Covers All 11 Chapters
,List of Figures
2.1 The neighborhoods Nh(q) and Nr(p) .....................................................................................................................13
2.2 Convex sets and nonconvex sets ............................................................................................................................. 23
2.3 The sets Nh(x), N h2(x) and Nqm (xk) .................................................................................................................. 25
2.4 The construction of the shrinking sequence .......................................................................................................... 29
3.1 The Cantor set .............................................................................................................................................................49
4.1 The graph of g on [an, bn]........................................................................................................................................ 59
4.2 The sets E and Ini . ....................................................................................................................................................63
4.3 The graphs of [x] and√(x) ........................................................................................................................................ 70
4.4 An example for α = 2 and n = 5.................................................................................................................. 72
4.5 The distance from x ∈ X to E.................................................................................................................................74
4.6 The graph of a convex function f.......................................................................................................................... 76
4.7 The positions of the points p, p + κ, q — κ and q ........................................................................................... 77
5.1 The zig-zag path of the process in (c)...............................................................................................................105
5.2 The zig-zag path induced by the function f in Case (i)......................................................................... 108
5.3 The zig-zag path induced by the function g in Case (i).......................................................................... 109
5.4 The zig-zag path induced by the function f in Case (ii) ...................................................................... 109
5.5 The zig-zag path induced by the function g in Case (ii) ........................................................................ 110
5.6 The geometrical interpretation of Newton’s method ......................................................................................111
8.1 The graph of the continuous function y = f (x) = (π — |x|)2 on [—π, π]. ........................................ 186
8.2 The graphs of the two functions f and g ........................................................................................................ 197
8.3 A geometric proof of 0 < sin x ≤ x on (0, π ]. ............................................................................................ 199
8.4 The graph of y = | sin x|......................................................................................................................................199
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8.5 The winding number of γ around an arbitrary point p ..............................................................................202
8.6 The geometry of the points z, f (z) and g(z) ................................................................................................. 209
9.1 An example of the range K of f ....................................................................................................................... 219
9.2 The set of q ∈ K such that (∇f 3)(f —1(q)) = 0 ............................................................................................ 220
9.3 Geometric meaning of the implicit function theorem......................................................................................232
9.4 The graphs around the four points ......................................................................................................................233
9.5 The graphs around (0, 0) and (1, 0) ................................................................................................................ 236
9.6 The graph of the ellipse X2 + 4Y 2 = 1..................................................................................................... 239
9.7 The definition of the function ϕ(x, t) ...................................................................................................................243
9.8 The four regions divided by the two lines αx1 + βx2 = 0 and αx1 — βx2 = 0 ......................... 252
10.1 The compact convex set H and its boundary ∂H ......................................................................................... 256
10.2 The figures of the sets Ui, Wi and Vi ....................................................................................................................................................................... 264
10.3 The mapping T : I2 → H ..................................................................................................................................... 269
10.4 The mapping T : A → D....................................................................................................................................... 270
10.5 The mapping T : A◦ → D0........................................................................................................................................................................................................ 271
10.6 The mapping T : S → Q ...................................................................................................................................... 277
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,List of Figures viii
10.7 The open sets Q0.1, Q0.2 and Q ........................................................................................................................ 278
10.8 The mapping T : I3 → Q3.................................................................................................................................. 280
10.9 The mapping τ1 : Q2 → I2..................................................................................................................................................................................................... 288
10.10 The mapping τ2 : Q2 → I2..................................................................................................................................................................................................... 289
10.11 The mapping τ2 : Q2 → I2..................................................................................................................................................................................................... 289
10.12 The mapping Φ : D → R2 \ {0} ........................................................................................................................ 296
10.13 The spherical coordinates for the point Σ(u, v) .............................................................................................. 300
10.14 The rectangles D and E ........................................................................................................................................ 302
10.15 An example of the 2-surface S and its boundary ∂S ................................................................................ 304
10.16 The unit disk U as the projection of the unit ball V.....................................................................................325
10.17 The open cells U and V ..........................................................................................................................................326
10.18 The parameter domain D....................................................................................................................................... 332
10.19 The figure of the Möbius band..............................................................................................................................333
10.20 The “geometric” boundary of M..........................................................................................................................335
11.1 The open square Rδ((p, q)) and the neighborhood N√2δ ((p, q)) .............................................................. 350
B.1 The plane angle θ measured in radians.............................................................................................................365
B.2 The solid angle Ω measured in steradians ........................................................................................................366
B.3 A section of the cone with apex angle 2θ........................................................................................................366
, List of Tables
6.1 The number of intervals & end-points and the length of each interval ḟor each En......................................... 121
9.1 Expressions oḟ x around ḟour points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
9.2 Expressions oḟ y around ḟour points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
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,List of Figures
2.1 The neighborhoods Nh(q) and Nr(p) .....................................................................................................................13
2.2 Convex sets and nonconvex sets ............................................................................................................................. 23
2.3 The sets Nh(x), N h2(x) and Nqm (xk) .................................................................................................................. 25
2.4 The construction of the shrinking sequence .......................................................................................................... 29
3.1 The Cantor set .............................................................................................................................................................49
4.1 The graph of g on [an, bn]........................................................................................................................................ 59
4.2 The sets E and Ini . ....................................................................................................................................................63
4.3 The graphs of [x] and√(x) ........................................................................................................................................ 70
4.4 An example for α = 2 and n = 5.................................................................................................................. 72
4.5 The distance from x ∈ X to E.................................................................................................................................74
4.6 The graph of a convex function f.......................................................................................................................... 76
4.7 The positions of the points p, p + κ, q — κ and q ........................................................................................... 77
5.1 The zig-zag path of the process in (c)...............................................................................................................105
5.2 The zig-zag path induced by the function f in Case (i)......................................................................... 108
5.3 The zig-zag path induced by the function g in Case (i).......................................................................... 109
5.4 The zig-zag path induced by the function f in Case (ii) ...................................................................... 109
5.5 The zig-zag path induced by the function g in Case (ii) ........................................................................ 110
5.6 The geometrical interpretation of Newton’s method ......................................................................................111
8.1 The graph of the continuous function y = f (x) = (π — |x|)2 on [—π, π]. ........................................ 186
8.2 The graphs of the two functions f and g ........................................................................................................ 197
8.3 A geometric proof of 0 < sin x ≤ x on (0, π ]. ............................................................................................ 199
8.4 The graph of y = | sin x|......................................................................................................................................199
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8.5 The winding number of γ around an arbitrary point p ..............................................................................202
8.6 The geometry of the points z, f (z) and g(z) ................................................................................................. 209
9.1 An example of the range K of f ....................................................................................................................... 219
9.2 The set of q ∈ K such that (∇f 3)(f —1(q)) = 0 ............................................................................................ 220
9.3 Geometric meaning of the implicit function theorem......................................................................................232
9.4 The graphs around the four points ......................................................................................................................233
9.5 The graphs around (0, 0) and (1, 0) ................................................................................................................ 236
9.6 The graph of the ellipse X2 + 4Y 2 = 1..................................................................................................... 239
9.7 The definition of the function ϕ(x, t) ...................................................................................................................243
9.8 The four regions divided by the two lines αx1 + βx2 = 0 and αx1 — βx2 = 0 ......................... 252
10.1 The compact convex set H and its boundary ∂H ......................................................................................... 256
10.2 The figures of the sets Ui, Wi and Vi ....................................................................................................................................................................... 264
10.3 The mapping T : I2 → H ..................................................................................................................................... 269
10.4 The mapping T : A → D....................................................................................................................................... 270
10.5 The mapping T : A◦ → D0........................................................................................................................................................................................................ 271
10.6 The mapping T : S → Q ...................................................................................................................................... 277
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,List of Figures viii
10.7 The open sets Q0.1, Q0.2 and Q ........................................................................................................................ 278
10.8 The mapping T : I3 → Q3.................................................................................................................................. 280
10.9 The mapping τ1 : Q2 → I2..................................................................................................................................................................................................... 288
10.10 The mapping τ2 : Q2 → I2..................................................................................................................................................................................................... 289
10.11 The mapping τ2 : Q2 → I2..................................................................................................................................................................................................... 289
10.12 The mapping Φ : D → R2 \ {0} ........................................................................................................................ 296
10.13 The spherical coordinates for the point Σ(u, v) .............................................................................................. 300
10.14 The rectangles D and E ........................................................................................................................................ 302
10.15 An example of the 2-surface S and its boundary ∂S ................................................................................ 304
10.16 The unit disk U as the projection of the unit ball V.....................................................................................325
10.17 The open cells U and V ..........................................................................................................................................326
10.18 The parameter domain D....................................................................................................................................... 332
10.19 The figure of the Möbius band..............................................................................................................................333
10.20 The “geometric” boundary of M..........................................................................................................................335
11.1 The open square Rδ((p, q)) and the neighborhood N√2δ ((p, q)) .............................................................. 350
B.1 The plane angle θ measured in radians.............................................................................................................365
B.2 The solid angle Ω measured in steradians ........................................................................................................366
B.3 A section of the cone with apex angle 2θ........................................................................................................366
, List of Tables
6.1 The number of intervals & end-points and the length of each interval ḟor each En......................................... 121
9.1 Expressions oḟ x around ḟour points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
9.2 Expressions oḟ y around ḟour points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
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