# Winter 2021 — 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__.2021” means “__February__ 1st 2021”).
This website uses MathML for rendering some maths formulæ.
For correct rendering of this website, the use of Mozilla Firefox is recommended.

## Dates

The **first lecture** takes place on Thursday, 07.10.2021.

Please observe that the first lecture takes place in room NT·02·008 in Kopernikusgasse 24/II.

*Lecture part:*- Th., 12:15–13:45, lecture hall BMT·EG·138
^{[1]} *Exercise part:*- within the time slot of the lecture on the following dates: 21.10.2021, 04.11.2021, 18.11.2021, 02.12.2021, 16.12.2021, 20.01.2022.

^{[1]}: changed room on 06.10.2021 (NT·02·008).
Changed room (HS H “Ulrich Santner”) on the following dates: 28.10.2021, 04.11.2021, 11.11.2021, 18.11.2021, 25.11.2021, 02.12.2021, 09.12.2021.

Note that at the time of writing, entering any building of the TU Graz is allowed only for persons who are in possession of a valid 3-G certificate. Please be prepared to provide proof of your 3-G status at all times during your stay at a TU Graz building.

## Syllabus

This course is aimed at students entering the master's programme *“Advanced Materials Science”* and having a background in chemistry.
Familiarity with mathematics at the level of the lectures *Mathematik für ChemikerInnen 1* and *Mathematik für ChemikerInnen 2* will be assumed, although important concepts will be reviewed during the lectures.

A guiding theme of the 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.

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

The *“VU”* format of this course (lecture with integrated exercises) entails that regular participation is expected and has an impact on the final grade.
At the time of writing, it is planned to hold the lecture and exercises in person.
If, however, during the course of the semester, the TU Graz advises to switch to online teaching, this course is prepared to follow suit and may switch to a Webex-based format.

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 three aspects: **(Q)** *bi-weekly* electronic submission of solutions to exercises, **(P)** *bi-weekly* presentation of solutions to exercises and **(A)** active participation in the course.
Details on these three aspects are expounded below.
Given your score in each of the three aspects, your final score η is calculated via the following formula:

η
= 40% · **(Q)**
+ 40% · **(P)**
+ 20% · **(A)**.

(Here your score for each of the three 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) |

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 longer than October 31st is necessary for receiving a grade for this course.

#### (Q): Solving exercises

Solutions to exercises are to be submitted *online* via the TeachCenter.

The exercise sheets for the *(Q) aspect* (“quizzes”) will be graded with points.
These sheets are made available on a bi-weekly basis (alternating with the exercise sheets for the (P) aspect).
Unless otherwise specified you are to use the forms provided with the sheets as to make grading easier.
The solutions 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 give a failing grade in case of violations to the above statute shall remain reserved.

#### (P): Presenting solutions

The exercise sheets for the *(P) aspect* are to be solved and written solutions are to be submitted *online* via the TeachCenter.
These sheets are made available on a bi-weekly basis (alternating with the exercise sheets for the (Q) aspect).
Solutions for these exercise sheets are then to be presented by the students during the exercise part of the course taking place during the lecture time slot (see above).
Points are awarded per presentation and your final score for this grading aspect is computed as the sum of your ~~four~~ three best presentation grades.
Note that points are only awarded if you have both submitted your solutions online and presented them during class.

#### (A): Active participation

Your active participation during the course is evaluated throughout the course. This consists of regular attendance of both lecture and exercise part as well as asking/answering questions (for instance, during the lecture or during presentations of your fellow students).

Please take note that simply showing up for the course is not enough to score points here by itself.

## 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.

Please note that all of the above books (excluding only Jänich's *Analysis*) 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 recordings

The recordings of the lectures are available on TUbe. Please ensure that you have successfully logged in in order to have access.

## Lecture notes

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

The animation of a mass on a spring shown during the first lecture can be downloaded here as a Mathematica notebook or here as a CDF file. (To use the former, you need Wolfram Mathematica; to use the latter you only need the Wolfram CDF player, which can be downloaded for free.)

The animation illustrating the relation of determinants with areas and volumes can be downloaded here as a Mathematica notebook or here as a CDF file. Videos of the animations in action can be downloaded here (2D) and here (3D).

The notes written on a tablet during the lecture on the 02.12.2021 can be downloaded here as PDF.

## Exercise sheets

Sheet | Due date … | To be discussed on … | |
---|---|---|---|

Sheet 1 | 12.10.2021 | ||

Sheet 2 | 19.10.2021 | 21.10.2021 | |

Sheet 3 | 26.10.2021 | ||

Sheet 4 | 02.11.2021 | 04.11.2021 | |

Sheet 5 | 09.11.2021 | ||

Sheet 6 | 16.11.2021 | 18.11.2021 | |

Sheet 7 | 25.11.2021 | ||

Sheet 8 | 02.12.2021 | 02.12.2021 | |

Sheet 9 | 09.12.2021 | ||

Sheet 10 | 16.12.2021 | 16.12.2021 | |

Sheet 11 | 13.01.2022 | ||

Sheet 12 | 20.01.2022 | 20.01.2022 | |

Sheet 13 | 27.01.2022 |