Exam (elaborations)
Exam (elaborations) TEST BANK FOR Principles of Mathematical Analysis
- Course
- Institution
Exam of 387 pages for the course TEST BANK FOR Principles of Mathematical Analysis at UM (error)
[Show more]
Seller
Follow
COURSEHERO2
Content preview
, A Complete Solution Guide toPrinciples of Mathematical Analysis
by Kit-Wing Yu, PhD
kitwing@hotmail.com
Copyright c 2018 by Kit-Wing Yu. All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photo-
copying, recording, or otherwise, without the prior written permission of the author.
ISBN: 978-988-78797-0-1 (eBook)
ISBN: 978-988-78797-1-8 (Paperback)
,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 h (x) and Nqm (xk ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2
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, π2 ]. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 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 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 : I 2 → 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 : I 3 → Q3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
10.9 The mapping τ1 : Q2 → I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
10.10The mapping τ2 : Q2 → I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
10.11The mapping τ2 : Q2 → I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
10.12The mapping Φ : D → R2 \ {0}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
10.13The spherical coordinates for the point Σ(u, v). . . . . . . . . . . . . . . . . . . . . . . . . 300
10.14The rectangles D and E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
10.15An example of the 2-surface S and its boundary ∂S. . . . . . . . . . . . . . . . . . . . . . 304
10.16The unit disk U as the projection of the unit ball V . . . . . . . . . . . . . . . . . . . . . . 325
10.17The open cells U and V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
10.18The parameter domain D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
10.19The figure of the Möbius band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
10.20The “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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller COURSEHERO2. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $9.42. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
82215 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now