Covers All 11 Chapters
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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),2 N h (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 2 (0, π ]. ..................................................... 199
8.4 The graph of y = | sin x| ..................................................................................... 199
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 (∇f3)(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 :→ I3Q3. ....................................................................................... 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 :→ DR2 \ {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 for each En
....................................................................................................................................................................................... 121
9.1 Expressions of x around four . . . . . . . . . . . . . . . . . . . . . . . . . 234
points. . . . . .
9.2 Expressions of y around four . . . . . . . . . . . . . . . . . . . . . . . . . 235
points. . . . . .
ix
,mailto:
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),2 N h (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 2 (0, π ]. ..................................................... 199
8.4 The graph of y = | sin x| ..................................................................................... 199
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 (∇f3)(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
vii
,List of Figures viii
10.7 The open sets Q0.1, Q0.2 and Q........................................................................... 278
10.8 The mapping T :→ I3Q3. ....................................................................................... 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 :→ DR2 \ {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 for each En
....................................................................................................................................................................................... 121
9.1 Expressions of x around four . . . . . . . . . . . . . . . . . . . . . . . . . 234
points. . . . . .
9.2 Expressions of y around four . . . . . . . . . . . . . . . . . . . . . . . . . 235
points. . . . . .
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