Jacobs University, Fall 2018
Class: Mon 11:15 - 12:30, Lecture Hall Research I; Wed 11:15 - 12:30, Lecture Hall Research I
Tutorial Session: Tue 19:15 - 20:30, Room 120 in Research I
Syllabus (as of Sep. 1, 2018) 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.
There will be two exams, a midterm and a final. The midterm will cover the material from the first half of the course and the final will cover all material with emphasis on the second half. Note that this class uses gradescope for grading exams, see here for more information.
Midterm Exam: Wed, Oct. 24, 2018, 11:15 - 12:30, Lecture Hall Research I
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. 10 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. 13, 2018, 9:00 - 11:00, Lecture Hall Research III
The exam begins at 9:00 sharp, so please come a few minutes earlier to be seated at 9:00. The final is about all material up to and including the Dec. 5 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!
Each week on Monday (with exceptions) there will be a homework assignment. These are an integral part of the coursework and working on the exercise sheets consistently is the best preparation for the exams. Note that
| Date | Sheet Number | Due Date |
|---|---|---|
| Sep. 10, 2018 | Sheet 1 | Sep. 17, 2018 |
| Sep. 17, 2018 | Sheet 2 | Sep. 24, 2018 |
| Sep. 24, 2018 | Sheet 3 | Oct. 01, 2018 |
| Oct. 01, 2018 | Sheet 4 | Oct. 08, 2018 |
| Oct. 08, 2018 | Sheet 5 | Oct. 15, 2018 |
| Oct. 29, 2018 | Sheet 6 | Nov. 05, 2018 |
| Nov. 05, 2018 | Sheet 7 | Nov. 12, 2018 |
| Nov. 12, 2018 | Sheet 8 | Nov. 19, 2018 |
| Nov. 19, 2018 | Sheet 9 | Nov. 26, 2018 |
| Nov. 26, 2018 | Sheet 10 | Dec. 03, 2018 |
Will be updated while class is progressing.
Click on the date to download the lecture notes of this day.
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. 03, 2018 | Polynomials (roots, factorization) HW I.1 (Polynomial Functions); RHB 1.1 |
| Sep. 05, 2018 | Polynomials (polynomial interpolation); Binomial Expansion HW I.2 (Binomial Theorem); RHB 1.5, 1.6 |
| Sep. 10, 2018 | Limits (sequences and their convergence properties) HW III.1 (Convergence of a Sequence) |
| Sep. 12, 2018 | Limits (sup, inf, limsup, liminf), Continuity (definition, properties, examples, intermediate value theorem) HW III.1 (Convergence of a Sequence), HW III.3 (Continuous Functions, The Intermediate Value Theorem) |
| Sep. 17, 2018 | Continuity (maximum theorem, limits of functions); Infinite Series (partial sums, difference method) HW III.3 (The Maximum Theorem, Limit of a Function), HW III.2 (beginning of the chapter); RHB 4.1, 4.2 |
| Sep. 19, 2018 | Infinite Series (absolute convergence, rearrangements, convergence criteria) HW III.2 (Criteria for Convergence, Absolute Convergence); RHB 4.3 |
| Sep. 24, 2018 | Infinite Series (Cauchy product); Power Series (radius of convergence, power series of exponential function) HW III.2 (The Cauchy Product of Two Series; for deeper understanding: Exchange of Infinite Series and Limits), HW I.2 (Exponential Function), HW III.3 (Monotone and Inverse Functions); RHB 4.4, 4.5 |
| Sep. 26, 2018 | Inverse Functions (logarithm); Landau Symbols; Complex Numbers HW parts of I.3 and parts of I.5, HW III.3 (Monotone and Inverse Functions); RHB 3.1-3.3 |
| Oct. 01, 2018 | Complex Numbers ctd HW I.5 (except the Euler's product part); RHB 3.3-3.5 |
| Oct. 03, 2018 | German Unity Day, no classes |
| Oct. 08, 2018 | Differentiation (definition, differentiation rules, examples) HW II.1, III.6 (first section); RHB 2.1.1 - 2.1.4 |
| Oct. 10, 2018 | Differentiation (examples, differentiation of power series, implicit differentiation and parametric representation, theorems of Rolle, Lagrange, Cauchy, L'Hospital) HW III.6 (only parts of it); RHB 2.1.5 - 2.1.7, 2.1.10, 4.6, 4.7 |
| Oct. 15, 2018 | Differentiation (L'Hospital, Taylor expansion) HW II.2 (parts of it), III.6 (The Rules of de L'Hospital), III.7 (Taylor Series); RHB 4.6, 4.7 |
| Oct. 17, 2018 | Differentiation (Taylor expansion, minimization and maximization problems, Newton's method) HW II.2, III.7 (Taylor Series); RHB 2.1.8, 4.6, 27.1.4 |
| Oct. 22, 2018 | Reading Day, no classes |
| Oct. 24, 2018 | Midterm Exam (here are some extra exercises for practice, and here are the solutions; let me know if you find or suspect mistakes/typos) |
| Oct. 29, 2018 | Newton Method and Convergence Rate; Integration (Definition of Riemann integral) HW III.5 (only some parts of Definitions and Criteria of Integrability); RHB 27.2, 2.2.1 |
| Oct. 31, 2018 | Day of Reformation, no classes |
| Nov. 05, 2018 | Integration (mean-value theorem, Fundamental Theorem of Differential Calculus, integration by inspection, substitution) HW II.4 (Primitives, parts of Applications, parts of Integration Techniques), HW III.5 (Integrable Functions, Inequalities and the Mean Value Theorem), III.6 (only parts of The Fundamental Theorem of Differential Calculus); RHB 2.2.2, 2.2.3, 2.2.5, 2.2.7 |
| Nov. 07, 2018 | Integration (integration by parts, areas, curve length, recurrence relations, rational functions and partial fractions) HW II.4 (parts of Integration Techniques), HW II.5 (Integration of Rational Functions); RHB 1.4, 2.2.4, 2.2.6, 2.2.8, 2.2.9, 2.2.13 (Finding the length of a curve) |
| Nov. 12, 2018 | Integration (power series, Taylor series, improper integrals) HW II.4 (Taylor's Formula with Remainder), III.5 (Integration of Infinite Series), III.6 (Derivatives of Infinite Series) HW III.7 (Differentiation and Integration, Taylor Series), HW III.8; RHB 2.2.10 |
| Nov. 14, 2018 | Sequences of Functions (uniform convergence) HW III.4 (The Limit of a Sequence of Functions) |
| Nov. 19, 2018 | Differential Equations: Introduction and types of integrable ODEs HW II.7; RHB 14.2.1, 14.2.7 (take a look at other nearby chapters for more methods to find solutions) |
| Nov. 21, 2018 | Differential Equations: Linear homogeneous and inhomogeneous equations HW II.8; RHB 15-15.1.3; 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. 26, 2018 | Final remarks about ODEs; Fourier series concerning Fourier series we are deviating from the books here; a 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. 28, 2018 | Fourier series see above for book references |
| Dec. 03, 2018 | Fourier series see above for book references |
| Dec. 05, 2018 | Fourier series and PDEs see above for book references |
| Dec. 13, 2018 | Final Exam (here are some extra exercises for practice, and here are the solutions; let me know if you find or suspect mistakes/typos) |