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), N2h (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 2
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 X 2 + 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 αx 1 — 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
@LECTSOLUTIONSSTUVIA
,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 for each En.................................121
9.1 Expressions of x around four points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
9.2 Expressions of y around four points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
ix
@LECTSOLUTIONSSTUVIA
,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), N2h (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 2
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 X 2 + 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 αx 1 — 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
@LECTSOLUTIONSSTUVIA
,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 for each En.................................121
9.1 Expressions of x around four points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
9.2 Expressions of y around four points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
ix
@LECTSOLUTIONSSTUVIA
