diff --git a/changes-draft.html b/changes-draft.html index 6a510c3..c69b50a 100644 --- a/changes-draft.html +++ b/changes-draft.html @@ -1,289 +1,4 @@ This file is a draft of the new changes for http://www.jirka.org/ra/changes.html

-??? ??th 2018 edition, Version 5.0 (edition 5, 0th update): -

-The motivation for this revision is to improve readability of existing material -rather than adding much new material. -To this end, -39 new figures were added (so 65 total) -there are several new examples, -as well as reorganizing and expanding explanations throughout. -Furthermore, -99 new exercises were added bringing the total to 528 total -(plus two had to be replaced). -

-A List of Notations is added at the back, giving a description and -a page number for the most relevant definition or use for the notations used -in the book. -

-There are the following more major additions: -A short new subsection in 0.3 on relations. -Two new subsections in 6.2 on limits of derivatives, and on power series. -I always felt like chapter 6 ends too abruptly if 6.3 is not covered. This -adds a nice simpler application of swapping of limits with much easier proof -than Picard. -A short new subsection on limits of functions in 7.5, as this is -really used in chapter 8 of volume II. -Section 4.3 was expanded with a discussion on Taylor series, as -well as the second derivative test. -Throughout the book, some material that was in remarks, examples, and exercises -but was used often, was formalized into several new propositions. -

-Some exercises, examples, propositions were added, some theorems became -propositions, a few definitions, theorems, propositions, lemmas, corollaries, -and examples in 0.3, 1.4, 2.5, 3.4, 4.1, 4.2, 5.2, 7.1, 7.2, 7.3, 7.4 were -renumbered. Figure numbers have changed. -Existing exercise numbers are the same, except -exercises 2.5.1, 7.2.5 were replaced, -7.2.12 asks for the reverse implication (that was an erratum, as this was -already proved in the text), -in exercise 3.3.11 we require that the example is continuous, -exercise 4.4.6 was simplified very slightly (the original was a typo), -and due to new material, exercise 3.6.2 now asks for more, and exercise -6.2.7 is easier. Exercise 7.5.9 is easier with a new proposition. -A couple of other exercises had errata fixed (0.3.5, 1.4.3, -5.4.2, 7.1.5). -Other than this, the new edition is essentially backward compatible as usual. - -

-A detailed list of changes: -

    -
  1. Identify book as Volume I on the title page, and refer to Volume II -in the introduction. -
  2. In the PDF the pages have been made slightly longer so that we can lower -the page count to save some paper. -
  3. To be more consistent with what is a Theorem and what is a Proposition, -demote Theorems 2.1.10, 2.3.5, 2.3.7, 3.4.6, 5.2.2 to Propositions. -Also change Theorem 4.2.2 to a Lemma as that's more appropriate. -Numbering didn't change. -
  4. Change the look of the figures to match the Volume II and -to better visually distinguish them from the surrounding text. -
  5. Change the "basic analysis result" to $x \leq \epsilon$ -for all $\epsilon > 0$ implies $x \leq 0$. This better fits the mantra that -in analysis we prove inequalities, and separates out the idea that to show -$x=0$ one proves $x \leq 0$ and $x \geq 0$. -
  6. Add a short paragraph about naming of Theorem vs Proposition vs Lemma vs -Corollary, to answer a common question. -
  7. Add a subsection on relations, equivalence relations, -and equivalence classes. This renumbers the following -propositions, definitions, etc... -
  8. Add figure for the sets $S$ and $T$ in 0.3 -
  9. Add figure for direct/inverse images in section 0.3. -
  10. Add figure for showing ${\mathbb{N}}^2$ is countable. -
  11. Add exercises 0.3.21, 0.3.22, 0.3.23, 0.3.24, 0.3.25. -
  12. Add figure for least upper bound definition. -
  13. Add note about uniqueness of sups and infs. -
  14. In proposition 1.1.8, add the two very commonly used properties -as parts (vi) and (vii). -
  15. Add explicitly proposition 1.1.11 about an ordered field with -LUB property also having GLB property. -
  16. Add link to Dedekind's Wikipedia page. -
  17. In exercise 1.1.6, removed the "In particular, $A$ is infinite". There -is no point in going into the distinction and it just confuses students. -
  18. Add exercises 1.1.11, 1.1.12, 1.1.13, 1.1.14. -
  19. Add footnote on impossibility of tuned pianos and rational roots -
  20. In proposition 1.2.2 simplify matters by changing the statement to not -assume that $x \geq 0$. The original statement is given in the paragraph -below as a remark. -
  21. Add figure to proof of the density of $\mathbb Q$ in section 1.2. -
  22. Add exercise 1.2.14, 1.2.15, 1.2.16, 1.2.17. -
  23. Change title of 1.3 to include "bounded functions". -
  24. Add figure for a bounded function, its supremum and its infimum -in section 1.3. -
  25. Add exercises 1.3.8, 1.3.9. -
  26. Add Proposition 1.4.1 (which moves Theorem 1.4.1 to 1.4.2), which is the -characterization of intervals that we often use later, so better to formalize -it. Proof is still an exercise. -
  27. We never defined/open closed for unbounded intervals, although later on -we make a big deal about a closed and bounded interval. To be more in line -with general usage, define what "unbounded closed" and -"unbounded open" intervals. -
  28. In exercise 1.4.6, be more explicit about what the intersection of closed -intervals is, and explicitly mention boundedness. -That, is say the intersection is $\cap_{\lambda I} [a_\lambda,b_\lambda]$. -
  29. Add exercise 1.4.10. -
  30. In Proposition 1.5.1, add the inequalities for all representations as well, -since we use these facts later. Also add the detail of the proof as it is -perhaps not as obvious to every reader. -
  31. Mark exercise 1.5.6 as challenging and add a longer hint. The real tricky -part is to get a bijection rather than two injections which is easier. -
  32. Add exercise 1.5.8, which is really required in the proof, so that -we do not require things from chapter 2. Be more explicit about its use -in the proof. -
  33. Add figure on cantor diagonalization in section 1.5. -
  34. Add more detail in proof of Proposition 1.5.3, to see how we use the -unique representation. -
  35. Add exercises 1.5.7, 1.5.9 -
  36. Add a very short example of a tail of a sequence in 2.1. -
  37. Add a diagram to proof of Proposition 2.1.15. -
  38. Simplify the proof of squeeze lemma as suggested by Atilla Yıllmaz. -
  39. Add example of showing $n^{1/n}$ going to 1 as a more subtle example of -the use of the ratio test. -
  40. Simplify/symmetrize the proof of product of limits is the limit of the -product. (Thanks to Harold Boas) -
  41. Show the convergence/unboundedness of $\{ c^n \}$ in a somewhat -a more elementary way without Bernoulli's inequality. -(Thanks to Harold Boas) -
  42. Add exercises 2.2.13, 2.2.14, 2.2.15, 2.2.16. -
  43. Add two figures in 2.3 for liminf and limsups, one for a random example, -and one for the given example. -
  44. Expand the discussion of infinite limits and liminf/limsup for unbounded -sequences. Add a proposition about unbounded monotone sequences, and a -proposition connecting the definition of liminf/limsup to the previous -definition for bounded sequences. -
  45. Add exercises 2.3.15, 2.3.16, 2.3.17, 2.3.18, 2.3.19. -
  46. Add figure to the example of geometric series with 1/2. -
  47. Make the geometric series into a Proposition as we use it quite a bit. -Also use geometric series as an example for the divergence if terms do not -go to 0, that is when $r \notin (-1,1)$. -
  48. Mention the ``infinite triangle inequality'' in text in 2.5, -I always do in class. -These two things renumber the subsequent examples, propositions, etc... in 2.5 -
  49. Replace exercise 2.5.1. The exercise was proved in example 0.3.8 -and already used previously. -
  50. Add exercises 2.5.14, 2.5.15, 2.5.16, 2.5.17. -
  51. Add a sentence and notation to the figure about possible -non-convergence at the endpoints of the radius of convergence. -
  52. Add exercises 2.6.13, 2.6.14. -
  53. Add a note and a footnote on the other common notations for the various -limits of restrictions. -
  54. Add Corollary after 3.1.12 for the absolute value, which shifts the -numbering of propositions and examples by one in 3.1. -
  55. Add exercises 3.1.15, 3.1.16. -
  56. Expand example 3.2.10 a little bit, and add a figure for the example. -
  57. Add exercises 3.2.17, 3.2.18, 3.2.19. -
  58. Add figure for definition of absolute minimum and maximum. -
  59. Add corollary 3.3.12 whose proof is the existing exercise 3.3.7. -
  60. In exercise 3.3.11 add the missing continuity hypothesis. -Otherwise the exercise is too easy (it is already easy). -
  61. Add exercise 3.3.14, 3.3.15, 3.3.16, 3.3.17. -
  62. Swap examples 3.4.2 and 3.4.3, they make a lot more sense in that order. -
  63. In Section 3.4 add a very short application of the continuous extension. -
  64. Add figure for the idea of the proof of the product rule, that is, -a picture of the identity given as hint. -
  65. Add exercises 3.4.15, 3.4.16, 3.4.17. -
  66. Add exercise 3.5.9. -
  67. Strengthen Proposition 3.6.2 to include limits at infinity, -which means that Exercise 3.6.2 asks for a bit more -since two new statements must be proved. -
  68. Add exercises 3.6.12, 3.6.13, 3.6.14, 3.6.15. -
  69. Actually prove the use of intermediate value theorem in proof of -corollary 3.6.3. -
  70. Add figure to Example 3.6.5. -
  71. Add figure to Example 3.6.7. -
  72. Add examples 4.1.3, 4.1.4 which moves everything down a number in 4.1. -
  73. Add link to Schwarz and Bunyakovsky and give a short note on the name -in a footnote. -
  74. Add exercises 4.1.13, 4.1.14, 4.1.15. -
  75. Reorganize the proof of Mean value theorem a little bit, add some -motivation for the proof, and move the figure up -earlier as it gives an idea for the proof. -
  76. Make Example 4.2.8 into a Proposition since that's what it really is. -Then we can refer to it rather than the exercise that proves it later. -
  77. The proof of exercise 4.2.9 was a little too challenging. In essence -one reproves Cauchy's mean value theorem anyway, so add that as a theorem, -and add an exercise to prove it. This causes some renumbering in 4.2. -
  78. Add a proposition about extension of derivatives to the boundary as that -is a in fact quite useful and has a very quick and straightforward -proof which is left as exercise. -
  79. Add small note about measuring speed with aircraft and mean value theorem. -
  80. Add some motivation to the proof of Darboux's theorem, and -add a figure. -
  81. Add exercises 4.2.13, 4.2.14, 4.2.15. -
  82. Add two figures for Taylor's theorem section (4.3). -
  83. Mention Taylor series and connection to power series in 4.3. -
  84. Add quick application of Taylor's theorem to prove second derivative test. -Proposition 4.3.3. -
  85. Add exercises 4.3.9, 4.3.10, 4.3.11. -
  86. Rewrite proof of Lemma 4.4.1, and use clearer variable names. -
  87. Add figure to Example 4.4.5. -
  88. Modify Exercise 4.4.6 very slightly, replace ``interval'' with -``open interval''. The distinction is irrelevant for how one proves it -and considering other types of intervals makes the proof longer. -
  89. Add figure to the proof of Proposition 5.1.7. -
  90. Add figure to Proposition 5.1.10. -
  91. Add figure to Example 5.1.12. -
  92. Add proposition on the sub/super additivity as proposition 5.2.5, -so all other propositions, theorems, and lemmas shift by one in 5.2. -
  93. In the monotonicity proposition, state it for upper and lower integrals -as well, we prove that anyway, it fits better with the style of exposition -in this book, and it can be useful in proofs. -
  94. Add proposition for the integrability of monotone functions. -We use this later, it is better to just refer to a proposition -than an exercise, and it is also genuinely useful. -
  95. Add exercise 5.2.17. -
  96. Add figure to proof of the fundamental theorem of calculus in 5.3. -
  97. Add remarks about other definitions of logarithm and the exponential, -and about the uniqueness and existence following from a subset -of the given conditions. -
  98. Add exercise 5.4.11. -
  99. Improve the exposition of the summability of the sinc function in -example 5.5.12 and add another figure to the example to show the bound. -
  100. Add figure for integral test for series in 5.5. -
  101. Add figure to example 6.1.4. -
  102. Add figure to definition uniform convergence in 6.1. -
  103. Add exercise 6.1.12, 6.1.13, 6.1.14. -
  104. Add subsection to 6.2 on swapping of limit of functions and derivatives -for continuously differentiable functions. -This makes exercise 6.2.7 much easier as we essentially do the main -bit as a theorem. There is a new figure in this subsection. -
  105. Add subsection to 6.2 on convergence, differentiation, and integration -of power series. -
  106. Change hint in 6.2.1 to be simpler, $|x|^{1+1/n}$ works but it is a -bit messy to prove all the details. -
  107. Add exercises 6.2.15, 6.2.16, 6.2.17, 6.2.18, 6.2.19, 6.2.20, 6.2.21. -
  108. Add remark about weaker solutions to ODEs using the integral equation. -
  109. Use the more common interior notation in 6.3, and in 7.6. -
  110. Add exercises 6.3.7, 6.3.8, 6.3.9. -
  111. Improve triangle inequality figure in 7.1. -
  112. Add example of complex numbers to 7.1, and an example of a sphere, -that renumbers the rest of the examples and propositions in 7.1. -
  113. Add exercises 7.1.9, 7.1.10, 7.1.11, 7.1.12, 7.1.13. -
  114. Improve the open set figure in 7.2. -
  115. Add Propositions 7.2.11 and 7.2.12 that codify some of the -subspace topology things we keep using. This renumbers the rest of the -definitions, examples, and propositions in 7.2. -
  116. Simplify proof of Proposition 7.2.15, as the conclusion was already -proved in exercise in 1.4, and is formalized in Proposition 1.4.1. -
  117. Replace exercise 7.2.5, the conclusion was already proved -in exercise 1.4.3 (in more generality, in fact). -
  118. In Exercise 7.2.12 the implication goes the other way (erratum in -earlier versions), as is needed in the text. -
  119. Add figures to Propositions 7.2.9, 7.2.13, 7.2.15, and 7.2.26. -
  120. Add $(0,\infty)$ and $[0,\infty)$ as an examples of an open and -closed sets in ${\mathbb{R}}$ to Example 7.2.5. -
  121. Add footnote about empty sets and connectedness. -
  122. Add exercises 7.2.15, 7.2.16, 7.2.17, 7.2.18. -
  123. Add figure to definition of convergence in 7.3. -
  124. Add example to 7.3 of $C([0,1],{\mathbb{R}})$ where convergence is the -uniform convergence. This renumbers the following examples, propositions, -etc... -
  125. Add remark that pointwise convergence does not come from a metric. -
  126. Add example for convergence in the complex numbers. -
  127. Add exercises 7.3.13, 7.3.14. -
  128. Add an example (in fact a set of 4 examples) of compact and noncompact -sets on the real numbers in 7.4. This again renumbers the remaining -propositions, etc... -
  129. Add proposition that $C([a,b],\R)$ is a complete metric space. -
  130. Add proposition that a closed subset of a complete metric space is -complete, that is used later. -
  131. Add remark at the end of 7.4 about Cauchy depending on the actual metric -and not just on the topology, along with an exercise working through the -counterexample. -
  132. Add an example for the Lebesgue covering lemma, finding a $\delta$ for a -cover. -
  133. Add figures to proof of Proposition 7.4.9, Lebesgue covering lemma, and -Theorem 7.4.11. -
  134. Add exercises 7.4.17, 7.4.18, 7.4.19, 7.4.20. -
  135. Add figure for Lemma 7.5.7. -
  136. Add a proposition 7.5.12 on continuity of functions defined by integration. -Makes exercise 7.5.9 simpler, but it seemed to that most students missed the -subtlety, and we use this result later a few times. -
  137. Add exercises 7.5.11, 7.5.12, 7.5.13, 7.5.14, 7.5.15, 7.5.16, 7.5.17, - 7.5.18. -
  138. Make notation more in line with the rest of the chapter in 7.6. -
  139. Move all exercises to the Exercises subsection 7.6 to be consistent -with the rest of the book. -
  140. Add exercise 7.6.11. -
+No new changes since 5.0 diff --git a/changes2-draft.html b/changes2-draft.html index 86cf756..6af5888 100644 --- a/changes2-draft.html +++ b/changes2-draft.html @@ -1,93 +1,4 @@ This file is a draft of the new changes for http://www.jirka.org/ra/changes2.html

-??? ??th 2018 edition, Version 2.0 (edition 2, 0th update): -

- -Numbering of definitions, examples, propositions changed in 8.1, 8.3, 10.1. -Numbering of exercises is unchanged, except for 9.1.7 which was replaced - due to erratum. - -

    -
  1. New section 10.7 on change of variables. -
  2. New chapter 11 on Arzela-Ascoli, Stone-Weierstrass, power series, -and Fourier series. -
  3. A List of Notations is added at the back as in volume I. -
  4. In the PDF the pages have been made slightly longer so that we can lower -the page count to save some paper. -
  5. Add figure showing vector as an arrow and discussion about this - for those that do not remember it from vector calculus. -
  6. Add a paragraph about simple algebraic facts such that $0v=v$. -
  7. Add footnote about linear independence for arbitrary sets in 8.1. -
  8. Add example that span of $t^n$ is ${\mathbb{R}}[t]$. -
  9. Add remark about proving a set is a subspace. -
  10. We also use the words "linear operator" for $L(X,Y)$, -and it is for $L(X)$ that we say "linear operator on $X$", -so update the definition appropriately. -
  11. Add convexity of $B(x,r)$ as a proposition since we use it so often. -
  12. Add exercise 8.1.19 -
  13. Proposition 8.2.4 doesn't need $Y$ to be finite dimensional, - same in the exercise 8.2.12, so no need to assume it. -
  14. In Proposition 8.2.5, emphasize where the finiteness of dimension is - needed. -
  15. Use $GL(X)$ as notation for invertible linear operators. -
  16. Give more detail on why mapping between matrices and linear operators is -one to one once a basis is fixed. -
  17. Add a commutative diagram to the independence on basis discussion. -
  18. Reorder the definition of sign of a permutation to be more logical. -
  19. Add short example of permutation as transpositions. -
  20. Add exercises 8.2.14, 8.2.15, 8.2.16, 8.2.17, 8.2.18, 8.2.19 -
  21. Add figure to definitions 8.3.1 and 8.3.8. -
  22. Add Proposition 8.3.6, which was conspicuously missing. -
  23. Add figure for differentiable curve and its derivative. -
  24. Add figure to exercises 8.3.5 and 8.3.6. -
  25. Add exercise 8.3.14 -
  26. Add graph to figure in example 8.4.3 (and adjust the formulas) -
  27. As application of continuous partials imply $C^1$ -
  28. Add exercises 8.4.7, 8.4.8, 8.4.9, 8.4.10 -
  29. Fix up statement of the inverse function theorem in 8.4. -
  30. Add a couple of figures to proof of the inverse function theorem. -
  31. Add a figure to the implicit function theorem. -
  32. Add a short paragraph about the famous Jacobian conjecture. -
  33. make the remark at the end of 8.5 into an actual "remark" -
  34. Add observation about solving a bunch of equations not just for $s=0$ for - the implicit function theorem. -
  35. Add exercises 8.5.9, 8.5.10, 8.5.11 -
  36. Add figure to 8.6 -
  37. In 8.6 cleanup the argument in the proposition and use only -positive $s$ and $t$ for simplicity. -
  38. Add exercises 8.6.5, 8.6.6, 8.6.7 -
  39. Refer to the new proposition 7.5.12 about the continuity in 9.1 -
  40. Add figure to example in 9.1 -
  41. Exercise 9.1.7 replaced due to erratum. The replacement shows - the same issue that the previous wrong exercise tried to. -
  42. Add exercise 9.1.8 -
  43. Reorder the introduction of 9.2 a bit, and fix an erratum in that - derivative at the endpoints was not really defined for mappings. -
  44. Add figure to examples 9.2.2, 9.2.3, 9.2.11, 9.2.13, 9.2.18 -
  45. Add figure for definition of a function against arc-length measure. -
  46. Add figure to proof of path independence implies antiderivative in 9.3. -
  47. Add figure to proof that integral over closed paths being zero - means that the integral is path independent in 9.3. -
  48. Add figure to definition 9.3.5 -
  49. Change hint to exercise 9.3.8. -
  50. Add Example 10.1.16 of compact support with a figure, following - examples/propositions in 10.1 are renumbered. -
  51. Explicitly mention monotonicity of outer measures right - after the definition (it is a rather easy exercise), and also - allowing finite sequences of rectangles in the definition - (a new exercise). -
  52. Add figure to definition of outer measure. -
  53. Clean up proof of Proposition 10.3.2. -
  54. Add exercises 10.3.11, 10.3.12 (and a figure), 10.3.13. -
  55. Add corollary for the Riemann integrability theorem showing that it is - an algebra, that min and max of two functions are Riemann integrable and - so is the absolute value. -
  56. Add exercises 10.4.6, 10.4.7, 10.4.8, 10.4.9, 10.4.10, 10.4.11 -
  57. Add exercises 10.5.5, 10.5.6, 10.5.7 -
  58. In 10.6, add figure for positive orientation and a figure illustrating - the three types of domains. -
  59. Many minor improvements in style and clarity, plus several -small new example throughout. -
  60. Fix errata. -
+No new changes since 2.0 diff --git a/cover.png b/cover.png index 93f1ae6..3b3339a 100644 Binary files a/cover.png and b/cover.png differ diff --git a/cover.xcf b/cover.xcf index 6411e0b..394bca5 100644 Binary files a/cover.xcf and b/cover.xcf differ diff --git a/cover2.png b/cover2.png index efeec8f..f115949 100644 Binary files a/cover2.png and b/cover2.png differ diff --git a/cover2.xcf b/cover2.xcf index 581416b..4708dd4 100644 Binary files a/cover2.xcf and b/cover2.xcf differ