII | 59 |

XXXV | 60 |

XXXVI | 63 |

XXXVII | 64 |

XXXIX | 68 |

XL | 70 |

71 | |

XLII | 73 |

XLIII | 74 |

XLV | 76 |

XLVI | 78 |

XLVII | 79 |

XLVIII | 81 |

XLIX | 83 |

L | 85 |

LI | 86 |

LII | 88 |

LIII | 89 |

LV | 92 |

LVI | 95 |

LVII | 101 |

LVIII | 105 |

LIX | 108 |

LX | 112 |

LXI | 113 |

LXIII | 115 |

LXIV | 117 |

LXV | 119 |

LXVI | 120 |

LXVIII | 121 |

LXX | 125 |

LXXI | 128 |

LXXXI | 140 |

LXXXII | 142 |

LXXXIII | 143 |

LXXXIV | 144 |

LXXXV | 147 |

LXXXVI | 150 |

LXXXVII | 152 |

LXXXVIII | 154 |

LXXXIX | 157 |

XC | 158 |

XCI | 160 |

XCII | 161 |

XCIII | 163 |

XCIV | 164 |

XCV | 165 |

XCVI | 167 |

XCVII | 168 |

XCVIII | 169 |

XCIX | 170 |

C | 171 |

CI | 174 |

CII | 175 |

CIII | 179 |

CIV | 180 |

CV | 181 |

CVIII | 182 |

CIX | 183 |

CX | 184 |

CXI | 186 |

CXII | 190 |

CXIII | 192 |

CXIV | 195 |

CXV | 197 |

CXVI | 199 |

CXVII | 201 |

CXX | 203 |

CXXI | 205 |

CXXII | 207 |

CXXIII | 208 |

CXXIV | 210 |

CXXVI | 212 |

CXXVII | 214 |

CXXVIII | 215 |

CXXIX | 216 |

CXXX | 218 |

CXXXI | 219 |

CXXXII | 222 |

CXXXIII | 224 |

244 | |

246 | |

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### Common terms and phrases

applies arbitrary associated bases basis bracket bundle called chapter clearly closed commute completely components connection Consider constant construct continuous contraction convention coordinate system curve defined definition determinant differential dimension discussion distance element equal equation equivalence Euclidean example Exercise exists expression exterior derivative fact fiber figure fixed follows function geometry given gives identity important independent indices integral invariant inverse involve Lie algebra Lie group linearly manifold mathematical matrix means metric natural neighborhood one-form operator original p-form parallel parameter physics possible properties prove region relation representation restriction result rotation rule scalar Show simply solution special relativity sphere spherical structure subgroup submanifold Suppose symmetric tangent tangent vector tensor theorem theory transformation unique usual vanish vector field vector space volume zero