LEADER 04677nam 2200445 450 001 9910554485503321 005 20230808201951.0 010 $a0-691-21989-3 024 7 $a10.1515/9780691219899 035 $a(OCoLC)1257077763 035 $a(OCoLC)1262307709 035 $a(EXLCZ)994100000011984764 100 $a20211023d2021 uy 0 101 0 $aeng 135 $aur||#|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aVisual differential geometry and forms $ea mathematical drama in five acts /$fTristan Needham 210 1$aPrinceton, New Jersey :$cPrinceton University Press,$d[2021] 210 4$dİ2021 215 $a1 online resource (xxviii, 502 pages) $c235 b/w illus 327 $tFrontmatter --$tContents --$tPrologue --$tAcknowledgements --$tACT I The Nature of Space --$t1 Euclidean and Non-Euclidean Geometry --$t2 Gaussian Curvature --$t3 Exercises for Prologue and Act I --$tACT II The Metric --$t4 Mapping Surfaces: The Metric --$t5 The Pseudosphere and the Hyperbolic Plane --$t6 Isometries and Complex Numbers --$t7 Exercises for Act II --$tACT III Curvature --$t8 Curvature of Plane Curves --$t9 Curves in 3-Space --$t10 The Principal Curvatures of a Surface --$t11 Geodesics and Geodesic Curvature --$t12 The Extrinsic Curvature of a Surface --$t13 Gauss?s Theorema Egregium --$t14 The Curvature of a Spike --$t15 The Shape Operator --$t16 Introduction to the Global Gauss?Bonnet Theorem --$t17 First (Heuristic) Proof of the Global Gauss?Bonnet Theorem --$t18 Second (Angular Excess) Proof of the Global Gauss?Bonnet Theorem --$t19 Third (Vector Field) Proof of the Global Gauss?Bonnet Theorem --$t20 Exercises for Act III --$tACT IV Parallel Transport --$t21 An Historical Puzzle --$t22 Extrinsic Constructions --$t23 Intrinsic Constructions --$t24 Holonomy --$t25 An Intuitive Geometric Proof of the Theorema Egregium --$t26 Fourth (Holonomy) Proof of the Global Gauss?Bonnet Theorem --$t27 Geometric Proof of the Metric Curvature Formula --$t28 Curvature as a Force between Neighbouring Geodesics --$t29 Riemann?s Curvature --$t30 Einstein?s Curved Spacetime --$t31 Exercises for Act IV --$tACT V Forms --$t32 1-Forms --$t33 Tensors --$t34 2-Forms --$t35 3-Forms --$t36 Differentiation --$t37 Integration --$t38 Differential Geometry via Forms --$t39 Exercises for Act V --$tFurther Reading --$tBibliography --$tIndex 330 $aAn inviting, intuitive, and visual exploration of differential geometry and formsVisual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton?s geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner.Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss?s famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein?s field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell?s equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan?s method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book.Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught. 606 $aMathematics 606 $aGeometry, differential 606 $aDifferential forms 615 0$aMathematics. 615 0$aGeometry, differential. 615 0$aDifferential forms. 676 $a510 686 $aSK 370$2rvk 700 $aNeedham$b Tristan$0442049 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910554485503321 996 $aVisual differential geometry and forms$92815623 997 $aUNINA