# Winter 2022 — Mathematics for advanced materials science (VU)

The instructor for this course is Marc Technau (E-Mail).

The contents of this website is subject to change.
Please mind the time stamp at the bottom of this website.
Dates are written in DD.MM.YYYY format (e.g., “01.__02__.2022” means “__February__ 1st 2022”).

## Dates

The **first lecture** takes place on Thursday, 06.10.2022.

*Lecture:*- Th., 12:15–13:45, room NT·02·008 in Kopernikusgasse 24/II.

## Syllabus

This course is aimed at students entering the master's programme *“Advanced Materials Science”* and having a background in chemistry.
Given the time constraints, basic familiarity with mathematics at the level of the lectures *Mathematik für ChemikerInnen 1* and *Mathematik für ChemikerInnen 2* will be assumed.
However, there is a substantial overlap with the aforementioned courses.

(The following syllabus is tentative and may change; the contents of the course is presently being re-evaluated.)

A guiding theme for this course is the desire for solving certain **differential equations** that arise in the study of physical phenomena.
We shall start with examples of such equations whose solution naturally motivates complex numbers.

After reviewing fundamentals of **complex numbers** and the exponential function, we study the **Laplace transform**. This is a certain integral transform for which complex numbers play a prominent role, and which constitutes a useful tool for solving certain classes of differential equations.

Moreover, we shall give a review of selected aspects of **linear algebra** (in preparation of more advanced topics; see below): basic vector arithmetic, linear maps and their relation with matrices, determinants, elements of analytic geometry in two and three dimensions (distances, dot product, cross product) and a brief discussion of eigenvalues and eigenvectors.

This leads naturally to spectral theory of certain differential operators and we discuss the model example for this: **Fourier expansion** of periodic functions.
As the attendees will have realised at that state of the lecture, these Fourier expansions turn out to be one of the most important technical tools for understanding certain phenomena in, for instance, solid state physics, where crystals are modelled as periodically repeating structures of atoms or molecules and the study of propagation of waves and heat in them is of considerable interest.

We return to the mathematics of **wave and heat propagation** towards the end of the course by studying the relevant partial differential equations.
However, before being able to do so, we review selected aspects of the fundamentals of calculus in higher dimensions (with a strong bias towards at most three dimensions) starting with **differentiability** and moving on towards integration.
Our ultimate goal here are the classical **integral theorems** of Kelvin–Stokes, Gauß and Green–Riemann which are ubiquitous in the study of electrodynamics.

Time permitting, the course then finishes after the aforementioned discussion of some selected **partial differential equations** (we shall restrict our discussion to linear second-order equations though).

The lecture-part of the course is aimed at discussing the topics listed above and show how to work with them. The exercise-part of the course is meant to give the attendees some practical skills in carrying out related calculations. The focus here is less on the theory and more on actual problem solving.

## Online teaching

It is planned to record the lecture-part of the course (see below). Please take note that, although the camera equipment is primarily focused at recording the lecturer and the blackboard area, it may not be possible to prevent that you appear on the recordings (sound or picture). By attending the lecture you automatically agree to these circumstances and agree to the publication of the recordings. In case you feel that a part of the recordings constitutes an infringement towards your personal rights, please contact Marc Technau (in well-founded cases, the relevant parts of the recordings can be cut out).

## Grading policy

The final grade is derived from two aspects:

**(E)***bi-weekly*electronic submission of solutions to exercises, and**(W)**- score from the written exam.

Details on these two aspects are expounded below. Given your score in each of the two aspects, and assuming that you have scored at least 30% of the points in the written exam, your final score η is calculated via the following formula:

η
= 60% · **(E)**
+ 40% · **(W)**.

(Here your score for each of the two aspects is understood to be the quotient of the points you have accumulated in that particular aspect and the maximum number of points attainable there.) Your grade is then determined from η and the following table:

Range for η |
Grade | |
---|---|---|

85% ≤ η |
„Sehr gut“ | (1) |

70% ≤ η < 85% |
„Gut“ | (2) |

60% ≤ η < 70% |
„Befriedigend“ | (3) |

50% ≤ η < 60% |
„Genügend“ | (4) |

η < 50% |
0% ≤ „Nicht genügend“ | (5) |

**Important note #1:** If you score less than 30% in the written exam, then η = 0 (that is, your grade is „nicht genügend“).

**Important note #2:** Showing up for the lectures is not necessary for passing the course. To pass, you need only hand in (correct) solutions to sufficiently many exercises and score at least 30% in the written exam. (The last part obviously requires physical attendance.)

Submission of solutions for exercises and having enrolled for the course via TUG▪online implies the receipt of a grade for the course, unless the enrollment in cancelled before the end of October 31st via TUG▪online. Conversely, staying enrolled beyond October 31st is necessary for receiving a grade for this course.

#### (E): Solving exercises

Exercise sheets will be posted on this website every Thursday if there was a lecture during that week.
Solutions to *every other* sheet (2nd sheet, 4th sheet, 6th sheet etc.) are to be submitted *online* via the TeachCenter by Thursday of the following week.
The solutions will then be graded.

The solutions you submit must be your own. In this regard, please ensure take into account the plagiarism statute of the TU Graz (see here for an English version). The right to award a failing grade in case of violations to the above statute shall remain reserved.

The exercise sheets not due for submission (1st sheet, 3rd sheet etc.) also contain material important for the written exam.
It is *strongly* suggested that participants of the course also try to work on these exercise sheets.

#### (W): Written exam

There are *two chances* for taking the written exam: **09.02.2023** and **23.02.2023**, both taking place from 12:15 until 13:45 (90 minutes) in room NT·02·008 in Kopernikusgasse 24/II.

If one takes part in both exams, then the score achieved in the second exam replaces the score from the previous one (regardless if it is higher or lower).

The students are allowed to bring two DIN-A4 sheets with notes (both sides of a sheet may contain notes).
Further aids (e.g., calculators, smart watches, phones etc.) are *not admissible*.

## Literature

- G. Bärwolff.
*Höhere Mathematik für Naturwissenschaftler und Ingenieure*. Berlin: Springer, 3rd edition, 2017. - C. Elsholtz, J. Hatzl, C. Heuberger, J. Pöschko.
*Mathematik für ChemikerInnen 1*. Lecture notes, 2019. - C. Elsholtz, C. Heuberger, J. Pöschko.
*Mathematik für ChemikerInnen 2*. Lecture notes, 2020. - K. Jänich.
*Analysis für Physiker und Ingenieure*. Berlin: Springer, 4th edition, 2001. - K. Jänich.
*Mathematik 1*. Berlin: Springer, 2nd edition, 2005. - K. Jänich.
*Mathematik 2*. Berlin: Springer, 2nd edition, 2011. - E. Kreyszig. Advanced engineering mathematics. New York: Wiley, 10th edition, 2020.
- M. Stone and P. Goldbart. Mathematics for physics. A guided tour for graduate students. Cambridge: Cambridge University Press, 2009. (Pre-publication PDF available here.)

### Comments

Please note that all of the above books (excluding only Jänich's *Analysis* and the book by Kreyszig) are freely available online. (For downloading the Springer books you should be accessing the internet from the university's IP range.)

The books of Jänich are arguably quite advanced in that they aim at expounding a high-brow view on the subject matter (although they start from scratch).
They do, however, have a quite unique style of giving a *real* explanation of *why* various results are true in a very insightful fashion that is hard to find anywhere else.
They are especially recommended.

The book of Bärwolff contains essentially all of the material of this course (and more), but takes a rather more calculative approach that may serve some readers better.

The present course shall attempt to combine aspects of both approaches.

## Lecture notes

The lecture notes are for *personal use only* and are not to be distributed!
Here are the various chapters:

- Chapter 0:
*Basics* - Chapter 1:
*Complex numbers* - Chapter 2:
*The Laplace transform* - Chapter 3:
*Linear algebra* - Chapter 4:
*Fourier analysis* - Chapter 5:
*Differentiability* - Chapter 6:
*Topics in differentiability* - Chapter 7:
*Integration* - Chapter 8:
*Differential equations*

Note that chapter 0 was not discussed during class. It contains (amongst other things) some tips and tricks on how to compute derivatives and certain integrals.

The complete lecture notes are available here as one PDF file.

## Exercise sheets

The following list will be updated regularly. New sheets generally become available every Thursday (exceptions may occur) and written solutions for exercise sheets with even numbers are due on the following Thursday (to be handed in online via the TeachCenter). There will probably be 13 exercise sheets.