Chains, forms, and duality: A novice’s guide from vector calculus to manifolds John Kerl February 3, 2008 1 Contents Contents 2 Preface 11 Goalsforthecourse .................................. .......... 11 Goalsforthepaper................................... .......... 12 Structureofthepaper ................................ ........... 12 When I say map,Imean........................................... 13 Specificityofdimension . ............ 13 Theexemplarmodel ................................... ......... 13 Themes ............................................. ...... 15 Acknowledgements ................................... .......... 16 1 Preliminaries from analytic geometry 17 1.1 Notation........................................ ........ 17 1.2 Trigonometric functions . ............. 17 1.3 The dot product and its applications . ............... 19 1.3.1 The dot product and projected length . ............ 19 1.3.2 Projection and perpendicular operators . .............. 20 1.3.3 Point-normalform .............................. ........ 21 1.4 Coordinates..................................... ......... 22 1.4.1 Coordinates, area, and the determinant . .............. 22 1.4.2 Changeofcoordinates . ......... 24 1.4.3 Somecoordinatesystems . ......... 24 1.4.4 Stereographic projection on S1 ............................... 24 1.4.5 Stereographic projection on Sn ............................... 26 1.5 Typesoffunctions ................................ .......... 29 1.5.1 Paths: R to Rm ........................................ 29 1.5.2 Maps: Rm to Rn ....................................... 30 1.5.3 Scalar functions: Rn to R .................................. 31 1.5.4 Levelsets ..................................... ...... 31 2 1.5.5 Images of paths under maps: pushforwards . ............. 32 1.5.6 Images of maps under functions: pullbacks . .............. 32 1.6 A gallery of curves and surfaces, part 1 . ................ 32 2 Preliminaries from vector calculus 34 2.1 Goalsforvectorcalculus. ............. 34 2.2 Differentiation .................................. .......... 35 2.2.1 Derivatives of functions from R to R ............................ 35 2.2.2 Derivativesofpaths ....... ....... ....... ....... ......... 35 2.2.3 Derivatives of functions from Rn to R ........................... 37 2.2.4 Derivatives of maps from Rm to Rn ............................ 39 2.2.5 Derivatives for images of paths under maps . .............. 41 2.2.6 Derivatives for images of maps under functions . ................ 42 2.2.7 The chain rule and the Jacobian . .......... 43 2.2.8 Divergenceandcurl ............................. ........ 43 2.3 Afirstpassattangentspaces . ............ 45 2.4 Integration ..................................... ......... 46 2.4.1 Integralsofpaths.............................. ......... 46 2.4.2 Integrals of scalar functions . ............. 46 2.4.3 Integralsofmaps............................... ........ 47 2.5 Changeofcoordinates ............................. ........... 48 2.5.1 Change of coordinates for derivatives . ............... 48 2.5.2 Change of coordinates for single-variable integrals .................... 48 2.5.3 Change of coordinates for integrals of scalar functions .................. 50 2.6 Integrationtheorems . ............ 52 2.6.1 Pieces for the integration theorems . ............. 52 2.6.2 The fundamental theorem of calculus . ............. 53 2.6.3 The second fundamental theorem of calculus . .............. 54 2.6.4 Green’stheorem ................................ ....... 54 2.6.5 ClassicalStokes............................... ......... 55 2.6.6 Divergencetheorem ............................. ........ 55 3 2.6.7 Fundamental theorem of calculus for line integrals . .................. 55 2.6.8 Cauchy’stheorem ............................... ....... 56 2.7 Lagrangemultipliers . ............ 57 2.8 ODEsinonevariable................................ ......... 59 2.9 ODEsintwovariables ............................... ......... 60 2.10PDEs ............................................. 62 2.11 Limitations of vector calculus . ................ 63 2.11.1 xxx0 ......................................... 63 2.11.2 xxx1 ......................................... 63 2.11.3 xxx2 ......................................... 63 3 Preliminaries from topology and analysis 64 3.1 Connectedness?.................................. .......... 64 3.2 Compactness?.................................... ......... 64 3.3 Homeomorphisms and diffeomorphisms . .............. 64 3.4 Implicitfunctiontheorem . ............. 64 3.5 Inversefunctiontheorem. ............. 66 4 Preliminaries from algebra 67 4.1 Algebraicaxioms ................................. .......... 67 4.1.1 Groups and semigroups . ........ 67 4.1.2 Normalizer and normal closure of a subgroup . .............. 67 4.1.3 Groupactions.................................. ....... 68 4.1.4 Groupactionexample . ........ 68 4.1.5 Rings ......................................... 69 4.1.6 Fields ........................................ ..... 69 4.1.7 Modulesandvectorspaces . ......... 69 4.1.8 Algebras...................................... ...... 70 4.1.9 Gradedalgebras ................................ ....... 71 4.2 Categories...................................... ......... 72 4.2.1 Definitions .................................... ...... 72 4.2.2 Examplesofcategories. ......... 73 4 4.2.3 Functors...................................... ...... 73 4.2.4 Homfunctors................................... ...... 75 4.3 Freeness........................................ ........ 77 4.3.1 Definitions .................................... ...... 77 4.3.2 Freegroups .................................... ...... 78 4.3.3 Freeabeliangroups. ......... 78 4.3.4 Freeproductofgroups. ......... 79 4.3.5 Freemodules ................................... ...... 79 4.4 Quotients, projections, and sections . .................. 81 4.5 Sequences, complexes, and homological algebra . ................... 84 4.5.1 Sequences..................................... ...... 84 4.5.2 Trapezoiddiagrams ............................. ........ 84 4.5.3 Non-complex, non-exact sequences . ............. 85 4.5.4 Exactsequences ................................. ...... 86 4.5.5 Complexes ..................................... ..... 86 4.5.6 [xxxmergew/theabove]Exactsequences. ............ 87 4.5.7 Techniquesforexactsequences . ............ 88 4.5.8 [xxxmergew/above]Complexes . .......... 88 4.5.9 Zigzaglemma................................... ...... 89 4.6 Linearalgebra................................... .......... 90 4.6.1 Notation...................................... ...... 90 4.6.2 Tobefiled ...................................... 90 4.6.3 Basisanddualspace............................. ........ 90 4.6.4 Geometric interpretation of dual . ............. 91 4.6.5 Dualanddoubledual ............................. ....... 92 4.6.6 Explicit computation of dual basis . ............ 92 4.6.7 Change-of-basis matrices; covariance and contravariance................. 93 4.6.8 Pullbacks ...................................... ..... 94 4.6.9 Pairings ...................................... ...... 95 4.6.10 Symmetric and skew-symmetric matrices . .............. 96 4.7 Tensors ......................................... ....... 98 5 4.7.1 Tensor products of vector spaces . ............ 98 4.7.2 Explicit representation of the tensor product; tensor product as vector space . 99 4.7.3 Decomposability of tensors . ........... 101 4.7.4 Tensor products of dual spaces; tensors as k-linear functions . 102 4.7.5 Explicit representation of multilinear functions . ...................103 4.7.6 Mixedtensors.................................. ....... 104 4.7.7 Examplesoftensors .............................. ....... 105 4.7.8 Pullbacks ...................................... 107 4.7.9 Tensoralgebras................................ ........ 108 4.7.10 Permutations acting on multilinear functions . ..................109 4.7.11 Symmetric and alternating tensor spaces . ................110 4.7.12 Symmetric and alternating tensor algebras . .................112 4.7.13 Some explicit computations with tensors . ................115 4.7.14 Change of basis; more about covariance and contravariance ............... 116 4.7.15 Contractions ................................. ........ 119 4.7.16 Tensor products over free modules . ............. 120 5 Topological manifolds 121 5.1 CWcomplexes ..................................... ....... 121 5.2 Classificationofsurfaces . .............. 121 5.3 Homotopy and the fundamental group . ............. 121 5.4 Homologies ...................................... ........ 121 5.5 Singularhomology ................................ .......... 122 5.6 Cellularhomology ................................ .......... 122 5.7 Homotopy equivalence and homotopy invariance . .................122 5.8 Deformationretracts . ............ 122 5.9 Seifert-vanKampen ............................... .......... 122 5.10 Mayer-Vietoris for homology . .............. 123 5.11Coveringspaces................................. ........... 123 5.12TopoTBD........................................ ....... 124 5.13Cubesandchains................................. .......... 125 6 5.14 The boundary operator ∂ ......................................125 5.15Homology ....................................... ........ 125 6 Smooth manifolds 126 6.1 Manifolds ....................................... ........ 126 6.1.1 Manifolds ..................................... 126 6.1.2 Coordinates and parameterizations . .............