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. |
Contents
I | 1 |
III | 5 |
IV | 9 |
V | 11 |
VI | 13 |
VII | 16 |
VIII | 20 |
IX | 23 |
LXXIII | 130 |
LXXIV | 131 |
LXXV | 132 |
LXXVI | 134 |
LXXVIII | 135 |
LXXIX | 136 |
LXXX | 137 |
LXXXI | 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 |
XLIV | 74 |
XLVI | 76 |
XLVII | 78 |
XLVIII | 79 |
XLIX | 81 |
L | 83 |
LI | 85 |
LII | 86 |
LIII | 88 |
LIV | 89 |
LVI | 92 |
LVII | 95 |
LVIII | 101 |
LIX | 105 |
LX | 108 |
LXI | 112 |
LXII | 113 |
LXIV | 115 |
LXV | 117 |
LXVI | 119 |
LXVII | 120 |
LXIX | 121 |
LXXI | 125 |
LXXII | 128 |
LXXXII | 140 |
LXXXIII | 142 |
LXXXIV | 143 |
LXXXV | 144 |
LXXXVI | 147 |
LXXXVII | 150 |
LXXXVIII | 152 |
LXXXIX | 154 |
XC | 157 |
XCI | 158 |
XCII | 160 |
XCIII | 161 |
XCIV | 163 |
XCVI | 164 |
XCVII | 165 |
XCVIII | 167 |
C | 168 |
CI | 169 |
CII | 170 |
CIV | 171 |
CV | 174 |
CVI | 175 |
CVII | 179 |
CVIII | 180 |
CIX | 181 |
CXII | 182 |
CXIII | 183 |
CXIV | 184 |
CXV | 186 |
CXVII | 190 |
CXVIII | 192 |
CXIX | 195 |
CXX | 197 |
CXXI | 199 |
CXXII | 201 |
CXXV | 203 |
CXXVI | 205 |
CXXVII | 207 |
CXXVIII | 208 |
CXXIX | 210 |
CXXXI | 212 |
CXXXII | 214 |
CXXXIII | 215 |
CXXXIV | 216 |
CXXXV | 218 |
CXXXVI | 219 |
CXXXVII | 222 |
CXXXVIII | 224 |
244 | |
246 | |
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Common terms and phrases
affine connection antisymmetric arbitrary basis vectors calculus called canonical form Cartesian commute components congruence coordinate basis coordinate system defined definition differential equations differential forms differential geometry dimension dual element Euclidean space example Exercise exterior derivative fiber bundle figure follows function ƒ geodesic gives GL(n Hamiltonian identity integral curves invariant inverse isotropy group Killing vector Killing vector fields left-invariant Lie algebra Lie bracket Lie derivative Lie dragging Lie group linear combination linearly independent manifold mathematical matrix metric tensor n-dimensional neighborhood notation one-form one-parameter subgroup open set operator p-form parameter permutation physics properties prove real numbers region rotation scalar Show simply solution spacetime sphere structure submanifold subspace symmetric tangent space tangent vector tensor field theorem theory topology transformation two-form unique vanish vector field vector space volume-form zero