COVERS ALL 11
CHAPTERS
, @SOLUTIONSSTUDY
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 N h(x), N
2
h (x) and Nq m (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 In i . ..................................................................................................................... 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, 2 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 : I3→ Q3. .......................................................................................................... 280
10.9 The mapping τ1 : Q→ 2 I2 ......................................................................................................................................................................... 288
10.10 The mapping τ2 : Q→ 2 I2 ......................................................................................................................................................................... 289
10.11 The mapping τ2 : Q→ 2 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 R δ((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
CHAPTERS
, @SOLUTIONSSTUDY
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 N h(x), N
2
h (x) and Nq m (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 In i . ..................................................................................................................... 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, 2 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 : I3→ Q3. .......................................................................................................... 280
10.9 The mapping τ1 : Q→ 2 I2 ......................................................................................................................................................................... 288
10.10 The mapping τ2 : Q→ 2 I2 ......................................................................................................................................................................... 289
10.11 The mapping τ2 : Q→ 2 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 R δ((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
