Jacobs University, Fall 2017
Class: Mon 11:15 - 12:30, East Hall 4; Wed 11:15 - 12:30, East Hall 4
Tutorial Session: Wed 19:00 - 20:15, East Hall 3
Syllabus (as of Sep. 1, 2017) available here. (Note that the Syllabus will not be updated, the most recent information can be found on this website.)
This class does not follow one particular textbook, but takes some material from several different ones:
See the Syllabus.
Midterm Exam: Wed, Oct. 18, 2017, 11:15 - 12:30, Lecture Hall Research I (when you come in through the main entrance, go straight to the end of the hallway).
The exam begins at 11:15 sharp, so please come a few minutes earlier to be seated at 11:15. The midterm is about all material up to the Oct. 09 session. You only need to bring a pen, no calculators or notes! Please be aware of the University policy regarding missing exams: Policies for Undergraduate Studies. In particular, in case you are missing an exam due to illness (only with doctor's note) or an emergency, you have to notify me (by email) before the beginning of the exam!
Final Exam: Thu, Dec. 7, 2017, 13:15 - 14:45, Lecture Hall Research III
The exam begins at 13:15 sharp, so please come a few minutes earlier to be seated at 13:15. The final is about all material up to and including the Nov. 29 session, with an emphasis on the material of part II of this course. You only need to bring a pen, no calculators or notes! Please be aware of the University policy regarding missing exams: Policies for Undergraduate Studies. In particular, in case you are missing an exam due to illness (only with doctor's note) or an emergency, you have to notify me (by email) before the beginning of the exam!
Homework Sheets are to be handed in individually at the beginning of class on Mondays. Alternatively, they can be deposited in a mailbox at the main entrance of Research I at some point before class on Monday (right next to the entrance there is a mailbox labeled "Advanced Calculus"). No late hand ins except with a doctor's note!
| Date | Sheet Number | Due Date |
|---|---|---|
| Sep. 11, 2017 | Sheet 1 (new; with slight correction in Problem 3) | Sep. 18, 2017 |
| Sep. 18, 2017 | Sheet 2 (new; with slight correction in Problem 6) | Sep. 25, 2017 |
| Sep. 25, 2017 | Sheet 3 | Oct. 02, 2017 |
| Oct. 02, 2017 | Sheet 4 | Oct. 09, 2017 |
| Oct. 09, 2017 | Sheet 5 | Oct. 16, 2017 |
| Oct. 23, 2017 | Sheet 6 | Nov. 06, 2017 |
| Nov. 6, 2017 | Sheet 7 | Nov. 13, 2017 |
| Nov. 13, 2017 | Sheet 8 | Nov. 20, 2017 |
| Nov. 20, 2017 | Sheet 9 | Nov. 27, 2017 |
| Nov. 27, 2017 | Sheet 10 | Dec. 04, 2017 |
RHB = Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering
HW = Hairer, Wanner - Analysis by its History
(Note that the book references given below offer only a rough orientation. Sometimes, only parts of a particular chapter are covered in class.)
| Date | Topics |
|---|---|
| Sep. 04, 2017 | Polynomials (roots, factorization) RHB 1.1; HW I.1 (Polynomial Functions) |
| Sep. 06, 2017 | Polynomials ctd. (polynomial interpolation); Binomial Expansion RHB 1.5, 1.6; HW I.2 (Binomial Theorem) |
| Sep. 11, 2017 | Limits (sequences, sup, inf, limsup, liminf); Continuity HW III.1 (Convergence of a Sequence), HW III.3 (Continuous Functions) |
| Sep. 13, 2017 | Continuity ctd. (intermediate value and maximum theorem, limits of functions) HW III.3 (Continuous Functions, The Intermediate Value Theorem, The Maximum Theorem, Limit of a Function) |
| Sep. 18, 2017 | Infinite Series (partial sums, difference method, absolute convergence, convergence criteria, Cauchy product) RHB 4.1, 4.2, 4.3, 4.4; HW III.2 (Criteria for Convergence, Absolute Convergence, The Cauchy Product of Two Series) |
| Sep. 20, 2017 | Power Series (radius of convergence, power series of exponential function) RHB 4.5; HW I.2 (Exponential Function), HW III.2 (Exchange of Infinite Series and Limits) |
| Sep. 25, 2017 | Power Series ctd. (Landau symbols); Inverse Functions (logarithm), Complex Numbers RHB 4.5, 3.1-3.5; HW III.3 (Monotone and Inverse Functions), I.3, I.5 |
| Sep. 27, 2017 | Complex Numbers ctd; Basics of Differentiation RHB 3.1-3.5, 2.1.1, 2.1.2; HW I.5, II.1 (The Derivative, Differentiation Rules), III.6 (first section) |
| Oct. 02, 2017 | Differentiation (differentiation rules, examples, differentiation of power series, implicit differentiation and parametric representation) RHB 2.1.1 - 2.1.7; HW II.1, III.6 (first section) |
| Oct. 04, 2017 | Differentiation (theorems of Rolle, Lagrange, Cauchy, L'Hospital) RHB 2.1.10, 4.7; HW III.6 (parts of The Fundamental Theorem of Differential Calculus, and The Rules of de L'Hospital) |
| Oct. 09, 2017 | Differentiation (Taylor expansion, minimization and maximization problems) RHB 2.1.8, 4.6; HW II.2, III.7 (Taylor Series) |
| Oct. 11, 2017 | Differentiation (Newton's method) RHB 27.1.4, 27.2; HW II.2 (towards the end) |
| Oct. 16, 2017 | Review and question session |
| Oct. 18, 2017 | Midterm Exam |
| Oct. 23, 2017 | Integration (Riemann integral, mean-value theorem, Fundamental Theorem of Differential Calculus, integration by inspection) RHB 2.2.1, 2.2.2, 2.2.3, 2.2.5; HW III.5 (only some parts of Definitions and Criteria of Integrability, Integrable Functions, Inequalities and the Mean Value Theorem), III.6 (only parts of The Fundamental Theorem of Differential Calculus), II.4 (Primitives) |
| Oct. 25, 2017 | Integration (substitution, integration by parts, areas, curve length, recurrence relations) RHB 2.2.7, 2.2.8, 2.2.9, 2.2.13 (Finding the length of a curve); HW II.4 (parts of Applications, Integration Techniques) |
| Oct. 30, 2017 | Reading Day, no classes |
| Nov. 01, 2017 | Integration (rational functions and partial fractions, integration of power series, Taylor series, improper integrals) RHB 1.4, 2.2.10; HW II.5 (Integration of Rational Functions), III.7 (Differentiation and Integration, Taylor Series), III.8 |
| Nov. 06, 2017 | Sequences of Functions (uniform convergence); Differential Equations HW III.4 (The Limit of a Sequences of Functions), III.5 (Integration of Infinite Series), III.6 (Derivatives of Infinite Series) |
| Nov. 08, 2017 | Differential Equations HW II.7 |
| Nov. 13, 2017 | Differential Equations HW II.8 |
| Nov. 15, 2017 | Differential Equations HW II.8; if you would like to know more about differential equations maybe a good start would be the first chapter (specifically the first section about "Phase Spaces") of "Arnol'd - Ordinary Differential Equations" |
| Nov. 20, 2017 | Fourier series the class material is mostly taken from the German book "Forster - Analysis 1 (Chapter 23)"; another good reference is "Tao - Analysis 2 (Chapter 5)"; see also RHB 12, mostly for applications; the exposition in Courant's book (Chapter IX) is also very nice |
| Nov. 22, 2017 | Fourier series |
| Nov. 27, 2017 | Fourier series |
| Nov. 29, 2017 | Fourier series |
| Dec. 04, 2017 | |
| Dec. 06, 2017 | |
| Dec. 07, 2017 | Final Exam |