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COMPLETE SOLUTION GUIDE TO
PRINCIPLES OF MATHEMATICS ANALYSIS
BY KIT-WING YU | ALL 1-11 CHAPTERS
COVERDED
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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), N2 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 (0, π ]. .................................................................................................. 199
2
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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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
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, 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
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COMPLETE SOLUTION GUIDE TO
PRINCIPLES OF MATHEMATICS ANALYSIS
BY KIT-WING YU | ALL 1-11 CHAPTERS
COVERDED
---
,--
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), N2 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 (0, π ]. .................................................................................................. 199
2
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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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
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, 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
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