Jacobs University, Spring 2020
This module is the second in a sequence introducing mathematical methods at university level in a form relevant for study and research in the quantitative natural sciences, engineering, Computer Science, and Mathematics. The emphasis in these modules lies in training operational skills and recognizing mathematical structures in a problem context. Mathematical rigor is used where appropriate. However, a full axiomatic treatment of the subject is done in the first-year modules "Analysis I" and "Linear Algebra". The lecture comprises the following topics
The class material is similar to the following textbook:
Other good books are:
A nice review book on Linear Algebra is:
Chapter 1: Some Extra Part I Topics
1.1: Binomial Expansion
1.2: Infinite Series
1.3: Power Series
1.4: Taylor Series
1.5: Improper Integrals
Chapter 2: Multivariable Calculus: Derivatives
2.1: Partial Derivatives
2.2: Total Derivative
2.3: Critical Points
2.4: Lagrange Multipliers
2.5: Vector-valued Functions
2.6: Vector Operators
Chapter 3: Ordinary Differential Equations
3.1: Basic Introduction
3.2: Some Types of Integrable ODEs
3.3: Qualitative Properties of ODEs
Chapter 4: Linear Algebra
4.1: Review
4.2: Determinant
4.3: Eigenvalues and Eigenvectors
4.4: Eigenspaces
4.5: Diagonalization
4.6: Special Types of Matrices
4.7: Matrix Decompositions
4.7.1: LU Decomposition
4.7.2: QR Decomposition
4.7.3: Singular Value Decomposition
The grade is only based on the final exam.
There will be one final exam (May 26, 2020) and one make-up final exam (Aug 31, 2020), see here for the full final exam schedule.
More information on the make-up final exam:
The make-up final exam takes place on Monday, August 31, 2020, from 11:00-13:00, in Eastwing (Campus Center); see here for the make-up exam schedule. The exam will be offered only in person (and not online). Everything regarding grading, the exam topics, and the exam format (see also the practice exams) from the remarks below still applies to the make-up exam.
More information on the final exam:
How will the exam be conducted? Since this is a large class, administration decided not to provide lecture halls for those who would like to take the final exam in person. Thus, the final exam will be offered online, with online proctoring. This is centrally organized using a software called LPLUS and online proctors from the company PRUEFSTER. The central organization of this is largely out of my hands, so please take a look at the central SRO website, where you can also find contact persons for questions about the setup. You will be provided opportunities to test the software before the exam. (See this interesting newspaper article that highlights the different opinions on online exams: Link.) For this class, the final exam will be provided as a pdf attachment in LPLUS. This means, all exam questions will be on the pdf, and you will have to solve them on paper. When the exam is finished, you would take scans or pictures of your hand-written notes, and upload these notes to the LPLUS software. Please read all the information regarding online exams that were sent to you by email carefully. In particular, make sure you know how to take good pictures of your notes with your phone. (And make sure you try out the test exam; when I was taking it my browser was constantly crashing.) According to admin, the instructor or TAs cannot be available during the exam to answer questions. I apologize for that.
How will the exam be graded? Since LPLUS is not so suitable for math exams, I will use gradescope for grading the exams, as I have done in the past. You will get an email with a link (to a password protected area) as soon as your exam is graded. There, you will see your graded exam with detailed comments. Gradescope allows to create categories for each step of the solution, so it is ensured that everyone gets the same amount of (partial) points for the same work. Moreover, there is an option to ask for regrading in case you think there is a mistake in the grading. If you prefer that gradescope is not used for your exam, you can just opt out by email to the instructor.
A note on grading: For almost every exam question in mathematics, it is extremely important to show your work and write down each step of the solution carefully. This is the most important part of the solution, and it is more important than the final answer. If you do every step of the solution right, but make a small computational mistake somewhere in the middle (say, a minus sign, or a factor 2 missing), you will still receive almost full points for the solution. (An exception is if a mistake simplifies the rest of the problem very much; this will then have to be taken into account for the grading, i.e., there might be a higher point deduction.)
Do I have to take this online exam? If you feel uncomfortable with the online proctoring or the online exam in general, or if you feel like you have been affected by the consequences of the pandemic in such a way that you would not like to take the exam now, you can get excused by email to the instructor. In this case, there will be a make-up exam in August, currently scheduled for August 31. I expect that the make-up exam will be offered as a usual in-person exam, but of course I cannot promise this at the moment.
What topics is the final exam about? Generally, the final exam is about all topics covered in class. However, since we do not have time to cover Fourier series/transform and systems of linear ODEs in detail and with practice exercises, those topics will not be on the exam. (But note that these topics are both very interesting and very important, so you should still follow the lecture notes and read up on them.)
How will the exam look like? Here and here are practice exams. (The solutions can mostly be found on moodle and the quiz solutions below, so no extra solutions for this practice exam will be provided.) Keep in mind that this practice exam does not cover all exam-relevant topics. For the final exam, no aids will be allowed, in particular no notes and no calculators. You should know the standard definitions. If an exam question is about a more advanced topic with a complicated definition, such a definition would be stated in the exercise. (E.g., you need to know what the Taylor expansion is, but if there would be a question about the remainder term, I would state its definition in the exercise.) What I wrote below at the beginning of the semester about the homework exercises of course still applies: "In order to motivate you to do the exercises, two problems from each the homework assignments, the moodle exercises, and the quizzes (so 6 problems in total) will appear in the final exam. Doing many of the exercises is the best preparation for the exam!" There will be some additional problems similar in spirit to the ones you have practiced, but a bit different. A few points of the exam will be designated bonus points.
Which homework/moodle/quiz questions are relevant for the exam? Generally, all moodle and quiz questions, and most homework questions are relevant for the final exam. The following homework questions are NOT suitable for a final exam: all bonus problems; HW 1 Problems 1, 2, 6; HW 2 Problem 1, remainder convergence in Problem 3 (but computation of the Taylor series is relevant); HW 4 Problem 5.
An essential component for doing well in this class is to work on practice exercises. Math (as well as almost all sciences) is about problem solving! During this course lots of possibilities for solving exercises are provided:
In order to motivate you to do the exercises, two problems from each the homework assignments, the moodle exercises, and the quizzes (so 6 problems in total) will appear in the final exam. Doing many of the exercises is the best preparation for the exam!
(Model solutions can be found on moodle.)
| Date | Sheet Number | Due Date |
|---|---|---|
| Feb 10, 2020 | (Sheet 1) Sheet 1 corrected version | Feb 24, 2020 |
| Feb 24, 2020 | Sheet 2 | Mar 09, 2020 |
| Mar 16, 2020 | Sheet 3 | Mar 30, 2020 |
| Apr 04, 2020 | Sheet 4 | Apr 20, 2020 |
| Apr 20, 2020 | Sheet 5 | May 04, 2020 |
| May 9, 2020 | Sheet 6 | no due date |
Please go to Jacobs moodle, login, and select the Calculus and Linear Algebra II class to view the exercises and the solutions. Below are pdfs of the exercises.
| Date | Sheet Number | Due Date |
|---|---|---|
| Feb 10, 2020 | Moodle Calculus review exercises | |
| Feb 10, 2020 | Moodle exercises 1 | Feb 17, 2020 |
| Feb 17, 2020 | Moodle exercises 2 | Feb 24, 2020 |
| Feb 24, 2020 | Moodle exercises 3 | Mar 02, 2020 |
| Mar 02, 2020 | Moodle exercises 4 | Mar 09, 2020 |
| Mar 09, 2020 | Moodle exercises 5 | Mar 16, 2020 |
| Mar 16, 2020 | Moodle exercises 6 | Mar 23, 2020 |
| Mar 23, 2020 | Moodle exercises 7 | Mar 30, 2020 |
| Mar 30, 2020 | Moodle exercises 8 | Apr 20, 2020 |
| Apr 06, 2020 | Moodle Linear Algebra review exercises | |
| Apr 20, 2020 | Moodle exercises 9 | Apr 27, 2020 |
| Apr 27, 2020 | Moodle exercises 10 | May 04, 2020 |
| May 04, 2020 | Moodle exercises 11 | May 11, 2020 |
| May 11, 2020 | Moodle exercises 12 | no due date |
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. RHB refers to the book by Riley, Hobson, Bence.)
| Date | Topics |
|---|---|
| Feb. 03, 2020 | Binomial expansion RHB Ch. 1.5 Binomial expansion, Ch. 1.6 Properties of binomial coefficients. (If you are interested into more material on probability, take a look at RHB Ch. 30, especially 30.3.2, 30.4.1, 30.5.1, 30.5.3 for the chapters connecting to what was discussed in class.) |
| Feb. 05, 2020 | Infinite series (partial sums, comparison and ratio tests); power series (radius of convergence, derivatives and integrals) RHB Ch. 4.1 Series, 4.2 Summation of series, 4.3 Convergence of infinite series, 4.5 Power series |
| Feb. 10, 2020 | Taylor Series RHB Ch. 4.6 Taylor Series |
| Feb. 12, 2020 | Applications of Taylor series to Newton's method; improper integrals; 15 minute quiz (quiz solutions) RHB Ch. 27.1.4 Newton-Raphson method, 27.2 Convergence of iteration schemes, 2.2.10 Infinite and improper integrals, 4.3.2 Convergence of a series containing only real positive terms (see Integral Test) |
| Feb. 17, 2020 | More on improper integrals; partial derivatives; directional derivatives RHB Ch. 2.2.10 Infinite and improper integrals; 5.1 Definition of the partial derivative |
| Feb. 19, 2020 | Total derivative This topic is not so well explained in RHB Ch. 5.2 The total differential and total derivative. A better explanation is in the book Folland - Advanced Calculus, in Chapter 2.2 Differentiability in Several Variables |
| Feb. 24, 2020 | More on surfaces and tangent planes; theorems for multi-variable differentiation; the chain rule See geogebra for nice visualizations of surfaces; vector algebra of planes: RHB Ch. 7.7.2 Equation of a plane; the connections between total, partial and directional derivatives are nicely explained in Folland - Advanced Calculus, in Chapter 2.2 Differentiability in Several Variables; RHB Ch. 5.5 The chain rule |
| Feb. 26, 2020 | Topics and applications in multi-variable differentiation (exact/inexact differentials, Taylor series, change of variables); 15 minute quiz (quiz solutions) RHB Ch. 5.3 Exact and inexact differentials; 5.7 Taylor's theorem for many-variable functions; 5.6 Change of variables; see also parts of Chapters 2.3 and 2.7 in Folland's book (more advanced than the treatment in class) |
| Mar. 02, 2020 | Topics and applications in multi-variable differentiation (change of variables, Leibniz integral rule); critical points RHB Ch. 5.12 Differentiation of integrals; beginning of Ch. 5.8 Stationary values of many-variable functions; see also the beginning of Ch. 2.8 in Folland's book (more advanced than the treatment in class) |
| Mar. 04, 2020 | no class (Jacobs Career Fair) |
| Mar. 09, 2020 | critical points; intro Lagrange multipliers RHB Ch. 5.8 Stationary values of many-variable functions; see also parts of CH. 2.8 in Folland's book (more advanced than the treatment in class) |
| Mar. 11, 2020 | Lagrange multipliers; vector-valued functions; 15 minute quiz (quiz solutions) RHB Ch. 5.9 Stationary values under constraints; RHB Ch. 10.1 Differentiation of vectors, Folland parts of Ch. 2.9 |
| Mar. 16, 2020 | Lagrange's method; Total derivative; Jacobian matrix RHB Ch. 10.2 Integration of vectors, 10.4 Vector functions of several arguments, 10.6 Scalar and vector fields; recall Ch.s 8.2, 8.3 and 8.4 in RHB for the basics on linear maps and matrices; see also Ch. 2.10 in Folland's book for a more advanced treatment. Here is a nice tool to visualize the gradient of a function: notice in particular that the gradient always shows in the direction of the largest directional derivative, and thus is always orthogonal to the level-sets. |
| Mar. 18, 2020 | Vector calculus (differentiability, div, grad, curl); Ordinary differential equations RHB Ch. 10.7 Vector operators; RHB parts of Ch.s 14.1 General form of solution, 14.2 First-degree first-order equations (note: the book discusses many more solution techniques) |
| Mar. 23, 2020 | Ordinary differential equations: Second order and qualitative behavior RHB discusses the topic only from a much more general perspective in Ch. 15.1 Linear equations with constant coefficients (which we are going to deal with later). If you would like to know more about differential equations a good start would be the first chapter "Phase Spaces" of "Arnol'd - Ordinary Differential Equations". The book is generally much more advanced, but the first chapter is extremely nice and accessible. |
| Mar. 25, 2020 | The determinant and its properties; 15 minute quiz (quiz solutions) The book "Linear Algebra" by Leduc is a good reference. In the notes, we covered most of what is described in the chapter "The Determinant; Method 2 for defining the determinant". |
| Mar. 30, 2020 | The determinant and its properties (in particular Laplace expansion) The book "Linear Algebra" by Leduc is a good reference. In the notes, we covered most of what is described in the chapter "The Determinant; Method 2 for defining the determinant" and "Laplace Expansions for the Determinant". |
| Apr. 01, 2020 | The determinant (Cramer's rule, invertibility, Leibniz formula) Cramer's rule: in RHB Ch. "8.18.2 N simultaneous linear equations in N unknowns" there is a small section on Cramer's rule, see also the chapter "Cramer's Rule" in Leduc's book. Invertibility: Leduc, "The classical adjoint of a square matrix", RHB Ch. "8.10 The inverse of a matrix". Leibniz formula: Leduc "Definitions of the determinant" |
| Apr. 06, 2020 | no class (Spring Break) |
| Apr. 08, 2020 | no class (Spring Break) |
| Apr. 13, 2020 | no class (Spring Break) |
| Apr. 15, 2020 | Eigenvalues and eigenvectors; 15 minute quiz (quiz solutions) See this video for why you should know about eigenvalues; they might show up on unexpected occasions and even particles have them! References: Leduc: "Definition and Illustration of an Eigenvalue and an Eigenvector", "Determining the Eigenvalues of a Matrix", parts of "Determining the Eigenvectors of a Matrix". RHB: parts of Ch.s 8.13 and 8.14 (the material in that book is organized a bit differently than the lecture notes). |
| Apr. 20, 2020 | Eigenvalues and eigenvectors; Eigenspaces Leduc: "Determining the Eigenvectors of a Matrix", parts of "Eigenspaces". Also RHB: parts of Ch.s 8.13 and 8.14 (the material in that book is organized a bit differently than the lecture notes). |
| Apr. 22, 2020 | Diagonalization Leduc: "Diagonalization". RHB Ch. 8.16 Diagonalisation of matrices. |
| Apr. 27, 2020 | Normal matrices RHB Ch. 8.7 The complex and Hermitian conjugates of a matrix, Ch. 8.12.7 Normal matrices, Ch. 8.13.1 Eigenvectors and eigenvalues of a normal matrix |
| Apr. 29, 2020 | Hermitian (self-adjoint), symmetric, unitary, and orthogonal matrices; 15 minute quiz (quiz solutions) RHB Ch. 8.12.3 Symmetric and antisymmetric matrices, Ch. 8.12.5 Hermitian and anti-Hermitian matrices, Ch. 8.13.2 Eigenvectors and eigenvalues of Hermitian and anti-Hermitian matrices, second half of Ch. 5.8 Stationary values of many-variable functions |
| May 04, 2020 | Unitary and orthogonal matrices; LU decomposition RHB Ch. 8.12.4 Orthogonal matrices, Ch. 8.12.6 Unitary matrices, Ch. 8.13.3 Eigenvectors and eigenvalues of a unitary matrix, see also some parts of Ch. 8.16 Diagonalisation of matrices. (More on the Jordan normal form can be found in any advanced Linear Algebra book, e.g., "Gockenbach - Finite-Dimensional Linear Algebra".) |
| May 06, 2020 | LU decomposition continued; QR decomposition These topics are not covered in RHB or Leduc. But many other books on Linear Algebra cover them, e.g., "Gockenbach - Finite-Dimensional Linear Algebra". |
| May 11, 2020 | QR decomposition continued; Singular value decomposition RHB Ch. 8.18.3 Singular value decomposition |
| May 13, 2020 | Review Session Some more topics that would naturally follow the class material are: Fourier series, Fourier transform, and its application to ODEs, and application of eigendecompositions to linear ODEs. These are covered, e.g., in the respective chapters in RHB and Gockenbach. Short introductions to Fourier series and systems of ODEs can be found in the Advanced Calculus lecture notes. |
| May 26, 2020 | Final Exam (see here for the full final exam schedule); Here and here are practice final exams. |
| Aug 31, 2020 | Make-up Final Exam |