Agoston M.K. Computer Graphics and Geometric Modelling: Mathematics

Подождите немного. Документ загружается.

[Smal61] Smale, S., “Generalized Poincaré’s Conjecture in Dimensions Greater Than Four,” Ann. of

Math., 75(2), 1961, 391–406.

[Smal62] Smale, S., “On the Strutcture of Manifolds,” Amer. J. Math., 84, 1962, 387–399.

[Thom98] Thompson, Abigail, “Algorithmic Recognition of 3-Manifolds,” Bull. AMS, 35(1), 1998, 57–66.

[Wall68] Wallace, Andrew H., Differential Topology: First Steps, W. A. Benjamin, Inc., 1968.

Geodesics

[Lyus64] Lyusternik, L.A., Shortest Paths: Variational Problems, The MacMillan Co., New York, 1964.

Geometric Modeling

[AgoM05] Agoston, Max K., Computer Graphics and Geometric Modeling: Implementation and Algo-

rithms, Springer, 2005.

Linear Algebra

[BanW83] Banchoff, Thomas, and Wermer, John, Linear Algebra Through Geometry, Springer-Verlag,

1983.

[Brad75] Bradley, Gerald L., A Primer of Linear Algebra, Prentice-Hall, 1975.

[Divi75] Divinsky, Nathan, Linear Algebra, Page Ficklin Publ. Co., 1975.

[Fink72] Finkbeiner, Daniel T., II, Elements of Linear Algebra, W.H. Freeman and Co., 1972.

[HofK71] Hoffman, Kenneth, and Kunze, Ray, Linear Algebra, 2

Edition, Prentice-Hall, Inc., 1971.

[IpsM95] Ipsen, Ilse C.F., and Meyer, Carl D., “The Angle Between Complementary Subspaces,” Amer.

Math. Monthly, 102(10), December 1995, 904–911.

[John67] Johnson, Richard E., Linear Algebra, Prindle, Weber & Schmidt, Inc., 1967.

[Lips68] Lipschutz, Seymour, Linear Algebra, McGraw-Hill Book Co., 1968.

[NobD77] Noble, Ben, and Daniel, James W., Applied Linear Algebra, 2

Edition, Prentice-Hall, Inc.,

1977.

[Penr55] Penrose, R., “A Generalized Inverse for Matrices,” Proc. Cambridge Phil. Soc., 51, 1955,

406–413.

[RaoM71] Rao, C.R., and Mitra, S.K., Generalized Inverse of Matrices and Its Applications, John Wiley &

Sons, 1971.

Miscellaneous

[AppH77] Appel, K., and Haken, W., “The Solution of the Four-Color Map Problem,” Sci. Amer., 237,

1977, 108–121.

[BoeP94] Boehm, Wolfgang, and Prautzsch, Hartmut, Geometric Concepts for Geometric Design, A K

Peters, Ltd., 1994.

[Fari87] Farin, Gerald, editor, Geometric Modeling: Algorithms and New Trends, SIAM, 1987.

[Greg86] Gregory, J.A., editor, The Mathematics of Surfaces, Clarendon Press, Oxford, 1986.

[HilC99] Hilbert, D., and Cohn-Vossen, S., Geometry and the Imagination, 2

Edition, AMS Chelsea

Publ., 1999.

[Hoff89] Hoffmann, Christoph M., Geometric and Solid Modeling: An Introduction, Morgan Kaufmann

Publ. Inc., 1989.

[RogA90] Rogers, D.F., and Adams, J.A., Mathematical Elements for Computer Graphics, 2

Edition,

McGraw-Hill, 1990.

Numerical Methods

[AbdY96] Abdel-Malek, Karim, and Yeh, Harn-Jou, “Determining Intersection Curves Between Surfaces

of Two Solids,” CAD, 28(6/7), June/July 1996, 539–549.

[Alex96] Alexander, Dan, “Newton’s Method – Or Is It?,” FOCUS, Math. Association of America,

October 1996, 32–33.

[ConD72] Conte, S.D., and de Boor, C., Elementary Numerical Analysis: An Algorithmic Approach,

McGraw-Hill Book Co., 1972.

[DahB74] Dahlquist, G., and Bjorck, A., Numerical Methods, Prentice-Hall, Inc., 1974.

Bibliography 919

[ForM67] Forsythe, George E., and Moler, Cleve B., Computer Solution of Linear Algebraic Systems, Pren-

tice-Hall, Inc., 1967.

[Hild87] Hildebrand, F.B., Introduction to Numerical Analysis, 2

Edition, Dover Publications, Inc.,

1987.

Offset Curves and Surfaces

[FarN90a] Farouki, R.T., and Neff, C.A., “Analytic Properties of Plane Offset Curves,” CAGD, 7, 1990,

83–99.

Projective Geometry and Transformations

(motions, afﬁne maps, projectivities)

[Ayre67] Ayres, Frank Jr., Projective Geometry, Schaum’s Outline Series, McGraw-Hill Book Co., 1967.

[Blin98] Blinn, James F., “The Cross Ratio,” CG&A, 18(6), November/December 1998, 78–80.

[Fari95] Farin, Gerald, NURB Curves and Surfaces: From Projective Geometry to Practical Use, A.K.

Peters, 1995.

[Gans69] Gans, D., Transformations and Geometries, Appleton-Century-Crofts, 1969.

[PenP86] Penna, M.A., and Patterson, R.R., Projective Geometry and its Applications to Computer Graph-

ics, Prentice-Hall, 1986.

Quadrics

[PetR98] Peters, Jörg, and Reif, Ulrich, “The 42 Equivalence Classes of Quadratic Surfaces in Afﬁne n-

Space,” CAGD, 15(5), May, 1998, 459–473.

Real Analysis

[Berb66] Berberian, Sterling K., Measure and Integration, The Macmillan Co., 1966.

[Nata61] Natanson, I.P., Theory of Functions of a Real Variable, translated from the Russian by Leo F.

Boron, Frederick Ungar Publ. Co., 1961.

[Spie69] Spiegel, Murray R., Theory and Problems of Real Variables, Schaum’s Outline Series, McGraw-

Hill, Co., 1969.

Topology

(See also Algebraic and Differential Topology)

[Barr72] Barr, Stephen, Experiments in Topology, Thomas Y. Crowell Co., Apollo Edition, 1972.

[BurM71] Burgesser, H., and Mani, P., “Shellable Decompositions of Cells and Spheres,” Math. Scand.,

29, 1971, 197–205.

[ChiS66] Chinn, W. G., and Steenrod, N. E., First Concepts of Topology, Random House, 1966.

[Crom97] Cromwell, Peter R., Polyhedra, Cambridge Univ. Press, 1997.

[Eise74] Eisenberg, Murray, Topology, Holt, Rinehart and Winston, 1974.

[Free82] Freedman, M., “The Topology of Four-Dimensional Manifolds,” J. Differential Geom., 17,

1982, 357–453.

[Jäni84] Jänich, Klaus, Topology, Springer-Verlag, 1984.

[KirS69] Kirby, R.C., and Siebenmann, L.C., “On the Triangulation of Manifolds and the Hauptver-

mutung,” Bull. AMS, 75, 1969, 742–749.

[Lips65] Lipschutz, Seymour, General Topology, Schaum’s Outline Series, McGraw-Hill, 1965.

[LunW69] Lundell, Albert T., and Weingram, Stephen, The Topology of CW Complexes, Van Nostrand

Reinhold Co., 1969.

[Mois52] Moise, E.E., “Afﬁne Structures in 3-Manifolds: V,” Ann. Math., 56(2), 1952, 96–114.

[Radó25] Radó, T., “Über des Begriff der Riemannschen Fläche,” Acta. Sci. Math. Szeged. 2, 1925,

101–121.

[Schu68] Schubert, Horst, Topology, Allyn and Bacon, Inc., 1968.

920 Bibliography

Abelian group, 824

ﬁnitely generated, 831

free, 833

torsion-free, 831

Abelianization

of group, 829

Abstract simplicial complex, 334

geometric realization of, 334

induced by labeled complex, 336

simplex of, 334

vertex of, 334

Accumulation point, 210, 296

Adapted frame ﬁeld

for surface, 656

Adjoining cells, 390

Adjoint, 39

Admissible labeling, 342

Afﬁne closure, 20

Afﬁne hull, 20

Afﬁne part

of projective variety, 688

Afﬁne properties, 101

Afﬁne space, 675

Afﬁne transformation, 95, 153

determinant of, 99, 109

equations for, 99, 109

Afﬁnely equivalent ﬁgures, 101

Agreeing vector ﬁelds, 662

Algebra, 859

division, 859

graded, 878

of exterior forms, 266, 886

of real-valued multilinear maps, 879

Algebraic closure

of ﬁeld, 858

Algebraic element

minimum polynomial of, 856

of degree n, 856

over ﬁeld, 856

over subring, 841

Algebraic extension ﬁeld, 856

Algebraic geometry, 468, 674

Algebraic plane curve, 675

projective, 676

Algebraic topology, 327

central problem of, 327, 358

Algebraic variety

afﬁne, 468, 675

in P

(k), 676

projective, 676

Algebraically closed ﬁeld, 858

Algebraically dependent, 857

Algebraically independent, 857

Aliasing problem, 896

Almost all transforms, 800

Alternation map, 881, 885

Analysis situs, 321

Analytic branch, 749

Analytic continuation, 749

direct, 749

Analytic element, 748

center of, 748

disk of, 749

Index

922 Index

point of continuability of, 749

point of noncontinuability of, 749

radius of, 749

singular point of, 749

Analytic function, 893, 899

global, 749

Angle

between oriented hyperplanes, 28

between planes, 18

between u- and v-parameter curves,

667

between vectors, 6

of rotation, 68, 70, 107, 113

signed, 29

Angle-preserving map, 203

Angular defect

at point of surface, 604

Anticonic pair, 641

Anticonics, 641

Antiderivative, 891

Antipodal map, 431, 444, 454, 472, 489, 814

Antipodal point, 140, 141, 316, 423, 513, 549,

632, 814

Approximation

via Chebyshev polynomials, 911

via Legendre polynomials, 910

Arc-length parametrization, 561

induced, 562

Area

of parameterization, 594

of surface, 596

Arg, 898

Argument function

on complex numbers, 898

Ascending chain condition, 839

Associate

in ring, 837

Asymptotic direction, 611

Asymptotic line, 611, 671

equation for, 612

Atan2 function, 815

Attaching a space by a map, 301

Attaching map, 301

for cell, 390

Augmented frame, 197

Automorphism

over ﬁeld, 849

Axiom of parallels, 205

Axis of revolution, 23

Azimuth, 552

Back k-face

of simplex, 449

Barycenter, 36

of dual cell, 440

Barycentric coordinates, 36, 137

preserving, 109

Barycentric subdivision

ﬁrst, 381

nth, 381

Base

of topology, 290

Base curve

of ruled surface, 645

Base plane

for generalized frame, 197

Base point

of fundamental group, 416

of pointed space, 303

Base space

of bundle, 422

of vector bundle, 510

Basis

for abelian group, 833

of empty set, 854

of vector space, 853

orthogonal, 7

orthonormal, 7, 13

Betti, Enrico (1823–1892)

Betti number, 388, 389, 407, 408, 495

of polyhedron, 375

of simplicial complex, 374

Between two points, 4

Bézout, Etienne (1730–1783)

Bézout’s theorem, 683, 700, 710, 713, 763,

786, 803, 804

relation to fundamental theorem of

algebra, 711

Biholomorphic, 906

Bijection, 820

Bilinear form, 48

Bilinear map, 44, 862

associated quadratic map for, 46

degenerate, 46

discriminant of, 45

matrix of, 45

nondegenerate, 46

positive deﬁnite, 46

rank of, 46

signature of, 48

Index 923

Binary relation, 818

Binormal,

of space curve, 574

Birational equivalence

of afﬁne varieties, 771, 775

of curves, 775

of projective varieties, 780

Birational map

between afﬁne varieties, 771

between projective varieties, 780

Birationally equivalent, 771, 780

Blowing up

a singularity, 760

Blowups, 804

Bolzano, Bernard (1781–1848)

Bolzano-Weierstrass theorem, 306

Bonnet, Pierre Ossian (1819–1892), 659, 673

Bordered surface, 353

classiﬁcation of, 354

representation of, 354

Borel, Emile (1871–1956), 212

Boundary

in set, 214

induced orientation of, 528

of C

manifold, 466, 501

of topological manifold, 297

of polyhedron, 387

of set, 211, 292

of simplicial complex, 330

of surface, 353

relative, 214

Boundary map, 364, 393

of singular q-chains, 451, 539

Boundary operator

of singular k-chains, 274, 539

Boundary point

of manifold, 297

of set, 292

Bounded set, 212, 286

Branch, 748, 753

analytic, 749

determined by analytic elements, 749

of curve, 757

Branch line, 750

Branch point, 750

Branch representation, 754

Brianchon, Charles J. (1785–1864)

Brianchon

theorem of, 200, 714

Brouwer, Luitzen Egbertus Jan (1881–1967)

Brouwer ﬁxed point theorem, 445, 531

Buchberger Gröbner basis theorem, 740

Bundle

base space of, 422

ﬁber of, 422

isomorphic, 423

locally trivial, 423

over space, 422

product, 422

projection of, 422

total space of, 422

Bundle automorphism, 423

Bundle isomorphism, 423

Bundle map, 423

C, 851

•

function/map, 224, 502

•

structure, 501

standard, 501

function/map, 223, 231, 502, 890

at point, 231

rank of, 231, 471, 503

manifold, 466, 501

product, 502

parameterization, 460

structure, 501

on product, 502

(A), 223, 231

CAGD (Computer Aided Geometric Design),

459, 579, 586, 587, 638, 643, 676, 744

Canal surface, 638

center curve of, 638

characteristic circle of, 638

radius function of, 638

Canonical line bundle

over P

, 513

over P

, 549

Canonical vector bundle map

on induced vector bundle, 515

Cap product, 449

Cartan structural equations, 656, 658

Cartan, Élie (1869–1951), 649

Casson, Andrew, 340

Category, 452

Catenary, 670

Cauchy, Augustin Louis (1789–1857), 255

Cauchy integral formula, 902

Cauchy sequence, 288

Cauchy-Riemann equations, 899

924 Index

Cauchy-Schwarz inequality, 6, 7, 52, 318,

864, 907

Cell

closed, 390

dual, 440

free, 395

open, 389

Cell decomposition, 389

minimal, 396, 499

Cellular map, 392

Center

of analytic element, 748

of central conic, 169

of parameterization, 755

of place, 757

of quadratic transformation, 759

of rotation, 70, 107

Center curve

for canal surface, 638

Center of curvature

for normal curvatures, 599

for planar curve, 564, 565

Central conic section, 167

center of, 169

Central conic sections

confocal, 168

Central points

of ruled surface, 647

Central projection, 127, 137

generalized, 196

Chain complex, 363

oriented, 363

Chain map, 378

induced, 378

Chain rule, 221

Change of coordinates, 464, 560

Change of coordinates transformation, 464

orientation-preserving, 464

orientation-reversing, 464

Change of parameters transformation, 464

Change of variable theorem, 891

generalized, 261

Characteristic

of ﬁeld, 847

Characteristic circle

of canal surface, 638

Characteristic function

of set, 259, 815

Characteristic map

for cell, 390

Characteristic polynomial

of matrix, 871

of linear transformation, 871

Chasles, Michel (1793–1880), 130

Chebyshev, Pafnuty Lvovich (1821–1894)

Chebyshev polynomials, 911

Chebyshev-Gauss quadrature, 912

Christoffel, Elwin Bruno (1829–1900)

Christoffel symbols, 619

for connection, 661, 663

Circle

directrix of, 167

eccentricity of, 167

focus of, 167

Circular cone, 166

Class C

•

function of, 224, 890

Class C

function of, 223, 224, 231, 502, 890

Classiﬁcation

of bordered surfaces, 354

of closed surfaces, 351

of conics, 175

of noncompact surfaces, 354

of quadratic surfaces, 196

of quadrics, 194

of vector bundles, 550, 551

Clifford, William Kingdon (1845–1879),

130

Closed

in set, 213

Closed map, 293

Closed set, 209, 285, 291

as variety, 724

relative, 213

Closed surfaces

classiﬁcation of, 351

Closure

in set, 214, 292

of set, 210, 292

relative, 214

Closure ﬁnite, 391

Cobordant manifolds, 497

Cobordism, 497

Coboundary, 410

Coboundary map, 410

Cochain group, 410

Cocycle, 410

Codazzi, Delﬁno (1824–1873), 619

Codazzi equations, 658

Index 925

Codimension

of submanifold, 474

of variety, 792

of vector subspace, 854

Coefﬁcient

of monomial, 843

Coefﬁcients

of ﬁrst fundamental form, 591, 667

of second fundamental form, 612,

667

Cofactor

of matrix, 866

Cohomology group, 410

de Rham, 548

singular, 452

Cohomology ring, 411

Colatitude, 552

Collapse of CW complex, 395

elementary, 395

Collapsing subspace to point, 300

Collar, 528

closed, 528

Collinear points, 4, 141

Collinear vectors, 853

Colon ideal, 722

Column rank

of matrix, 867

Combinatorial topology, 323, 374

Combinatorially equivalent

simplicial complexes, 339

Commutative algebra, 674, 715

Commutative diagram, 816

Commutator

of group, 829

Commutator subgroup, 419, 829

Compact space, 304

Compact subset, 212

Compatibility equations, 619

Complex conjugate, 851

Complex line, 60

Complex manifold, 684, 750

Complex number

real part of, 851

imaginary part of, 851

modulus of, 851

Complex number ﬁeld, 851

Complex plane

extended, 898

k-dimensional, 60

Complex variety, 675

Component

of simplicial complex, 355

of space, 309

of subset of R

, 217

of tangent vector, 506, 508

of variety, 701

Component function

of vector ﬁeld, 269

Composite function, 820

Composition

of formal power series, 844

Concave

downward, 242

upward, 242

Concavity

of graph, 242

Condition number

of matrix, 908

Cone, 191

as ruled surface, 645

axis of, 166

circular, 166

double, 166

oblique circular, 166

of lines, 166

on space, 303

right circular, 166

vertex of, 166

Confocal conic sections, 168

Confocal quadrics, 194

Conformal

at point, 900

Conformal map, 203, 900

Congruent

modulo an ideal, 836

modulo an integer, 817

Congruent ﬁgures, 87

Congruent matrices, 45

Congruent transformation, 64, 87

Conic, 180

afﬁne, 170

classiﬁcation of, 175

constructing points on, 200

degenerate, 172, 174

ﬁtting to data, 185, 186, 189, 190

in projective plane, 173

in R

versus in P

, 182

nondegenerate, 129, 134, 174

projective, 173

tangent line of, 184

926 Index

Conic section, 166, 180

central, 167

degenerate, 166

directrix of, 167

eccentricity of, 167

focus of, 167

natural coordinate system for, 168

nondegenerate, 166

vertex of, 168

Connected space, 217, 308

Connected sum, 348

Connection, 661

compatible with metric, 662

Levi-Civita, 663

symmetric, 662

torsion-free, 662

Connection form

for frame ﬁeld, 653

Connectivity number

of polyhedron, 407

of simplicial complex, 407

Continuity

at point, 215, 293

(d,d¢)-, 287

sequential, 296

uniform, 216

Continuous function, 215, 293

Contractible space, 311

Contravariant order

of tensor, 878

Contravariant tensor, 507, 878

Converge

in metric space, 287

in topological space, 296

Convergence

pointwise, 288

uniform, 288

Convex

curve, 570

set, 30

Convex closure, 31

geodesic, 602

Convex combination, 35

Convex hull, 31

Convex linear polyhedron, 31

Convex surface, 602

Convexity

at point on surface, 602

Coordinate neighborhood,

501

Coordinate neighborhood cover

of P

(k), 686

Coordinate ring

of variety, 766

Coordinate system

choosing one for P

(k), 687

curvilinear, 478, 489

for projective line, 147

for projective plane, 150

left-handed, 22, 27

right-handed, 22, 27

skew, 111

view plane, 197

Coordinates

afﬁne, 150

change of, 148, 151, 152, 464

cylindrical, 553

extended afﬁne, 147

extended real, 147

for projective line, 147

for projective plane, 150

spherical, 552

Coset, 827

left, 827

right, 827

Cosine function, 901

Countability axioms, 297

Countable set, 820

Covariant derivative

along curve, 634

compatible, 661

for manifold, 661

of vector ﬁeld, 634, 650, 652

Covariant order

of tensor, 878

Covariant tensor, 507, 878

Cover, 212, 304

closed, 212, 304

ﬁnite, 304

locally ﬁnite, 315

open, 212, 304

Covering space, 424

from group action, 432

n-fold covering, 424

summary, 433

Covering transformation, 424, 430

Cowlick, 488

Critical point, 240, 490, 529

degenerate, 249

nondegenerate, 249, 490

Index 927

Critical value, 240, 490, 529

Cross product, 17

generalized, 51

how to remember it, 17

Crosscap, 350

Cross-ratio, 130

in projective space, 145, 149

Cross-section

nonzero, 510

of vector bundle, 510

support of, 511

zero, 510

Cross-sections

linearly independent, 511

Cube, 256

face of, 256

n-dimensional, 256

singular, 273

unit, 813

Cup product

on cohomology groups, 411

Curl

of vector ﬁeld, 272

Curvature

at vertex of polygonal curve, 572

difference between 2d and 3d case, 573

geodesic, 622

generalized, 578

geometric deﬁnition of, 564, 565, 569, 571,

573

of curve, 566, 573, 664

of regular curve, 567, 577

signed, 566, 569, 571

Curvature tensor, 663

Curvature vector

of curve, 566, 573

Curve, 298, 475

algebraic, 782

algebraic plane, 675

closed, 332, 475

convex, 570

curvature of, 566, 573, 664

curvature vector of, 566

geodesic path, 624

hyperelliptic, 789

interior of, 570

length of, 561

locally ﬂat, 574

normal curvature of, 605

offset, 586, 666

order of, 699

parallel, 586

parametric, 474

planarity condition for, 576, 578

polygonal, 332

projective algebraic plane, 676

rectiﬁable, 559

simple closed, 569

space, 573, 783

straight line condition for, 569

tangent vector of, 475

topological, 298

torsion of, 575

transformed, 761

unit tangent vectors for, 475

Cusp, 707

extraordinary, 587

of parallel curve, 587

ordinary, 587

Cutting and pasting, 333, 337, 390

CW complex, 390

closed n-cell of, 390

dimension of, 391

ﬁnite, 391

ﬁnite dimensional, 391

homology theory for, 393

inﬁnite dimensional, 391

irregular, 391

locally ﬁnite, 392

normal, 392

n-skeleton of, 391

open n-cell of, 390

properties of, 392

regular, 391

subcomplex of, 392

underlying space of, 391

Cyclic group, 828

Cyclide, 640

central, 642

degenerate, 642

Dupin’s deﬁnition of, 641

horned, 642

Maxwell’s deﬁnition of, 640

parabolic, 642

revolute, 642

ring, 640

spindle, 642

spine curve of, 640

string construction for, 641

Cycloid, 670

928 Index

Cylinder, 645

elliptic, 192

generalized, 645

geodesics of, 626

hyperbolic, 192

parabolic, 192

Cylindrical coordinates, 553, 655

Cylindrical frame ﬁeld, 655

•

metric, 283, 289

metric, 283

d-ball, 284

d-bounded, 286

d-closed, 285

d-diameter, 286

d-disk, 284

d-distance, 283

between sets, 286

from point to set, 286

d-neighborhood, 285

d-open, 285

d-sphere, 284

de Rham, Georges (1903–1990)

de Rham cohomology, 548

Deformation retract, 312

Deformation retraction, 311

Deglex order, 732

Degree

and curvature of curve, 571

geometric interpretation of, 443, 702

mod 2, 534

of algebraic element, 856

of hypersurface, 699

of intersection, 801

of map, 313, 443, 445, 532

of map at point, 531

of map over point, 531

of polynomial, 841, 843

of variety, 800

Degree lexicographic order, 732

Degree reverse lexicographic order, 732

Degrevlex order, 732

Dehomogenization

of homogeneous polynomial, 688

Dense set, 292

Derivation, 507

Derivative

of complex function, 899

of formal power series, 846

of function, 218

of map between manifolds, 481, 507,

509

of polynomial, 677, 846

of vector ﬁeld along curve, 650

Derived subgroup, 829

Descartes, René du Perron (1596–1650), 321,

460

Determinant

and volume, 262

of linear transformation, 869

of matrix, 866

Determinant map

of R

, 884

Developable surface, 648

history of, 649

Diagonal map, 819

Diagonalizable

matrix, 870

linear transformation, 870

Diagonalized linear transformation, 870

Diameter, 286

D-ic form, 46

Diffeomorphism, 232, 473, 503

, 232

local, 232

Differentiable function/map, 218, 471,

503

continuously, 224

rank of, 231, 471, 503, 509

between manifolds, 471, 503

Differentiable manifold, 466, 501

Differentiable structure, 501

obvious, 503

Differential

of differential form, 272, 539

of function, 537

of real-valued function, 269

Differential equation

existence and uniqueness of solution to,

895, 896

initial conditions for, 895

linear, 895

nth order, 895

solution of, 895

Differential equations

existence and uniqueness of solutions to,

894

ﬁrst order system of, 894