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This file is a draft of the new changes for https://www.jirka.org/ra/changes.html
-General changes:
-
-In definitions of limits (sequence, continuous limits, in metric spaces,
- etc), don't "cheat" and say "if a limit is unique." While it feels a
- little wordy since the first thing we do is prove that the limit is
- unique, I'm starting to feel that this may be contributing to confusion
- about proof writing to students.
-Do not shorten sequences to \(\{ x_n \},\) but always write out
- \(\{x_n\}_{n=1}^\infty.\) It seems to me this shorthand is
- causing more confusion than it is worth, especially with
- regards to the distinction between set and sequence, and also
- when working with subsequences.
-Do not shorten sequence limits to \(\lim\, x_n,\) but always write out
- \(\lim_{n\to\infty} x_n.\) It seems to me this shorthand is
- causing more confusion than it is worth, especially
- when subsequences are introduced.
-Do not shorten series to \(\sum\, x_n,\) but always write out
- \(\sum{n=1}^{\infty} x_n.\) It seems that it is making students
- forget a limit is involved.
-Uniform norm notation is changed to \(\|\cdot\|_K,\)
- where \(K\) is the set where the supremum is taken. I've had a number of
- complaints about the \(u\) notation not being very standard. And this goes
- better with my other books where I use the more standard notation.
-Add parentheses to the notation for "Riemann integrable" for consistency,
- that is, \(\mathscr{R}([a,b])\) instead of \(\mathscr{R}[a,b] .\)
-
-Specific larger changes:
-
-In 0.3, after definition of composition state as an "exercise"
- that compositions of bijections are bijections. This is actually a
- WebWorK exercise.
-Add Exercises 0.3.26 and 0.3.27 to prove the DeMorgan's laws and
- the pushforward/pullback propositions for infinite unions and
- intersections.
-Add Figure 1.3 on the set \(\{ \frac{1}{n} : n \in {\mathbb{N}} \}\)
- and its infimum (Corollary 1.2.5) (renumbers all the following figures in
- chapter 1).
-Simplify proof of Proposition 2.1.7 by just defining the \(B\) once
- rather than defining two different bounds.
-As per suggestions name 2.1.10 the "Monotone convergence theorem"
- and therefore make it a Theorem rather than a Proposition.
-Exercise 2.2.9, add hypothesis that \(x_n\not=x\) for all \(n.\)
- It is implied in that the limit makes sense,
- but it should be stated explicitly.
-Replace Example 2.4.3 with a more useful example, one that isn't Cauchy.
- Suggested by Harold Boas.
-Clean up the proof of Proposition 2.6.2, the Alternating series test.
- Mainly improve the readability by using the variable names more
- consistently, rewrite the end of the proof, and fix an erratum.
-Add Figure 2.8 showing graphically why the alternating sum converges.
-Add Figure 2.9 to show how the sample rearrangement of the alternating
- harmonic sum converging to 1.2 works. Renumbers the following figure
- chapter 2.
-Add Exercise 2.6.15 for Tonelli/Fubini for sums. This is too useful of
- a "variant" of reordering not to have it, plus we do use it in
- volume II in a proposition.
-Add Figure 3.1 for Example 3.1.6 where limit is different from
- value (renumbers all figures in chapter 3).
-In Corollaries 3.1.9, 3.1.10, 3.1.11, the hypothesis is only needed
- for all \(x \in S \setminus \{ c \},\) as we do in all the other
- results of this section. The way it is stated could
- be confusing, so change them to this hypothesis. (It is equivalent
- because one can always replace \(S\) with \(S \setminus \{ c \} \)
- of course. This affects
- Exercises 3.1.3 and 3.1.4, but it at worst makes them slightly less
- confusing and more straight forward.
-Replace Exercise 3.1.10. This exercise was almost exactly the
- same as 3.1.11.
-Rename section 3.3 to "Extreme and intermediate value theorems".
-Add a new Example 3.3.11 showing the existence of roots,
- so the old 3.3.11 becomes 3.3.12, and
- Corollary 3.3.12 becomes 3.3.13. This is a nice application,
- and ties in some prior results, and it is good to see it especially
- if 4.4 is not covered.
-Add Figure 3.8 for the Corollary 3.3.13 (was 3.3.12) where
- image of a continuous function is an interval.
-Add Figure 3.10 to visualize why the square root is not Lipschitz.
-Make the Lemma 4.2.2 be stated for an open interval \((a,b)\) since we
- don't need the endpoints and it could really just be confusing.
-Add Exercise 5.1.15
-In the proof of Proposition 5.2.2 mention boundedness to be completely
- rigorous.
-Add Exercise 5.2.18.
-In Exercise 5.3.7, add \(a+\epsilon \lt b-\epsilon\) to emphasize where
- things are well defined.
-Add Exercise 5.3.13.
-Add Figure 5.6 and Figure 5.7 giving the logarithm and the exponential,
- the later figures in chapter 5 are renumbered.
-Add Figure 6.7 to Example 6.2.9 to illustrate what is happening.
-Add Exercise 6.2.22.
-Add Figure 6.8 in subsection 6.3.1 to demonstrate a first order ODE
- as a slope field.
-In Example 6.3.3, refer back to the figure for the exponential which
- shows the slope field. Also add figure showing the exponential together
- with the first few iterates.
-In Remark 6.3.7, use \(x\) instead of \(t\) for the Heaviside function
- to make things less confusing.
-Add Figure 7.2, to clarify especially the end of Example 7.1.3. This
- renumbers the rest of figures in chapter 7.
-Add \(\{x\}\) as an example of a closed set to Example 7.2.5, but leave
- the proof to the online homework (it is rather simple).
-To be more consistent, and avoid overuse of the letter x for everything,
- use \(p\) instead of \(x\) in Propositions 7.3.11, 7.3.12, 7.3.13, 7.4.2,
- Exercises 7.3.1, 7.3.5, 7.3.7, and Definition 7.4.2
-Move the remark about subspaces not being complete after Proposition
- 7.4.5 as it makes more sense that way.
-In Definition 7.4.7, Example 7.4.8, and the proof of Proposition 7.4.9,
- use \(m\) instead of \(k\) to avoid overusing the letter \(k,\) to make
- it easier to talk about the proof.
-Add a paragraph after Proposition 7.4.9 emphasizing the difference
- between compactness and closedness in the sense that "compact" doesn't
- care about the ambient topology while "closed" most definitely does.
-Rewrite the \(n=1\) part of the proof of 7.4.13 (Heine-Borel) to be a
- little bit more like \(n=2\) part (use the proposition to reduce to
- closed and bounded interval) and just refer to Example 2.3.8 instead of
- repeating the argument.
-Change the mapping in Theorem 7.6.2 (Contraction mapping principle)
- to \(\varphi\) from \(f\) to avoid overusing \(f\) in this section.
-Mention that \(d\) is the uniform norm on \([-h,h]\) in the proof.
-
-Smaller changes:
-
-Many minor rephrasings and rewordings for added clarity.
-In the definition of \(S\) in the proof of Theorem 0.3.6,
- use \(n,\) to avoid overloading of \(m.\)
-Say that \(A\) is a set in Cantor to be a bit more precise.
-Simplify parts (iii) and (iv) of Definition 1.1.2.
-In proof of part (v) of 1.1.8, should just use the definition rather than
- part (ii) of the proposition.
-Improve wording around Proposition 1.2.2, remove some unnecessary words
- and explicitly state the version with \(|x|,\) which is a common
- statement.
-In the proof of 1.2.6, use \(c\) instead of \(b\) for the second
- inequality to avoid overloading \(b.\)
-Improve slightly the wording in the examples after definition of a limit
- of a sequence.
-After Proposition 2.1.10, make a remark about monotone sequences and
- boundedness above/below.
-Improve wording of Example 2.1.12.
-After definition of subsequences, give a little bit more detail of the
- example subsequence.
-Improve the recursive sequence (Newton's method) wording slightly.
-In the proof of Proposition 2.2.11, use \(x\) instead of \(L\) in the
- proof as \(L\) is used in the related Lemma 2.2.12 for something else.
-In the proof of Theorem 2.3.4, when defining the subsequence, suppose
- \(n_1,\ldots,n_{k-1}\) is defined and define \(n_k\). That way it
- is more consistent with the rest of the proof and should be easier
- to follow. Also say \(m \geq n_{k-1}+1\) instead of \(m \gt n_{k-1}\)
- to make it clearer where the \(+1\) comes from.
-Change the index variable in Proof of Proposition 2.3.6 from \(j\) to
- \(n\) for consistency.
-Simplify the remark after Definition 2.3.12 as it may be hard to parse.
-Be more precise with the hint and the indexing in Exercise 2.3.7. Also
- mention that it is just one of the possible proofs (I find it a cool proof).
-Remark 2.4.6 should refer to theorem not proposition. Also clarify that
- Cauchy completeness means that the limit should be back in the set.
- The remark is purposefully vague (to omit the gory details),
- it's not really a definition, nor a construction of the reals,
- but we don't want to be misleading.
-In Definition 2.5.1 don't define an extra variable \(x\) just for the
- limit.
-In Definition 2.5.14 be consistent with wording for absolute
- and conditional convergence. That is, change "is conditionally
- convergent" to "converges conditionally", and add both "converges
- conditionally" and "conditional convergence" to the index.
-In 2.5, when talking about the terms going to zero "fast enough"
- before the comparison test, this is about series with positive terms,
- so make that clear.
-Throughout, where appropriate, use \(i\) or another letter instead of \(j\)
- as that typesets a lot better with series and as powers.
- In some places this also changes the other indices.
-Rephrase Merten's theorem a tiny bit.
-In Exercise 2.6.11 part c, be more precise in the parenthetical remark
- about divergence.
-Reword Exercise 2.6.4 part a) to be a little easier to understand
- what is being asked
-Before Proposition 3.1.15, emphasize the meaning of it, that it means
- that the limit is "local."
-Below Definition 3.2.1, we make a statement about the converse not
- holding, but with no reference. An example is given in Example
- 3.2.13 so give a parenthetical reference to it.
-While fixing the labels in Figure 3.4 (was 3.3), make them smaller so that they
- don't run into each other and move them below the axis.
-Improve wording of Example 3.2.12.
-Before Lemma 3.3.1, emphasize that \([a,b]\) is a closed and bounded
- interval. In a related change, this was emphasized in the statement
- of the Min-Max/Extreme value theorem, but that was making it too
- wordy, so make a remark right after the theorem and simplify the
- statement.
-When defining absolute minimum/absolute maximum, say that these are
- what \(f(c)\) is (as is shown in the figure).
-Improve the wording of the examples 3.3.4, 3.3.5, 3.3.6 to emphasize
- which properties are satisfied and which are not.
-At the end of the proof of Proposition 3.3.10, when we claim the root
- by Bolzano, say it is in the open interval so that the claim lines up
- better with the theorem.
-Remove the "definition" of the phrase "uniformly continuous on \(X\)"
- as we only ever use it in a remark and it is not standard verbiage anyway.
-Rewrite the introductory paragraph to section 3.4.2 a bit, and
- make the statement of Lemma 3.4.5 slightly more precise.
-Clean up Figure 3.9 very slightly.
-In Example 3.4.10, name the second function \(g\) to make things
- hopefully a bit clearer.
-Reword Example 3.5.3 a little.
-Make caption to Figure 3.12 a bit more precise.
-Clean up Figure 4.1 very slightly.
-In Lemma 4.2.2, move the initial sentence of the proof to the end and
- reword it.
-After Theorem 4.2.4, say explicitly that the slope of the secant line
- is the mean value of the derivative, hence the name of the theorem.
-In subsection 4.2.4 (applications of the mean value theorem) add a short
- description of how the applications work: by getting rid of a limit.
-At the end of Example 4.2.12, be a little less wordy.
-Reword the paragraph in front of Corollary 4.4.3, now that we have the
- existence of roots as an explicit example in 3.3.
-In Exercise 4.4.2, remark that it is the same as Exercise 4.1.10, to avoid
- possibly assigning this type of problem twice.
-In Proof of Proposition 5.1.7, use (in addition to changing \(j\) to
- \(i\)) \(q\) instead of \(p\) since I just realized that this is a
- terrible name since the partition is \(P.\) Also changes Figure 5.2.
-Before Proposition 5.1.13, refer to Figure 5.1 for intuition of what
- the difference of the upper and lower sums measures.
-Explicitly mention in the beginning of the proof of Lemma 5.2.7
- that \(f\) is bounded.
-Mention in a footnote that people often say "converges" when they mean "converges
- pointwise"
-Improve the wording of Exercise 6.1.10.
-Reorganize the setup in proof of Proposition 7.2.11 a bit.
-Add a small note after Heine--Borel (7.4.14) to emphasize it does not
- hold in subspaces of \({\mathbb{R}}^n.\)
-In Exercise 7.5.2, reference the figure with the graph.
-Rewrite Proposition 7.4.9 to emphasize in which direction the
- implication goes.
-Use \(z\) instead of \(\tilde{y}\) in the proof of Proposition 7.5.12
-In 6.3 and 7.6, mention for completeness that
- \([h-x_0,x_0+h] \subset I\) in the statement of Picard's theorem.
-In 7.6, improve the wording of the proof of Picard's theorem.
-Improve the wording in part c) of Exercise 7.6.9. The point of applying
- the theorem is not to find that \(\sqrt{2}\) is a fixed point, that follows
- just from the formula. The point is that the theorem does apply.
- Add a note about why this is useful.
diff --git a/changes2-draft.html b/changes2-draft.html
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This file is a draft of the new changes for https://www.jirka.org/ra/changes2.html
-Many small improvements in wording.
-Add the addition/scaling meaning to the arrow figure (8.1), and
- mention it in text.
-In the definition of dimension, make it "\(d\) is the largest
- integer such that \(X\) contains a set of \(d\) linearly independent
- vectors" to make the definition be a priory well-defined instead of
- relying on the following paragraph. Consequently move the
- statement of the previous definition (there is a set of \(d\)
- linearly independent vectors but not a set of \(d+1\) independent
- vectors) to the paragraph afterwards.
-Relabel some of the indices in 8.1 and improve the typesetting at the
- same time. Use \(k\) more consistently when running
- over the indices, \(n\) for size of a basis, and \(m\) for the size of
- a set of vectors.
-Simplify the discussion about the sum of two linear combinations being a
- linear combination. It is a trivial point and the discussion was
- unnecessarily complicated.
-Move the remark about it being better to use the abstract definition
- to a more logical place after the \({\mathbb{R}}^n\) example.
-Make Remark 8.1.6 into a Proposition.
-Add comment about span being the smallest subspace containing \(Y\)
- after Proposition 8.1.11 to make that sentence about if \(Y\) is already a
- subspace a bit less mysterious.
-Add a note to Definition 8.1.12 that we usually consider the basis
- to be ordered, not just a set.
-Right after Proposition 8.1.14, add a short note about linearly
- independent set of \(d\) vectors (where \(d\) is the dimension) automatically
- spans, that's a particular case for \(m=d\) in the last item.
-The claim used in the proof is now an explicit part (ii). This renumbers
- Moves the old (ii) to (iii), but also make the old (iii) into an "in
- particular" as it really does not need its own item I don't think.
- Also reword parts of the proof of the proposition.
-Add the statements before Proposition 8.1.16 into the proposition. And
- add an explicit easy exercise (8.1.20) to prove the statements.
-Before Proposition 8.1.18, make more explicit this idea of looking at the
- nullspace to show injectivity, and put "nullspace" into the index.
-Add \([x,y]\) for a line segment to the notations index.
-Move Proposition 8.1.24 after the construction of convex hull as that
- makes more sense. This changes the numbers, the proposition is now 8.1.25
- and Example 8.1.25 is now Example 8.1.24. Add a short introduction
- to the proposition and make the proposition more
- consistent with the notation of the section.
-Always use \((1-t)x+t y\) in this order for convex combination.
-In subsection 8.1.4 use \(X\) and \(Y\) for vector spaces and \(U\)
- and \(V\) for subsets to be more consistent with the rest of the section.
-Add "Compare ..." for Exercises 8.1.4 and 8.1.16 to reference each other.
-Reword Exercise 8.1.18 to be slightly clearer.
-In 8.2 as in 8.1, avoid \(j\) as an index in some places.
-Add a short sentence that the inequality
- \(\|Ax\| \leq C \|x\|\) proves \(\|A\| \leq C.\)
-In introducing matrix multiplication use \(z_1,\ldots,z_n\) instead of
- \(c_1,\ldots,c_n\) as \(c_{i,j}\) is used in the same paragraph for
- something else. When doing the bound on the norm a bit later
- \(c=(c_1,\ldots,c_n)\) as our element in \({\mathbb{R}}^n.\)
-In Proposition 8.2.6, just move the "\(GL(X)\) is open" statement into
- part (i), since that what I always say when I write down part (i) anyway,
- that's where that statement belongs.
-Add figure to illustrate what we mean by determinant stretching space for
- a specific matrix. This renumbers figures in chapter 8.
-Be more precise in the statement of Proposition 8.2.10.
-Make Exercises 8.2.12 and 8.2.13 clearer by using different notations for
- the different norms.
-Add a note that Jacobian could confusingly also refer to the matrix,
- for this reason, switch to always saying "Jacobian determinant" in the book.
-Add Exercise 8.3.15 on the mean value theorem for functions of several
- variables.
-In Proposition 8.4.2, use \(p\) and \(q\) instead of \(x\) and \(y\)
- to avoid confusion given the relevant example that comes right after
- uses \(x\) and \(y\) for the coordinates in the plane.
-In the proof of 8.4.4, use \(c_0\) for the specific constant instead of
- \(c\) since we just said "for all \(c\)" in the sentence before.
-In the proof of 8.4.6, to emphasize that \(x\) is fixed use \(p\)
- instead. While at it, change \(k\) to \(h'\) since \(k\) is often
- used as an integer and to be consistent with the \(A'\) naming.
- In fact, use \(j\) and \(k\) in the same
- capacity as in 8.3 for consistency \(k\) for the component of \(f\)
- and \(j\) for the component of \(x.\)
-Move Exercise 8.4.10 to section 9.1, where it becomes 9.1.9
- and add a missing hypothesis.
-Add new Exercise 8.4.10 replacing the one that was moved.
-In 8.5, move the paragraph about contraction mapping principle after
- stating the theorem, it was kind of putting the cart before the horse the
- way it was stated.
-In Exercise 8.5.2 part b, add a note that the differentiability of \(f\)
- needs to be established.
-More consistent use of indices in section 8.6. Use \(\ell) and \(m\)
- for components of \(x,\) \(k\) for order and don't use \(j\) at all
- (the others are easier to read, typeset more nicely)
-In the proof of Proposition 8.6.2, don't use the notation
- \({\mathbb{R}}_{+}^2\) as it wasn't quite right. Just say the domain of
- \(g\) is \((0,\epsilon) \times (0,\epsilon) .\)
-Add Figure 9.2 for Exercise 9.1.6.
-Make the proof of Proposition 10.1.13 more precise and less handwavy.
-In Exercises 10.1.3 and 10.1.4, emphasize that the rectangles are closed.
-In the proof of Fubini change j to i and k to j. That way we don't have
- k going from 1 to K which is confusing.
-Add Figure 10.3 to proof of Proposition 10.1.13, renumbers the other
- figures in chapter 10.
-Add Figure 10.5 to example 10.2.1.
-Clarify the proof of Proposition 10.3.2, among other things, emphasize why
- \(\ell \geq 1 .\) Also change the labeling in the proposition and
- the claim to \(k\) to be more consistent with the labeling in the proof.
-Add Figure 10.7 to the proof of Proposition 10.3.2.
-Add reference to the new Tonelli/Fubini for sums exercise in
- Proposition 10.3.4, as really that's really the best way
- to justify the double series.
-Add a note that the example in 10.3.5 is uncountable if \(n \geq 2 .\)
-Be a little bit more precise in the end of Example 10.3.6 where we fix
- the erratum and so we add \(m^*([a,b]) \geq b-a \) as a conclusion
- of the argument.
-Add Figure 10.8 to Example 10.3.6.
-Add a couple of short in-text examples to 10.4 to illustrate the
- definition of oscillation and include a figure with the graph of
- \(\sin(\frac{1}{x})\) which is half of a figure from 3.1.
-Rework the proof of 10.4.3 (Riemann--Lebesgue) a little bit adding some
- detail. At one point it says unnecessarily \(M_j-m_j < 2 \epsilon\)
- instead of \(\epsilon\) so just say \(M_j-m_j < \epsilon.\)
- Also say a bit more about the estimate of volumes of
- \(R_{q+1},\ldots,R_p .\)
-Add alternate name Lebesgue--Vitali to the theorem. Not sure which is
- the more correct. Also add reference to the new Riemann--Lebesgue exercise
- from volume I to note that the naming may be confusing.
-Add Figure 10.12 to the proof of 10.4.3.
-In Exercise 10.4.1, add "(bounded)" after "anything".
- The definition of the integral won't make sense if the
- arbitrary boundary values are not bounded, and hopefully this makes
- the problem clearer and avoids students thinking about an unrelated
- technicality.
-Add exercises 10.4.12 and 10.4.13
-Add definition of "for almost every" and "almost everywhere"
- in section 10.5.
-Add Proposition 10.5.6 and 10.5.7 on basic properties
- of integration over Jordan measurable sets with proofs left as new
- exercises 10.5.8 and 10.5.9.
-Add Proposition 10.5.8, which is just the Exercise 10.5.3
- for integrating over type I domains in the plane. Since we use it later
- and it is generally useful, it is better to state it as a proposition.
-Add Figure 10.13 to Proposition 10.5.8.
-The old Proposition 10.5.6 on images of Jordan measurable sets
- becomes Proposition 10.5.9.
-Add Example 10.6.5, which renumbers the following example which is now
- 10.6.6. This is an example of the vortex vector field and
- how to apply Green's to a more complicated domain where we need to
- cut things up.
-Say a little about where harmonic functions come up and namedrop the
- Laplace equation.
-The proof of Theorem 11.2.4 didn't explicitly mention why
- the convergence is uniform, just assume the reader would notice.
- Add an argument.
-Proposition 11.2.7 needs to assume that \(Y\) is Cauchy-complete
- (erratum), then perhaps it is not completely clear why 11.2.8 is a
- corollary since we do not assume completeness, so add a note on why.
-In Proposition 11.3.1 say the series converges absolutely when
- \(\rho=\infty .\) That's what we prove anyway.
-Move the sentence defining the radius of convergence and the figure up
- in front of the proof, it might make more sense that way.
-Be more consistent with indices in subsection 11.3.4, replace \(j\) with \(m\) in
- the first part to be consistent with the second part. Also add
- comma between the indices as in some of the other places.
-Similarly for Exercise 11.3.1.
-Rename section 11.4 "Complex exponential and trigonometric functions"
- (remove the definite articles for simplicity)
-Rename section 11.5 "Maximum principle and the fundamental theorem of
- algebra" as we now talk about the maximum principle a bit more
- explicitly in text.
-State Lemma 11.5.1 more precisely and also state it for power series
- though still leaving the power series proof as an exercise.
-Make Remark 11.5.2 (maximum principle) into a theorem (still leave its
- proof as an exercise)
-In Corollary 11.7.2, add \(C([a,b],\mathbb{R})\) to the statement. It
- is clear from the proof, but it might be good to emphasize.
-Move Corollary 11.7.4 up a couple of paragraphs into the subsection
- 11.7.1 where it makes more sense.
-In Example 11.7.8, give any sort of set \(X\) for the polynomials,
- no need to restrict ourselves to \(\mathbb{R} \) only.
-In Example 11.7.9, emphasize that the algebra vanishes at no point.
-Expand Example 11.7.10, and add some commentary to introduce the theorem.
-Add figure to proof of Stone-Weierstrass to make the idea more
- transparent, and add a little bit more commentary to the proof.
-Add exercise 11.7.14.
-Make indices in the Fourier series section 11.8 more consistent.
-Move the definition of "symmetric partial sum" earlier to where we say
- we will sum the series like that.
-In the Fourier series section, add Example 11.8.3, computing and
- plotting the Fourier series for the heaviside function and a
- continuous saw function, which renumbers the rest of the examples,
- propositions and theorems in 11.8.
-In the figures of Fourier series, mark the x axis using multiples of pi.
-In Exercise 11.8.1, use \(k\) instead of \(n\) and add a remark in the
- footnote about what the coefficients are, to emphasize the \(n\)th
- coefficient is either zero or \(\frac{1}{n} .\)
-Add Exercise 11.8.13 to prove that continuous functions have no "minimum
- rate of decay" of the coefficients.