Riemann on Riemann sums

November 16, 2013

Though Riemann sums had been considered earlier, at least in particular cases (for example, by Cauchy), the general version we consider today was introduced by Riemann, when studying problems related to trigonometric series, in his paper Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe. This was his Habilitationsschrift, from 1854, published posthumously in 1868.

Riemann’s papers (in German) have been made available by the Electronic Library of Mathematics, see here. The text in question appears in section 4, Ueber den Begriff eines bestimmten Integrals und den Umfang seiner Gültigkeit. The translation below is as in

  • A source book in classical analysis. Edited by Garrett Birkhoff. With the assistance of Uta Merzbach. Harvard University Press, Cambridge, Mass., 1973. MR0469612 (57 #9395).

Also zuerst: Was hat man unter \displaystyle \int_a^b f(x) \, dx zu verstehen?

Um dieses festzusetzen, nehmen wir zwischen a und b der Grösse nach auf einander folgend, eine Reihe von Werthen x_1, x_2,\ldots, x_{n-1} an und bezeichnen der Kürze wegen x_1 - a durch \delta_1, x_2 - x_1 durch \delta_2,\ldots, b - x_{n-1} durch \delta_n und durch \varepsilon einen positiven ächten Bruch.  Es wird alsdann der Werth der Summe

\displaystyle S = \delta_1 f(a + \varepsilon_1 \delta_1) + \delta_2 f(x_1 + \varepsilon_2 \delta_2) + \delta_3 f(x_2 + \varepsilon_3 \delta_3) +\cdots \displaystyle +\delta_n f(x_{n-1} +\varepsilon_n \delta_n)

von der Wahl der Intervalle \delta und der Grössen \varepsilon abhängen.  Hat sie nun die Eigenschaft, wie auch \delta und \varepsilon gewählt werden mögen, sich einer festen Grenze A unendlich zu nähern, sobald sämmtliche \delta unendlich klein werden, so heisst dieser Werth \displaystyle \int_a^b f(x) \, dx.

In Birkhoff’s book:

First of all: What is to be understood by \displaystyle \int_a^b f(x)\,dx?

In order to establish this, we take the sequence of values x_1,x_2,\ldots, x_{n-1} lying between a and b and ordered by size, and, for brevity, denote x_1 - a by \delta_1, x_2 - x_1 by \delta_2,\ldots, b - x_{n-1} by \delta_n, and proper positive fractions by \varepsilon_i. Then the value of the sum

\displaystyle S = \delta_1 f(a + \varepsilon_1 \delta_1) + \delta_2 f(x_1 + \varepsilon_2 \delta_2) + \delta_3 f(x_2 + \varepsilon_3 \delta_3) +\cdots \displaystyle +\delta_n f(x_{n-1} +\varepsilon_n \delta_n)

will depend on the choice of the intervals \delta_i and the quantities \varepsilon_i. If it has the property that, however the \delta_i and the \varepsilon_i may be chosen, it tends to a fixed limit A as soon as all the \delta_i become infinitely small, then this value is called \displaystyle \int_a^b f(x) \, dx.

(Of  course, in modern presentations, we use \Delta_i instead of \delta_i, and say that the \delta_i approach 0 rather than become infinitely small. In fact, we tend to call the collection of data x_1,\dots,x_{n-1}, \varepsilon_1,\dots,\varepsilon_n a tagged partition of {}[a,b], and call the maximum of the x_{i+1}-x_i the mesh or norm of the partition.)

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AlgoRythmics

November 15, 2013

This link should take you to the YouTube channel of Algo-rythmics, or see their website.

Different sorting algorithms (bubble sort, insertion sort, quicksort, selection sort, shell sort) illustrated through folk dance.


Credit

November 5, 2013

L'Hospital

I recognize I owe much to Messrs. Bernoulli’s insights, above all to the young, currently a professor in Groningue. I did unceremoniously use their discoveries, as well as those of Mr. Leibniz. For this reason I consent that they claim as much credit as they please, and will content myself with what they will agree to leave me.

L’Hôpital, in the preface (page xiv) of his Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes (1696), the first calculus textbook, published anonymously. (A posthumous second edition, from 1716, identifies L’Hôpital as the author.)

L'Hospital-2 Read the rest of this entry »


Office Hours

October 28, 2013

This week, I will not be able to hold office hours on Thursday (my son does not have school on Hallowe’en). Instead, they will be on Wednesday, 3-4:30 pm.

2013-08-11 11.41.23


170 – Extra credit

September 20, 2013

Choose a mathematician whose work is somehow related to calculus. The connection may be loose (of course, Newton or Leibniz qualify, but so does John Nash). To avoid repetitions, email me your choice, and I’ll let you know whether your choice has not yet been claimed.

Write (well, type) an essay on their life and mathematical work. It may be short. Make sure to follow reasonable standards of style when citing references, etc. Due December 2 (after Thanksgiving break).


170 – Hippasus of Metapontum

September 16, 2013

Erroll Morris ran a series of essays on the New York Times a couple of years ago, on the topic of incommesurability. The whole series is highly recommended. You may particularly enjoy reading Part III, on Hippassus of Metapontum, the mythical Pythagorean who proved the irrationality of square root of two, and was killed by the other followers of the cult as a result.

The legend is told in many places; Morris lists a few in his essay. I first found it as an appendix to Carl Sagan‘s book version of Cosmos.


170 – Calculus I (Honors). Syllabus

August 25, 2013

Math 170 Section 8: Calculus I. Honors.

Instructor: Andres Caicedo.

Contact Information: See here.
Time: MWF 1:30-2:45 pm.
Place: Mathematics building, Room 139.
Office Hours: (I expect they will be on) Th 1:30-3:00 pm, or by appointment (email me a few times/dates you have available).

Text:

  1. Calculus, Whitman College (David Guichard and others). The text is distributed under a Creative Commons license. It can be downloaded from Whitman’s page.
  2. We will complement with exercises from Schaum’s Outlines Calculus, Frank Ayres Jr and Elliot Mendelson, Sixth Edition, McGraw Hill, 2012.
  3. There will be additional material, in the form of extracts from other books, articles from mathematical magazines, discussions online, and some group exercises from Applications of Calculus, Philip Straffin, ed., Resources for Calculus Collection, Vol. 3, MAA notes, Vol. 29, 1993; these will be provided as needed.

Please make sure each day you bring to class a copy of the pages that are being covered from the online text, according to the list below; this may be an electronic copy. (To be safe, bring at least the pages corresponding to the sections for that day, for the previous day, and for the next day, since I may cover a bit more than what is scheduled, and we may need to refer back to recently covered material.)

Contents: The department’s course description reads:

Definitions of limit, derivative and integral. Computation of the derivative, including logarithmic, exponential and trigonometric functions. Applications of the derivative, approximations, optimization, mean value theorem. Fundamental Theorem of Calculus, brief introduction to applications of the integral and to computations of antiderivatives.

Our emphasis is on understanding the theory, but we will also cover some applications. Roughly, I expect we will see Sections 1.1-8.1, 9.1-9.2. This list includes somewhat more than strictly required, giving us a little additional time in case we want or need to emphasize some topics. Please bookmark this post. I update it frequently with detailed week-to-week descriptions.

Detailed day to day description and homework assignments. All problems are from the Whitman calculus book unless otherwise explicitly specified:

  • August 26. Sections 1.1, 1.2. Homework: 1.1: 3, 6, 9, 12. 1.2: 1(d,e,f), 2(d,e,f), 5.
  • August 28. Sections 1.2, 1.3. Homework: 1.3: 4, 9, 14, 16.
  • August 30. Sections 1.3, 1.4. Homework: 1.4: 7, 11, 17, 19. Quiz 1, on sections 1.1-1.3.

The first homework set is due Wednesday, September 4. It consists of all the problems listed above. See below for the required format.

  • September 4. Sections 2.1, 2.2. Homework 1 is due today. Homework: 2.1: 1, 3, 4, 7. [The following problems from the Schaum’s book are strongly suggested, but not required: Chapter 1 (Supplementary problems): 10-16; Ch. 9 (Solved problems): 1-10; (Supplementary problems): 14-17.]
  • September 6. Sections 2.2, 2.3. Homework: 2.2: 2, 3. 2.3: Homework: 2, 5, 11-15, 17, 18. [The following problems from the Schaum’s book are strongly suggested, but not required: Ch. 7 (Solved): 9-12; (Suppl.): 16, 17, 22, 23.] Quiz 2.

The second homework set is due Monday, September 9. It consists of all the problems from the Whitman book listed on September 4 and 6. See below for the required format.

  • September 9. Section 2.3 (continued). Homework 2 is due today.
  • September 11. Section 2.4. Homework: 2, 4, 7 [Suggested problems from Schaum’s: Ch. 9 (Solved): 11; (Suppl.): 18, 19, 22-24.]
  • September 13. Section 2.5. Quiz 3, on Sections 2.3, 2.4. Homework: 3, 5, 7 [Suggested problems from Schaum’s: Ch. 8 (Solved): 2; (Suppl.): 4-8.]

The third homework set is due Monday, September 16. It consists of all the problems from the Whitman book listed on September 11 and 13. See below for the required format.

  • September 16. Sections 3.1-3.3. Errol Morris‘s essay on Hypassus of Metapontum. Homework 3 is due today. Homework: 3.1: 3, 4, 6. 3.2: 3-5, 8, 10. 3.3: 2, 4, 5. [Suggested problems from Schaum’s: Ch. 10 (Solved): 1, 2(4,6), 5-9; (Suppl.): 27-33, 43.]
  • September 18. Sections 3.4, 3.5. Homework: 3.4: 2, 4, 6, 9. [Suggested problems from Schaum’s: Ch. 10 (Solved): 2(7); (Suppl.): 35, 44.]
  • September 20. Group activity 1. Due Wednesday, September 25. Arbitrating disputes, by Philip Straffin, pp. 7-21 of Applications of Calculus. Extra credit project posted, due December 2.

The fourth homework set is due Monday, September 23. It consists of all problems from the Whitman book listed on September 16 and 18. See below for the required format.

  • September 23. Homework 4 is due today. Sections 4.1-4.3, and Schaum’s chapter 16. Review of basic trigonometric identities. Homework: Section 4.1: 3, 9, 11. [Suggested problems from Schaum’s: Ch. 16 (Solved): 1-7, 11-12; (Suppl.): 1-22. Ch. 17 (Solved): 1; (Suppl.): 21-24, 26.]
  • September 25. Group activity 1 is due today. Sections 4.3-4.5. Homework: Section 4.3: 3, 5, 6. Section 4.4: 2, 3, 4.
  • September 27. Midterm 1. It covers the material from the first 3 chapters of the book, with emphasis on Chapters 2 and 3.

The fifth homework set is due Monday, September 30. It consists of all problems from the Whitman book listed on September 23 and 25. See below for the required format.

The sixth homework set is due Monday, October 7. It consists of all problems from the Whitman book listed on September 30 and October 2. See below for the required format.

  • October 7. Sections 4.8, 4.9. Celestial mechanics and Kepler’s equation. Kepler’s laws of planetary motion state that the orbits of celestial bodies are ellipses; the prevailing theory was that orbits were perfect circles, this was based on the old theory of epicycles that goes back to Ptolemy, when it was still believed that the Earth was the center of the universe. Here is an illustration of how useless as an explanatory theory this is. Homework: Section 4.7: 2, 6, 10, 12, 16. Section 4.8: 2, 5, 6, 11, 14. [Suggested problems from Schaum’s: Ch. 25 (Suppl.): 8, 10, 11, 12, 15; Ch. 26 (Solved): 1, 2, 4, 6, 7; (Suppl.): 8, 11, 17-20.]
  • October 9. Sections 4.9, 4.10. L’Hôpital controversy (see also here). Homework: Section 4.9: 2, 5, 6. Section 4.10: 2, 5, 7, 14, 20. [Suggested problems from Schaum’s: Ch. 7 (Solved): 3, 7 (Suppl.): 17, 18; Ch. 27 (Solved): 1-4; (Suppl.): 7, 10.]
  • October 11. Quiz 5. Sections 4.10, 4.11. Hyperbolic functions.

The seventh homework set is due Monday, October 14. It consists of all problems from the Whitman book listed on October 7 and 9. See below for the required format.

  • October 14. Sections 5.1, 5.2. Homework: Section 5.1: 4, 6, 8, 10, 16. Section 5.2: 4, 6, 8, 10, 14. [Suggested problems from Schaum’s: Ch. 14 (Solved): 1, 4, 5, 7, 8; (Suppl.): 23, 26.]
  • October 16. Sections 5.3, 5.4, 5.5. Homework: Section 5.3: 2, 4, 6, 8, 9, 12. Section 5.4: 2, 4, 6, 8. [Suggested problems from Schaum’s: Ch. 14 (Solved): 2, 3, 6, 9; (Suppl.): 24. Ch. 15 (Solved): 1-8; (Suppl.): 11, 12; Ch. 17 (Solved): 14; (Suppl.): 29.]
  • October 18. Group activity 2. Due Wednesday, October 23. Somewhere within the rainbow, by Steven Janke, pp. 42-53 of Applications of Calculus.

The eight homework set is due Monday, October 21. It consists of all problems from the Whitman book listed on October 14 and 16. See below for the required format.

  • October 21. Section 6.1. Richard Feynman on the Scientific method. Homework: Section 5.5: 2, 7, 10, 15, 19, 23, 30, 32. [Suggested problems from Schaum’s: Ch. 15 (Solved): 9-10; (Suppl.): 13; Ch. 17 (Solved): 15.]
  • October 23. Group activity 2 is due today. Section 6.1.
  • October 25. Quiz 6. Section 6.1. Homework: 6.1: 3, 5, 8, 11, 15. [Suggested problems from Schaum’s: Ch. 14 (Solved): 1, 11, 12, 14, 18; (Suppl.): 26-30.]

The ninth homework set is due Monday, October 28. It consists of all problems from the Whitman book listed on October 21 and 25. See below for the required format.

  • October 28. Section 6.2. Homework: 6.2: 5, 6, 9, 15, 17, 19. [Suggested problems from Schaum’s: Ch. 20: All problems.]
  • October 30. Sections 6.2, 6.3. Homework: 6.3: 2, 3.
  • November 1. Midterm 2. It covers the material from the book up to and including section 6.2. Emphasis on the material from Chapter 4 on.

The tenth homework set is due Monday, November 4. It consists of all problems from the Whitman book listed on October 28 and 30. See below for the required format.

The eleventh homework set is due Monday, November 11. It consists of all problems from the Whitman book listed on November 4 and 6. See below for the required format.

  • November 11. Section 7.1. Riemann sums. Homework: 7.1: 2, 4, 6, 8.
  • November 13. Section 7.2. “The birth of the calculus,” produced by The Open University, and narrated by Jeremy Gray.
  • November 15. Group activity 3. Due Wednesday, November 20. Speedy sorting, by Steven Janke, pp. 223-238 of Applications of Calculus. A reason why we care about “fast” counting/sorting techniques, is usually called combinatorial explosion, see here. An illustration of sorting algorithms through folk dance. A visualization of 15 sorting algorithms.

The twelfth homework set is due Monday, November 18. It consists of all problems from the Whitman book listed on November 11. See below for the required format.

  • November 18. Section 7.2. Homework: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. [Suggested problems from Schaum’s: Ch. 24 (Solved): 1-5; (Suppl.): 8-26, 29, 31.]
  • November 20. Group activity 3 is due today. Section 7.3. Homework: 2, 3, 5. [Suggested problems from Schaum’s: Ch. 24 (Solved): 6; (Suppl.): 27, 28, 30, 32, 37; Ch. 25 (Solved): 1; (Suppl.): 13, 14, 17.]
  • November 22. Midterm 3. It covers the material from the book up to and including section 7.2. Emphasis on the material from Chapters 6 and 7.

The week of November 25-November 29 is Thanksgiving break (Happy thanksgiving!). The thirteenth homework set is due Monday, December 2. It consists of all problems from the Whitman book listed on November 18 and 20. The extra credit project is also due December 2.

  • December 2. Section 8.1. Homework: Section 8.1: 2, 6, 7, 10, 16, 18. [Suggested problems from Schaum’s: Ch. 24 (Suppl.): 34; Ch. 25 (Solved): 5; (Suppl.): 9; Ch. 26 (Suppl.): 9.]
  • December 4. Section 9.1. Mechanical devices to measure areas: A “tannery mechanical surface integrator“, and a Planimeter. Fermat’s method of quadratures. (Adequality.) (The link to Fermat’s method is to a write up by Fred Rickey. Other interesting short notes on his page can be found here. Translation of Fermat’s papers, including his paper on quadratures, can be found at a curious website, here.) Homework: Section 9.1: 3, 4, 8, 11.
  • December 6. Quiz 8. Section 9.2. Terminal velocity, Felix Baumgartner. Supplement: Falling raindrops, by Walter J. Meyer, pp. 101-111 of Applications of Calculus.

The fourteenth homework set is due Monday, December 9. It consists of all problems from the Whitman book listed on December 2 and 4. See below for the required format.

There is no assigned homework set for this week.

Homework: There is weekly homework, due Mondays at the beginning of lecture; you are welcome to turn in your homework early, but I will not accept homework past Mondays at 1:35 pm. The homework consists on the problems assigned during the previous week. It is a good idea to work daily on the homework problems corresponding to the material covered that day. A grader (Blake Oren) will check your homework for completeness, and grade carefully 1 or 2 questions (each homework is graded out of 10 points, completeness is worth 2 points). You should use it as a guide for what material to focus on, and what kind of skills are required from you. It is a very good idea to do all of the assigned homework. During office hours, you are welcome to ask about problems from the assigned sets (or any other problems you find interesting). Frequently, some (but not necessarily all) of the problems from the quizzes will be fairly close, if not outright identical, to homework problems.

Your homework must follow the format developed by the mathematics department at Harvey Mudd College. You will find that format at this link. If you do not use this style, your homework will be graded as 0.

Quizzes: There will be weekly quizzes, on the last 20 minutes of Friday’s lecture. Each quiz will evaluate, roughly, the material covered from Friday to Wednesday. You are not allowed to only show up about 20 minutes before the end of the lecture in order to take the quiz; if you show up only for the quiz, your score is 0. If you fail to take a quiz, it will be scored as 0. There are no make-up quizzes. The lowest score is dropped.

For each quiz, I will provide you with a page with the question(s) printed. You may use this page to solve the questions. You need to bring any additional pieces of paper you may require. Calculators, notes, textbooks are allowed. Most likely I will not have calculators, or pencils, etc, so bring your own.

Exams: There will be 3 in-class exams (dates to follow) and a comprehensive final exam.

  • Midterm 1: Friday, September 27.
  • Midterm 2: Friday, November 1.
  • Midterm 3: Friday, November 22.
  • Final exam: Wednesday, December 18, 2013; 2:30 – 4:30 pm. This exam is cumulative, including material from handouts and group activities.

The grade will be decided based on homework (17% of the total score), quizzes (17%), group work (16%), the three in-class exams (10% each), and a final exam (20%).

I will then grade on a linear scale:

  • If your final score is 90% or higher, you receive an A.
  • If it is between 80 and 89%, you receive a B.
  • If it is between 70 and 79%, you receive a C.
  • If it is between 60 and 69% you receive a D.
  • If it is 59% or lower, you receive an F.

Attendance: Not required, but encouraged. Any material covered in lecture may be used in quizzes and exams, even if it is not discussed in the textbooks. I will use this website to post any additional information, and encourage you to use the comments feature, but (in general) I will not post here standard content covered in the textbooks or in class. If you leave a comment, please use your full name, which will simplify my life filtering spam out.

Please pay attention to the Student Code of Conduct. Feel free to ask me if there are any questions.

I post links to supplementary material on Google+. Circle me and let me know if you are interested, and I’ll add you to my Calculus circle. As with this blog, I encourage you to comment there.

Twitter.