There are altogether five courses which we should look at.
Calculus, Advanced Calculus I, Advanced Calculus II, Analysis I, and Analysis on Metric Spaces.
For Calculus, Yan Loi and Wen Yin are currently working on course notes using Stewart's Calculus. There are not much to be changed in this first course.
I have a suggestion:
1. Include introduction of sequences and test of convergence. Students will see these again in Second Year Advanced Calculus II but they will see the proof of some of these tests.
For Advanced Calculus I, (Course contents will be entirely on Calculus of several variables, following second part of Stewart's Calculus). Yan Loi's excellent notes will be useful to anyone who might teach the course in future.
For more details of his notes, click here for Part I and click here for Part II . Victor will be teaching this next semester.
Current Syllabus for MA1102R :
Calculus. I split this into the following parts:
Part I : Real Numbers, Inequalities, rational and irrational numbers, absolute value, triangle inequality, functions and inverses, trigonometric functions.
Part II: Continuity, bounded functions
Part III: Differentiations, derivative, chain rule, implicit differentiation, Differentiation of inverse functions, Newton's Method
Part IV: Mean Value Theorem, Rolle's Theorem, Taylor's formula, L'Hopital's rule.
Part V: Integration: Definition and basic properties of Riemann integral, fundamental theorem of calculus, natural log and exponential, basic methods of integration, applications.
Comments and suggestions:
Newton's Method be removed. Insert sequences and series tests somewhere. Some other topics can be mentioned quickly since these are covered in A level.
From Wu Jie's course notes, it is clear that most of the things are done in a rigorous manner. This is similar to the old Analysis I course. Some of the things here should not be repeated in Analysis I in third year.
I feel that we should concentrate on Advanced Calculus II and Analysis I.
Analysis I has been moved to third year this coming semester. I feel that this is a wrong move. The topics in this course is overwhelming. It will be a very tough third year course to pass.
I currently have a copy of Soo Teck's notes on Analysis. Soo Teck feels that some students (who are not extremely strong) are able to appreciate the mathematical rigor offered by this course.
For those who are interested to take a look at Soo Teck's notes, please drop me a note.
Part of Analysis I should be moved to Advanced Calculus II. This will free up some time for the lecturer to concentrate on Integration.
Comments and suggestions :
PLAN A.
Remove Advanced Calculus II and Analysis I.
Introduce two new course code Mathematical Analysis I and II.
Mathematical Analysis I can be introduced in second year and II in third year (although I prefer to see this in second year).
Follow closely the text used by Soo Teck,
Reference : ``Introduction to Real Analysis'' by Bartle and Sherbert.
Mathematical Analysis I : Chapters 1 to 6 of Bartle and Sherbert.
Completeness property of R, Applications of the Supremum property, Sequences and their limits, Limit Theorems, Monotone sequences, Subsequences and the Bolzano-Weierstrass Theorem, The Cauchy Criterion, Limit of Functions, Limit Theorems, Continuous functions, Uniform Continuity, Derivative, Mean Value Theorem, L'Hopital's Rules, Taylor's Theorem..
Mathematical Analysis II : Chapters 7 to 9. (Chapters 10 and 11 if time permits).
Riemann Integral, Riemann integrable functions, fundamental theorem, Pointwise and uniform convergence, interchange of limits, absolute convergence, tests for convergence, series functions.
PLAN B.
Keep Advanced Calculus II and Analysis I but reshuffle the topics.
Currently one of the problems is that in Advanced Calculus II and Analysis I, the chapter on sequences is repeated. Wu Jie did a good job in designing Advanced Calculus II, with emphasis on rigor (This contradicts our first sentence in the syllabus of our Analysis I, which starts with `` This is a first rigorous course in Analysis...''). However, Wu Jie did mention about interchange of integral and sum, differentiation and sum. It would be strange because students would not have encountered uniform convergence and rigorous treatment of integral and differentiation at this stage (see Calculus and Advanced Calculus I). It appears that Wu Jie's course should come more appropriately after Soo Teck's Analysis I. In other words, Soo Teck's course is more in tune with the above proposed Mathematical Analysis I and Wu Jie's course is closer to the above proposed Mathematical Analysis II.
Proposal: Keep the old codes, reshuffle the two courses. Allow Soo Teck's course to stay in second year. Move Wu Jie's course to third year.
Proposed new syllabus for Analysis on Metric Spaces (By Denny Leung)
Convergent sequences. Continuity in terms of convergent sequences. Convergent sequences of functions. Uniform convergence.
Definition and basic properties of open & closed ball/sets. Interior, closure, boundary, accumulation points. Continuity in terms of open and closed sets.
(This part to me embodies the spirit of analysis. Anyone who understands this can learn the rest of the module on her own if necessary.)
Cauchy sequences and definition of completeness. Examples of complete spaces. Nested set property. (Contraction Mapping Theorem. Baire Category Theorem – these two optional).
Defintion of connected spaces. Examples of connected spaces. Continuous image of connected sets. Intermediate value theorem.
Equivalence of sequential compactness, Bolzano-Weierstrass property and completeness+total boundedness. Heine-Borel Theorem. Compactness. Equivalence of compactness & sequential compactness. Properties of compact sets. Continuous image of compact sets. Extreme value theorem. Uniform continuity.
In general, I think that the course should contain a lot of examples to illustrate the concepts. I would even sacrifice some of the topics in favor of examples if there’s not enough time to do everything.