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Oprea J. — Differential Geometry and Its Applications
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Название: Differential Geometry and Its ApplicationsАвтор: Oprea J. Аннотация: Differential geometry has a long, wonderful history it has found relevance in areas ranging from machinery design of the classification of four-manifolds to the creation of theories of natures fundamental forces to the study of DNA. This book studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole, it mixes geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. Differential geometry is not just for mathematics majors, it is also for students in engineering and the sciences. Into the mix of these ideas comes the opportunity to visualize concepts through the use of computer algebra systems such as Maple. The book emphasizes that this visualization goes hand-in-hand with the understanding of the mathematics behind the computer construction. Students will not only see geodesics on surfaces, but they will also see the effect that an abstract result such as the Clairaut relation can have on geodesics. Furthermore, the book shows how the equations of motion of particles constrained to surfaces are actually types of geodesics. Students will also see how particles move under constraints. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Издание: 2nd editionГод издания: 2007Количество страниц: 469Добавлена в каталог: 11.12.2011Операции: Положить на полку |
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Holonomy, preserved by isometry 281 Holonomy, via Maple 306 Hopf - Rinow theorem 225 Hopfs conjecture 179 Huygens 51 Hyperbolic helix 31 Hyperbolic plane 228 Hyperbolic plane, geodesic 233 Hyperboloid of 1- Hypersurface 402 418 Imaginary part 182 Immersed 137 Interior angles 288 Inverse function theorem 90 Involute 16 47 Involute of circle 47 Involute of cycloid 51 Involute of helix 17 Involute, curvature of 31 Isometry 235 Isometry, global 236 Isometry, preserves geodesics 238 Isoperimetric inequality 39 Isoperimetric problem 349 Isothermal coordinates 181 185 195 Isothermal parametrization 181 J transformation 230 Jacobi bisection theorem 295 Jacobi elliptic functions 138 Jacobi equation 300 302 303 305 342 Jacobi identity 404 Jacobi's theorem 360 Jacobian matrix 400 401 Killing form 425 kinetic energy 312 323 326 351 352 358 362 412 Kuen's surface 116 Lagrange identity 18 109 111 Lagrange multiplier 347 349 Laplace - Young equation 163 Laplace equation 179 183 Lassoing a cone 237 Least area 203 325 341 370 416 Leibniz rule 81 Lemniscate 46 141 Level set 82 Lie bracket 403 423 Lie bracket, properties of 404 Lie group 424 Liebmann's Theorem 130 Line 1 209 Line of curvature 93 112 114 117 120 214 281 Line of curvature, planar 114 Line of striction 76 Line, characterization of 4 23 Linear transformation 84 Linking number 22 local 43 Loxodrome 261 M 80 Manifold 398 Manifold, Einstein 428 Mann quote 430 map command 44 Maple Elliptic E correction xv 145 393 Mean curvature 107 109 181 408 Mean curvature of Enneper's surface 114 Mean curvature, and soap bubbles 175 Mean curvature, constant 173 175 Mean curvature, for isothermal coordinates 186 Mean curvature, vector field 408 416 Mean curvature, via Maple 150 Mechanical curvature 361 Mercator projection 241 259 Meridian 119 Meridian, as geodesic 213 Meromorphic 188 Metric 123 409 423 Metric coefficients 409 Metric, divergence of 426 Metric, multiple 423 Minimal ruled surface 167 Minimal surface 110 Minimal surface equation 165 172 326 Minimal surface equation, via Maple 199 Minimal surface of revolution 132 Minimal surface, adjoint of 193 Minimal surface, associated family 193 Minimal surface, examples 168 Minimal surface, Gauss curvature of 193 Modulus 137 183 Monge patch 70 78 Monkey saddle 180 Moving frame 278 Multiple metric 423 Myers's theorem 422 Mylar balloon 371 Mylar balloon, characterization of 378 Mylar balloon, parametrization of 374 Mystery curve 14 Natural boundary conditions 321 333 Natural equations 21 Neil's parabola 33 Newton's problem 4 370 Nodary 136 148 Nodoid 148 Norm 43 Normal coordinates 413 Normal curvature 92 120 210 Normal vector 78 Normal vector field 402 Normal, outward-pointing 93 Normal, principal 18 Optimal control problem 366 Orbits 361 Orientable 400 Orientation option 44 Osculating circle 26 Osculating plane 26 36 Pair of pants 152 Parabola evolute 32 Paraboloid 108 393 Paraboloid, elliptic 108 116 151 Paraboloid, hyperbolic 108 116 parallel 119 Parallel postulate 233 234 275 Parallel surface 112 Parallel translation 412 Parallel transport 281 Parallel vector field 280 411 Parallel vector field for Foucault pendulum 285 Parameter curve 67 Parametrization 1 68 Parametrization by arclength 14 16 Parametrization of line 2 Patch 68 Patch, Monge 70 Path connected 69 Pendulum 361 Pendulum, and spring 362 Pendulum, double 363 Periodic solution 360 Plane evolute 31 Plateau's problem 170 Poincar 228 Poincar 232 Poincare upper half space 423 Pole 299 Pontryagin maximum principle 368 370 potential energy 312 323 326 351 353 358 362 412 Potential function 338 Principal curvatures 93 109 Principal normal 18 Principal vectors 93 Procedure 43 Product rule 6 81 Pseudosphere 121-123 Pursuit curve 13 Real part 182 Recreate procedure 54 Recreate3d 55 Rectifying curve 36 Rectifying curve, characterization of 36 Regular 3 Regular mapping 68 Reparametrization 15 Reparametrization by arclength 15 Ricci curvature 420 422 424 Ricci curvature for surfaces 423 Ricci curvature, divergence of 427 Richmond's surface, WE representation 191 Riemann curvature 417 Riemann curvature, generalizes Gauss curvature 418 Riemann curvature, symmetries of 419 Riemannian connection 280 Ros's theorem 176 Roulette 134 135 Ruled minimal surface 167 Ruled surface 74 117 Ruling 75 Saddle surface 75 94 116 Scalar curvature 421 423 Scherk's fifth surface 170 Scherk's surface 165 Scherk's surface, via Maple 198 201 Scherk's surface, WE representation 191 Schwarz inequality 6 341 Schwarzschild solution 429 Second Bianchi identity 419 Second Derivative Test 118 Second fundamental form 407 Second fundamental form, properties of 407 Second structure equation 434 Second variation 343 Secondary variational problem 343 Sectional curvature 418 424 Sectional curvature, as Gauss curvature 418 Shape operator 84 107 108 279 406 Shape operator of cylinder 85 Shape operator of hypersurface 418 Shape operator of saddle 85 Shape operator of sphere 84 407 Shape operator of torus 85 Shape operator, and Gauss map 110 Shape operator, as symmetric transformation 88 Shape operator, formulas 108 Shape operator, shortest distance 4 6 302 312 324 350 370 Shape operator, zero 85 Sheet 75 76 97 116 Sheets 114 305 Shoulder 263 Shoulder, Maple procedure for 270 Shrinkable curve 287 Simplify correction xvi Smooth 68 Soap bubble 175 Soap film 163 Space evolute 36 spacecurve 44 Speed of curve 3 Sphere 84 398 401 407 Sphere, Gauss curvature of from forms 435 Sphere, sectional curvature of 418 Spherical curve 25 Spiral of Cornu 38 348 Spring-pendulum 362 State equations 368 Stereographic plane, geodesic 234 Stereographic projection 195 398 Stereographic sphere 229 Structure equation, first 432 Structure equation, second 434 Sturm - Liouville theorem 304 Subs command 44 surface 68 Surface area 164 414 Surface of Delaunay 133 137 Surface of revolution 72 119 Surface tension 161 Surface, closed 296 Surface, compact 70 Surface, convex 296 Surface, non-orientable 80 Surface, orientable 80 Surface, ruled 74 Surface, saddle 75 Suspension bridge 11 Symmetric matrix 88 Tangent developable 102 117 Tangent indicatrix 64 294 Tangent plane 77-79 Tangent space 398 Tangent vector field 402 Tantrix 64 294 Taut string 363 Tautochrone 9 61 Theorem Egregium 124 Torsion of curve 20 23 29 43 Torsion, geodesic 282 Torsion, total 93 Torus 73 120 Torus, flat 229 Total Gauss curvature 110 277 283 Total torsion 93 Total torsion of curve 22 Tractrix 121 Transition map 398 399 Transversality condition 318 320 Triangle angle sum 276 Trick to remember 17 tubeplot 45 Twist 22 Twisted cylinder 105 Umbilic 127 193 Umbilic point 95 Undetermined time problem 318 Undulary 136 Unduloid 136 142 Unduloid parametrization 144 Unduloid, geodesic 251 Unduloid, mean curvature of 147 Unit normal 79 80 82 Unit speed relation 222 Unrolling 236 237 Upper half-plane 228 Variation 313 414 Vector field 81 82 337 402 Vector field, parallel 280 411 Velocity vector 1 Viviani's curve 26 61 Wave equation 364 Wedge product of forms 432 Weierstrass - Enneper representation 188 189 206 Weierstrass - Enneper representation of the bat 207 Weierstrass - Erdmann condition 344 Weierstrass condition 340 Weierstrass excess function 336 341 357 Weingarten map 84 Witch of Agnesi 11 45
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