Course number: CH-202-A
Constructor University, Spring 2023
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 related material, 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 in class before the due date.
| Date | Sheet Number | Due Date |
|---|---|---|
| Feb 07, 2023 | Sheet 1 | Feb 14, 2023 |
| Feb 14, 2023 | Sheet 2 | Feb 21, 2023 |
| Feb 21, 2023 | Sheet 3 | Feb 28, 2023 |
| Feb 28, 2023 | Sheet 4 | Mar 7, 2023 |
| Mar 7, 2023 | Sheet 5 | Mar 14, 2023 |
| Mar 14, 2023 | Sheet 6 | Mar 21, 2023 |
| Mar 21, 2023 | Sheet 7 | Mar 28, 2023 |
| Mar 28, 2023 | Sheet 8 | Apr 11, 2023 |
| Apr 11, 2023 | Sheet 9See here for a picture regarding Problem 4. | Apr 18, 2023 |
| Apr 18, 2023 | Sheet 10 | Apr 25, 2023 |
| Apr 25, 2023 | Sheet 11 | May 02, 2023 |
| May 02, 2023 | Sheet 12 | May 09, 2023 |
| May 09, 2023 | 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. 03, 2023 | Organization and brief review of continuity and differentiability Any Analysis or Calculus textbook. |
| Feb. 07, 2023 | Brief review of Riemann integral and Taylor series See the lecture notes by Sloughter and the ones by Folland. |
| Feb. 10, 2023 | 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. 14, 2023 | Uniform covergence ctd; Convergence tests for series For uniform convergence, see the references from Feb 10. 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. 17, 2023 | Power series Power series: See lecture notes above and Session 7 (Advanced Calculus, Fall 2018). |
| Feb. 21, 2023 | Topological spaces, metric spaces, normed spaces, inner product spaces See parts of Chapter 1.1 in Kantorovitz. |
| Feb. 24, 2023 | 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. 28, 2023 | Total, partial and directional derivatives See Kantorovitz: Chapter 2.1. (Also Rudin: Chapter 9: "Differentiation"). |
| Mar. 03, 2023 | Total, partial and directional derivatives See Kantorovitz: Chapter 2.1. (Also Rudin: Chapter 9: "Differentiation"). |
| Mar. 07, 2023 | 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. 10, 2023 | The Hessian; Maxima and Minima See Kantorovitz: parts of Chapter 2.2 (Local Extrema). |
| Mar. 14, 2023 | 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", where the Implicit Function Theorem is proved after the Inverse Function Theorem. |
| Mar. 17, 2023 | Inverse and implicit function theorems (proofs) Same references as for previous class. |
| Mar. 21, 2023 | Partial Integrals (uniform continuity, Leibniz rules I and II) Kantorovitz Chapter 4.1 (up to Example 4.1.5). |
| Mar. 24, 2023 | Partial integrals (Examples, Leibniz rule III, and order of integration) Kantorovitz Chapter 4.1 (from Example 4.1.5 to Example 4.1.11). |
| Mar. 28, 2023 | The Riemann integral in several dimensions Reference: Kantorovitz Chapter 4.2 (until Theorem 4.2.4). |
| Mar. 31, 2023 | Properties of the Riemann integral; Normal domains; Change of variables Kantorovitz Chapter 4.2 (from Theorem 4.2.4 onwards). Note: This chapter covers many more interesting examples. |
| Apr. 04, 2023 | No class (spring break) |
| Apr. 07, 2023 | No class (spring break) |
| Apr. 11, 2023 | Polar/spherical/cylindrical coordinates; Curves and their length Kantorovitz Chapter 4.3 (first part). |
| Apr. 14, 2023 | Line integrals; Conservative vector fields Kantorovitz Chapter 4.3 (Line Integrals and beginning of Conservative Fields. |
| Apr. 18, 2023 | Potentials and conservative vector fields Kantorovitz Chapter 4.4 (and 4.3.9 Exact Differential Form and Potential). |
| Apr. 21, 2023 | Green's theorem Kantorovitz Chapter 4.4. |
| Apr. 25, 2023 | Surface integrals Kantorovitz Chapter 4.5 (first 4 pages). |
| April 28, 2023 | Divergence theorem, Stokes' theorem Kantorovitz Chapter 4.5. |
| May 02, 2023 | Divergence theorem, Stokes' theorem (Continuity equation, Maxwell's equations); Fourier series Kantorovitz Chapter 4.5. |
| May 05, 2023 | 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 09, 2023 | Complex Analysis (holomorphic functions, Cauchy's integral theorem) A good exposition is in Riley, Hobson, Bence, Chapter 24: Complex variables. Complex Analysis is its own field, and there are many good books about it, e.g., Stein and Shakarchi - Complex Analysis. |
| May 12, 2023 | Complex Analysis (Cauchy's integral formula, Laurent series, residue theorem) See previous session. |
| May 26, 2023 | Final exam |
| Aug 25, 2023 | Make-up Final exam |
