MATH 53353, Analytical Perspectives on the Calculus: Course Outline and Readings

Week 1: The Cultural Shift From Secondary to College Mathematics

Monday Elementary examples

Tuesday The modern mathematical perspective on the calculus.
Reading for Tuesday: Frank Quinn, "A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today," Notices of the AMS , January 2012.
Available at: http://www.ams.org/notices/201201/

Wednesday Limits from intuitive and rigorous perspectives.
Reading for Wednesday: W.A. Granville, Elements of the Differential and Integral Calculus , pp. 19-20
Available at: https://archive.org/details/elementsdiffere02goog
Walter Rudin, Principles of Mathematical Analysis , pp. 36-38 (to 3.8), and pp. 64-65.
Gilbert Strang, Calculus , pp. 78-83
Available at: http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/

Thursday Necessary background for cultural transition: trigonometry and the modern understanding of functions.
Reading for Thursday: MacTutor History of Mathematics Archive: The function concept
Available at: http://www-history.mcs.st-and.ac.uk/HistTopics/Functions.html


Link: Homework 1

From this point on, references to Rudin will correspond to the third edition.

Week 2: The Hidden Calculus of the Real Numbers

Monday HW1 DUE. Classes of infinity. The Continuum.
Viewing for Monday: Jeff Dekofsky, ``The Infinite Hotel Paradox.''
Available at: http://ed.ted.com/lessons/the-infinite-hotel-paradox-jeff-dekofsky
Reading for Monday: Walter Rudin, Principles of Mathematical Analysis , pp. 23-30 (to Metric Spaces).

Tuesday Basic Set Theory. DeMorgan's Laws.
Reading for Tuesday: Walter Rudin, Principles of Mathematical Analysis , pp. 30-36 (to Compact Sets).

Wednesday Dedekind cuts. Transcendental numbers.
Reading for Wednesday: John O'Connor, A First Analysis Course , Supplement 3: Dedekind Cuts.
Available at: http://www-history.mcs.st-and.ac.uk/~john/analysis/Lectures/A3.html
Walter Rudin, Principles of Mathematical Analysis , Appendix, pp. 17-21.

Thursday Limit points and limits, Construction of the real numbers via sequences.
Reading for Thursday: Walter Rudin, Principles of Mathematical Analysis , Cauchy Sequences, pp. 52-55.

Link: Homework 2

Week 3: Perspectives on Continuity

Monday HW2 DUE. Metric Topology.
Reading for Monday: John O'Connor, A First Analysis Course , Chapter 15: Metric Spaces.
Available at: http://www-history.mcs.st-and.ac.uk/~john/analysis/Lectures/L15.html
Suggested reading: Wikibooks: Topology/Metric Spaces (to, but not including, Convergence)
Available at: https://en.wikibooks.org/wiki/Topology/Metric_Spaces

Tuesday Open Sets. Continuity. Uniform continuity.
Reading for Tuesday: Walter Rudin, Principles of Mathematical Analysis , Limits of Functions and Continuous Functions, pp. 83-89.
Math Tutor (Czech Technical University, Prague), Uniform Continuity.
Available at: http://math.feld.cvut.cz/mt/txtb/3/txe3ba3g.htm

Wednesday Pointwise and uniform convergence of sequences of functions.
Reading for Wednesday: Walter Rudin, Principles of Mathematical Analysis , pp. 143-150 (to Theorem 7.12).

Thursday Illustrative examples.
Viewing for Thursday: Unit VII: Lec 5 | MIT Calculus Revisited: Single Variable Calculus
Available at: https://www.youtube.com/watch?v=iM4DRgFqPso

Link: Homework 3

Week 4: Perspectives on the Differential Calculus

Monday HW3 DUE. Discussion.

Tuesday Midterm Exam.

Wednesday Derivatives and tangent lines.
Reading for Wednesday: Walter Rudin, Principles of Mathematical Analysis , pp. 103-110.


Thursday Limits of ratios and infinitessimals. A Pathological example.
Reading for Thursday: Walter Rudin, Principles of Mathematical Analysis , pp. 152-154.
Tevian Dray, Corinne A. Manogue, "Putting Differentials Back Into Calculus", College Mathematics Journal , March 2010.
Available at: http://math.oregonstate.edu/bridge/papers/

Link: Homework 4
Courtesy of Project Gutenberg, S.P. Thompson's Calculus Made Easy.

Week 5: Perspectives on the Integral Calculus

Monday HW4 DUE. Discussion.

Tuesday The Method of Exhaustion and Developments to the 17th century
Reading for Tuesday: Luke Mastin, "Hellenistic Mathematics-- Archimedes"
Available at: http://www.storyofmathematics.com/hellenistic_archimedes.html
Muharrem Aktümen, Ahmet Kaçar, "Maplets for the Area of the Unit Circle: 1. The Method Using Regular Polygons", Journal of Online Mathematics and Its Applications , v. 7, 2007.
Available at: http://www.maa.org/external_archive/joma/Volume7/Aktumen/Polygon.html
T.L. Heath, translator, The Works of Archimedes , "The Quadrature of the Parabola," pp. 233-252.
Available at: https://archive.org/details/worksofarchimede029517mbp


Wednesday 17th century developments; The Fundamental Theorem, Newton, and Leibniz
Reading for Wednesday: Murray Bourne, "What did Newton originally say about Integration?", squareCircleZ, the intMath blog , July 22, 2010.
Available at: http://www.intmath.com/blog/mathematics/what-did-newton-originally-say-about-integration-4878

Thursday Cauchy, Riemann, and Darboux
Reading for Thursday: Walter Rudin, Principles of Mathematical Analysis , pp. 120-127.
Wikipedia, "The Darboux Integral."
Available at: https://en.wikipedia.org/wiki/Darboux_integral

Link: Homework 5

Week 6: The Fundamental Theorem

Monday HW5 DUE. Discussion.

Tuesday The Fundamental Theorem of Calculus.
Reading for Tuesday: Walter Rudin, Principles of Mathematical Analysis , pp. 133-134.


Wednesday Integral counterparts to the rules of differentiation. Discussion.

Thursday Final Exam.