Course number: CH-202-A
Jacobs University, Spring 2022
All the most recent information about class can be found on this website.
This module covers advanced topics from calculus that are part of the core mathematics education of every Physicist and also forms a fundamental part of the mathematics major. It features examples and applications from the physical sciences. The module is designed to be taken with minimal pre-requisites and is tightly coordinated with the parallel module Calculus and Elements of Linear Algebra II. The style of development strives for rigor, but avoids abstraction and prefers simplicity over generality.
For more applications and examples, see the lecture notes on the class website of Calculus and Linear Algebra II (Spring 2020).
The total grade is only based on the final exam. There is the possibility for bonus points (up to 10% in the percentage grading scheme), see below. Note that the module "Applied Mathematics" consists of this class and the "Numerical Software Lab". The total module grade is 2/3 the grade of this class and 1/3 the grade of the lab.
There will be one final exam and one make-up final exam. All topics discussed in class are relevant for the exams. Note that this class uses gradescope for grading exams, see here for more information. (Note: Just talk to the instructor if you prefer that gradescope is not used for your exam.)
For practice, here are some exams (with solutions) from the Spring 2020 class.
There will be regular homework assignment. These are an integral part of the coursework and working on the exercise sheets consistently is the best preparation for the exam. As an additional motivation to work on the homework sheets there are bonus points. These are obtained in the following way. An average homework grade in percentage scale is computed out of all but the two worst homework sheets; it is rounded up to the next integer. The bonus points are attributed in the following way:
| Total homework grade | Bonus |
|---|---|
| 0% - 10% and at least one homework sheet with at least one point for each exercise. | 1% |
| 11% - 20% | 2% |
| 21% - 30% | 3% |
| 31% - 40% | 4% |
| 41% - 50% | 5% |
| 51% - 60% | 6% |
| 61% - 70% | 7% |
| 71% - 80% | 8% |
| 81% - 90% | 9% |
| 91% - 100% | 10% |
The homeworks can be handed in only by uploading them to Moodle before the due date.
| Date | Sheet Number | Due Date |
|---|---|---|
| Feb 08, 2022 | Sheet 1 | Feb 15, 2022 |
| Feb 15, 2022 | Sheet 2 Sheet 2 (new) | Feb 22, 2022 |
| Feb 22, 2022 | Sheet 3 Sheet 3 (new) | Mar 1, 2022 |
| Mar 1, 2022 | Sheet 4 | Mar 8, 2022 |
| Mar 8, 2022 | Sheet 5 | Mar 15, 2022 |
| Mar 15, 2022 | Sheet 6 | Mar 22, 2022 |
| Mar 22, 2022 | Sheet 7 | Mar 29, 2022 |
| Mar 29, 2022 | Sheet 8 | Apr 05, 2022 |
| Apr 05, 2022 | Sheet 9See here for a picture regarding Problem 4. | Apr 19, 2022(extended to April 26, 2022) |
| Apr 19, 2022 | Sheet 10 | Apr 26, 2022 |
| Apr 26, 2022 | Sheet 11 | May 03, 2022 |
| May 03, 2022 | Sheet 12 | May 10, 2022 |
| May 10, 2022 | Sheet 13. See here for solutions (except exercise 3). | not for credit |
Chapter 1: Sequences and Series of Functions
1.1: Review of Differentiation, Integration, and Taylor's Theorem
1.2: Sequences of Functions
1.3: Power Series
1.4: Metric and Normed Spaces
Chapter 2: Derivatives
2.1: Total and Partial Derivatives
2.2: Higher Order Derivatives
2.3: The Inverse and Implicit Function Theorems
Chapter 3: Integrals
3.1: Partial Integrals
3.2: The Riemann Integral in R^n
3.3: Line Integrals
3.4: Green's Theorem
3.5: Surface Integrals
3.6: Divergence Theorem
Chapter 4: Fourier Series
Chapter 5: Complex Analysis
Will be updated while class is progressing.
Below, please click on the date to download the lecture notes of this day.
(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 |
|---|---|
| Feb. 01, 2022 | Organization and brief review of continuity and differentiability Any Analysis or Calculus textbook. |
| Feb. 04, 2022 | Brief review of Riemann integral and Taylor series See the lecture notes by Sloughter and the ones by Folland. |
| Feb. 08, 2022 | Taylor series ctd; uniform covergence For Taylor series, see the references from Feb 4. For uniform convergence, see, e.g., the lecture notes by Levermore (Sections 12.1 and beginning of 12.2) and Ramakrishnan (beginning of Section 11.1). |
| Feb. 11, 2022 | Uniform covergence ctd; power series (convergence tests) For uniform convergence, see the references from Feb 8. For infinite series, see, e.g., some of my old lecture notes: Session 2 (Calculus and Linear Algebra II, Spring 2020) and Session 6 (Advanced Calculus, Fall 2018). |
| Feb. 15, 2022 | Power series ctd; topological spaces Power series: See lecture note above and Session 7 (Advanced Calculus, Fall 2018). For topological spaces, see parts of Chapter 1.1 in Kantorovitz. |
| Feb. 18, 2022 | Metric spaces, normed spaces, inner product spaces See parts of Chapter 1.1 in Kantorovitz. |
| Feb. 22, 2022 | Compactness; The derivative as a linear map A more detailed exposition of compactness is in Chapter 1.2 in Kantorovitz. The derivative is introduced in Kantorovitz in Chapter 2.1 in two steps: first for real-valued functions in "The Differential", and only later for vector-valued functions in "The Differential of a Vector Valued Function". In class, we gave the latter more general definition right away. A good reference for the approach in class is also Rudin Chapter 9: "Differentiation". |
| Feb. 25, 2022 | Total, partial and directional derivatives See Kantorovitz: Chapter 2.1. (Also Rudin: Chapter 9: "Differentiation"). |
| Mar. 01, 2022 | Total, partial and directional derivatives See Kantorovitz: Chapter 2.1. (Also Rudin: Chapter 9: "Differentiation"). |
| Mar. 04, 2022 | Gradient; Higher derivatives and Taylor series in many variables See Kantorovitz: parts of Chapter 2.2. (Also Rudin: parts of Chapter 9: "Derivatives of Higher Order"). |
| Mar. 08, 2022 | The Hessian; Maxima and Minima See Kantorovitz: parts of Chapter 2.2 (Local Extrema). |
| Mar. 11, 2022 | Inverse and implicit function theorems Kantorovitz proves the Implicit Function Theorem first, and then the Inverse Function Theorem as a corollary. This is done in Chapters 3.1-3.3. In class, we rather follow the approach by Rudin in Chapter 9 "The Inverse Function Theorem". (The Implicit Function Theorem is proved afterwards.) |
| Mar. 15, 2022 | Inverse and implicit function theorems Same references as for previous class. |
| Mar. 18, 2022 | Implicit function theorem; Integrals in higher dimension Same references as for previous class. |
| Mar. 22, 2022 | Partial Integrals Kantorovitz Chapter 4.1 (up to Example 4.1.5). |
| Mar. 25, 2022 | Partial integrals and order of integration Kantorovitz Chapter 4.1 (from Theorem 4.1.6 to Example 4.1.11). |
| Mar. 29, 2022 (online) |
The Riemann integral in several dimensions See our MS Teams channel for the class recordings.Reference: Kantorovitz Chapter 4.2 (until Theorem 4.2.4). |
| Apr. 01, 2022 (online) |
Properties of the Riemann integral; Normal domains; Change of variables; Polar coordinates See our MS Teams channel for the class recordings.Reference: Kantorovitz Chapter 4.2 (from Theorem 4.2.4 onwards). Note: This chapter covers many more interesting examples. |
| Apr. 05, 2022 (online) |
Curves and line integrals Kantorovitz Chapter 4.3 (first part). |
| Apr. 08, 2022 (online) |
Conservative vector fields, exact differentials, potentials Kantorovitz Chapter 4.3 (Conservative Fields and parts of 4.3.9 Exact Differential Form and Potential). |
| Apr. 12, 2022 | No class (spring break) |
| Apr. 15, 2022 | No class (spring break) |
| Apr. 19, 2022 | Conservative vector fields Kantorovitz Chapter 4.4 (4.3.9 Exact Differential Form and Potential). |
| Apr. 22, 2022 | Green's theorem Kantorovitz Chapter 4.4. |
| Apr. 26, 2022 | Surface integrals Kantorovitz Chapter 4.5 (first 4 pages). |
| Apr. 29, 2022 | Divergence theorem, Stokes' theorem Kantorovitz Chapter 4.5. |
| May 03, 2022 | Divergence theorem, Stokes' theorem See the lecture notes of Sessions 23 and 24 and Kantorovitz Chapter 4.5. |
| May 06, 2022 | Fourier Series See the lecture notes from Sessions 21-24 from Advanced Calculus (Fall 2018). Good references for Fourier Series are: "Tao - Analysis 2" (Chapter 5), and Riley, Hobson, Bence Chapter 12 (mostly for applications); the exposition in Courant's book "Differential and Integral Calculus Volume I" (Chapter IX) is also very nice. |
| May 10, 2022 | Complex Analysis, holomorphic functions A good exposition is in Riley, Hobson, Bence, Chapter 24: Complex variables. |
| May 13, 2022 | Complex Analysis See references above. |
| May 20, 2022 | Final exam |
| Aug 25, 2022 | Make-up Final exam |
