## 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 |

CI | 171 |

CII | 174 |

CIII | 175 |

CIV | 179 |

CV | 180 |

CVI | 181 |

CVIII | 182 |

CIX | 183 |

CX | 184 |

CXI | 186 |

CXII | 190 |

CXIII | 192 |

CXIV | 195 |

CXV | 197 |

CXVI | 199 |

CXVII | 201 |

CXVIII | 203 |

CXIX | 205 |

CXX | 207 |

CXXI | 208 |

CXXII | 210 |

CXXIV | 212 |

CXXV | 214 |

CXXVI | 215 |

CXXVII | 216 |

CXXVIII | 218 |

CXXIX | 219 |

CXXX | 222 |

CXXXI | 224 |

244 | |

246 | |

### Common terms and phrases

affine connection arbitrary axial basis vectors calculus called canonical form chapter commute components congruence Consider const coordinate basis coordinate system d/dX defined definition differential equations differential forms differential geometry dimension dimensional dual eigenvalues element Euclidean space example Exercise exterior derivative fact fiber bundle figure follows function geodesic gives GL(n global gradient Hamiltonian hypersurface identity integral curves invariant inverse isotropy group Killing vector fields Lie algebra Lie bracket Lie derivative Lie dragging Lie group linear combination linearly independent manifold mathematical matrix metric tensor neighborhood notation one-dimensional one-form one-parameter subgroup open set operator orthonormal p-form parallel parallel-transport parameter permutation phase space physics properties prove real numbers region rotation scalar Show simply solution spacetime sphere spherical harmonics structure submanifold subspace symmetric tangent space tangent vector tensor field theorem theory three-dimensional topology transformation two-dimensional two-form unique vanish vector space volume-form zero