Advanced Mathematics

Gudrun Thäter and Jonathan Rollin talk about their plans for the
course Advanced Mathematics (taught in English) for mechanical
engineers at the Karlsruhe Institute of Technology (KIT). The
topics of their conversation are relevant in the mathematical
education for engineers in general (though the structure of
courses differs between universities). They discuss


how to embrace university mathematics,

how to study,

what is the structure of the educational program and

what topics will be covered in the first semester in
Karlsruhe.



For students starting an engineering study course it is clear,
that a mathematical education will be an important part.
Nevertheless, most students are not aware that their experiences
with mathematics at school will not match well with the
mathematics at university. This is true in many ways. Mathematics
is much more than calculations. As the mathematical models become
more involved, more theoretical knowledge is needed in order to
learn how and why the calculations work. In particular the
connections among basic ideas become more and more important to
see why certain rules are valid. Very often this knowledge also
is essential since the rules need to be adapted for different
settings.


In their everyday work, engineers combine the use of
well-established procedures with the ability to come up with
solutions to yet unsolved problems. In our mathematics education,
we try to support that skills insofar as we train certain
calculations with the aim that they become routine for the future
engineers. But we also show the ideas and ways how mathematicians
came up with these ideas and how they are applied again and again
at different levels of abstraction. This shall help the students
to become creative in their engineering career.


Moreover seeing how the calculation procedures are derived often
helps to remember them. So it makes a lot of sense to learn about
proofs behind calculations, even if we usually do not ask to
repeat proofs during the written exam at the end of the semester.


The course is structured as 2 lectures, 1 problem class and 1
tutorial per week. Moreover there is a homework sheet every week.
All of them play their own role in helping students to make
progress in mathematics.


The lecture is the place to see new material and to learn about
examples, connections and motivations. In this course there are
lecture notes which cover most topics of the lecture (and on top
of that there are a lot of books out there!). So the lecture is
the place where students follow the main ideas and take these
ideas to work with the written notes of the lecture later on.


The theory taught in the lecture becomes more alive in the
problem classes and tutorials. In the problem classes students
see how the theory is applied to solve problems and exercises.
But most importantly, students must solve problems on their own,
with the help of the material from the lecture. Only in this way
they learn how to use the theory. Very often the problems seem
quite hard in the sense that it is not clear how to start or
proceed. This is due to the fact that students are still learning
to translate the information from the lecture to a net of
knowledge they build for themselves. In the tutorial the tutor
and the fellow students work together to find first steps onto a
ladder to solving problems on the homework.


Gudrun and Jonathan love mathematics. But from their own
experience they can understand why some of the students fear
mathematics and expect it to be too difficult to master. They
have the following tips:


just take one step after the other, and do not give up too
early

discuss problems, questions and topics of the lecture with
fellow students - talking about mathematics helps to understand
it

teach fellow students about things you understand - you will
be more confident with your arguments, or find some gaps to fix

take time to think about mathematics and the homework
problems

sit down after the lecture, and repeat the arguments and
ideas in your own words in order to make them your own

use the problem classes and tutorials to ask questions



In the lecture course, students see the basic concepts of
different mathematical fields. Namely, it covers calculus, linear
algebra, numerics and stochastics. Results from all these fields
will help them as engineers to calculate as well as to invent.
There is no standard or best way to organize the topics since
there is a network of connections inbetween results and a lot of
different ways to end up with models and calculation procedures.
In the course in Karlsruhe in the first semester we mainly focus
on calculus and touch the following subjects:


Numbers

Methods of proof

Complex numbers

Sequences and convergence

Functions and continuity

Series

Differential calculus of one real variable

Integral calculus

Numerical integration

Elementary differential equations



All of these topics have applications and typical problems which
will be trained in the problem class. But moreover they are
stepping stones in order to master more and more complex
problems. This already becomes clear during the first semester
but will become more clear at the end of the course.

Literature and related information

K. F. Riley, M. P. Hobson, S. J. Bence: Mathematical Methods
for Physics and Engineering. Cambridge University Press.

K. F. Riley, M. P. Hobson: Foundation Mathematics for the
Physical Sciences. Cambridge University Press.

T. Arens, F. Hettlich, Ch. Karpfinger, U. Kockelkorn, K.
Lichtenegger, H. Stachel: Mathematik.Spektrum Akademischer
Verlag, Heidelberg (in German).

J. Stewart: Calculus, Early Transcendentals. Brooks/Cole
Publishing Company.

K. Burg, H. Haf, F. Wille: Höhere Mathematik für Ingenieure.
Volumes I-III. Teubner Verlag, Stuttgart (in German).

E. Kreyszig: Advanced Engineering Mathematics. John Wiley
& Sons.

E.W. Swokowski, M. Olinick, D. Pence, J.A. Cole: Calculus.
PWS Publishing Company. Boston.


Podcasts

F. Hettlich: Höhere Mathematik, Gespräch mit G. Thäter im
Modellansatz Podcast, Folge 34, Fakultät für Mathematik,
Karlsruher Institut für Technologie (KIT), 2014.

J. Eilinghoff: Analysis, Gespräch mit S. Ritterbusch im
Modellansatz Podcast, Folge 36, Fakultät für Mathematik,
Karlsruher Institut für Technologie (KIT), 2014.