Course Details | Important Documents |
Course Outline
Instructor: Patrick Boily, uOttawa Questions related to the course content should be asked on the course’s Piazza forum (that space is shared with the other section). Section-specific communications with the instructor should take place on the course’s Slack workspace (the registration link is available in the course kick-off email sent on January 9 and in the “Announcements” section on Brightspace). Exercise Sessions Schedule: Exercise Sessions Zoom link Important Dates: Note: the assignments must be typeset using LaTeX and uploaded as a single PDF document to Brightspace. |
A. Savage’s Course Notes
Textbooks: Course Documents: Assignments: Some of the video lectures are available without a soundtrack here. In case of divergence, the videos on the website (this page) take precedence. State of the course address, after the first two months (30:00) |
1. Preamble | 2. The Real Numbers |
Topics: Classical problems; calculus as an informal collection of solutions; the need for formalism. How to set up LaTeX for assignments. Video Lectures (0:37:33) |
Topics: Field and order axioms; mathematical induction; Archimedean property; lower/upper bound; infimum/supremum; completeness axiom; constructing and finding reals. Useful inequalities: triangle; Bernoulli; Cauchy-Schwartz. Finite sets; infinite sets; countable sets; uncountable sets; infinite subsets; countability of Q; uncountability of [0,1]; uncountability of R. Nested intervals theorem. Video Lectures (2:41:25) Problem Set Exercises: Q1-Q18 |
3. Sequences | 4. Limits and Continuity |
Topics: Sequences; limit; boundedness. Arithmetic of limits; squeeze theorem for limits. Monotone sequences; bounded monotone convergence theorem. Subsequences; subsequences of convergent sequences; Bolzano-Weierstrass theorem; boundedness and convergence of subsequences. Cauchy sequences; boundedness of Cauchy sequences; equivalence of convergence and Cauchy sequences in R. Euclidean spaces; norms; sequences and limits in R^d. Topology of R^d; open balls; complements; open sets; closed sets; interior points; isolated points; accumulation points; boundary points. Compactness; open cover; Heine-Borel theorem; sequential compactness. Video Lectures (3:39:54) Problem Set Exercises: Q19-Q44 |
Topics: Limit points in R (reprise); link between limit points and convergent sequences; limit of a function; uniqueness of limits; sequential definition of limits. Properties of limits; squeeze theorem for functions. Continuous functions; continuity and elementary operations; composition of continuous functions; continuous image of a compact set, uniform continuity. Intermediate value theorem; fixed point theorem; Max/Min theorem; continuous image of a closed and bounded interval. Video Lectures (2:15:43) Problem Set Exercises: Q45-Q65 |
5. Calculus | 6. Sequences of Functions |
Topics: Definition of the derivative; differentiability and continuity; differentiation rules; Carathéodory’s theorem; chain rule. Relative extrema; Rolle’s theorem; mean value theorem; functions with constant derivatives. Higher-order derivatives; Taylor’s theorem. Monotone functions; 1st derivative test; change of sign of the derivative; Darboux’ theorem. Integrals as areas; integral of a continuous function over a closed and bounded interval; partitions and refinements; lower/upper sums; Riemann-integrable functions; conventions; Riemann’s criterion; familiar formulas. Riemann-integrability of monotone functions. Properties of Riemann integrals: additivity; linearity; triangle inequality; comparisons; mean value theorem for integrals; Riemann-integrable fonction with uncountable discontinuities. Composition theorem. 1st fundamental theorem of calculus; 2nd fundamental theorem of calculus. Integration by parts; 1st substitution theorem; 2nd substition theorem; squeeze theorem for integrals. Video Lectures (4:18:56) Problem Set Exercises: Q66-Q87 |
Topics: Sequences of functions; pointwise convergence; uniform convergence; Cauchy property for sequences of functions. Limit interchange theorems; sequences of uniformly continuous functions; sequences of differentiable functions; sequences of Riemann-integrable functions. Video Lectures (1:10:24) Problem Set Exercises: Q88-Q92 |
7. Series | Exercise Sessions |
Topics: Series of real numbers; Cauchy criterion for series; comparison test; alternating series test; ratio test; root test; absolute convergence; rearrangement. Series of functions; convergence; Cauchy criterion; Weierstrass M-test; Abel’s test. Power series; radius of convergence; Fourier series. Video Lectures (TBD) Problem Set Exercises: Q93-Q100 |
Jan 15: Q2-Q4 + Archimedean Property Variants (1:02:39), [PDF] Jan 20: Q5-Q7 (0:45:58), [PDF (including Q8)] Jan 22: Q10-Q12 (0:51:31), [PDF] Jan 27: Q13-Q15 (0:45:14) [PDF] Jan 29: Q16-Q18, Q20 (0:43:39) [PDF (including Q19)] Feb 03: Q21-Q23, Q26 (0:57:59) [PDF (including Q24, Q25)] Feb 05: Q27-Q30 (0:53:11) [PDF] Feb 10: Q31-Q34 (1:02:31) [PDF] Feb 12: Q35-Q37 (42:39) [PDF] Feb 24: Q38-Q43 (1:11:43) [PDF] Feb 26: Q44-Q47 (1:06:50) [PDF (including Q48)] Mar 03: Q49-Q53 (no video today) [PDF] Mar 05: Q55-Q59 (1:02:09) [PDF (including Q54, Q60)] Mar 10: Q61-Q65 (00:00) Mar 12-Mar 18: Q66-Q72 (00:00) Mar 19-Mar 25: Q73-Q79 (00:00) Mar 26-Apr 01: Q80-Q87 (00:00) Apr 02-Apr 08: Q88-Q92 (00:00) Apr 09-Apr 15: Q93-Q100 (00:00) |