After successful completion of the course, students are able to... [Esteemed readers, ask the maintainers of TISS why this sentence has to be here, don't ask me!]

After successful completion of the course, students are able, for topics from numerical mathematics, linear algebra and analysis mentioned under ‘Subject of course’, to determine whether problems are well-defined, in selected cases if they are solvable, and in concrete instances to obtain a theoretically justified computational solution and possibly to give a geometric interpretation thereof. They are able to check whether certain properties hold, to decide whether the application of certain methods makes sense, and to give sound reasons for why their results are correct. They are able to define important mathematical concepts and to explain interconnections between them. Moreover, they are able to master elementary mathematical models and employ these models properly in suitable circumstances, as well as to understand mathematical language and formalism so that they can independently study mathematical textbooks and work their way into new subject areas (as needed, for example, for applications in physics or physical chemistry).

Mathematical methods and models from the following subjects:

• Numerical mathematics: polynomials interpolation, quadrature of functions, approximative calculation of roots of functions of one variable, approximative solutiosn for 1st order ordinary differential equations via Runge-Kutta-type methods, comparison of the quality of numerical methods and assessment of the order of convergence for certain such algorithms.
• Linear algebra: basic operations using vectors and scalars, matrix vector calculus, Euclidean inner products, angle, cross product, lineare (in)dependence, generating (spanning) systems, basis, dimension (simple cases), linear maps, kernel, image, rank, coordinatisation of linear maps via matrices and change of coordinates, rank-nullity-theorem, systems of linear equations, Gaussian elimination, rank, determinant and inverse of a matrix, geometric interpretation of eigenvectors and eigenvalues of linear maps and matrices, computation in the finite-dimensional case, diagonalisability, applications to HMO-theory
• Multi-variate real analysis: partial derivatives and their continuity, derivative of a function using its Jacobian and linear approximation, existence of a scalar potential of a vector field, computation of the derivative of compound maps via the chain rule (matrix product), (local) derivatives of implicitly given functions, directional derivative, higher order derivatives by iterated differentiation, Taylor expansion, checking scalar fields for local extrema via application of certain sufficient and necessary conditions to critical points, global extrema in simple cases, extrema with equational side conditions via Lagrange multipliers, simple integrals over n-cubes and normal domains, line integrals of vector and scalar fields, volumes of bent spacial domains (curves, surfaces, manifolds) with given parametrisation, relationship between independence of line integrals of vector fields from the chosen path to the existence of a scalar potential

In the lecture the mathematical contents are presented mainly by data projection;
this includes presentation of general theoretical arguments, discussion of concrete problem cases, showcasing of (computational) solutions for concrete questions.

From the students' side: active attendance during the lecture (livestream); preparation of written notes; reassess, consolidate and expand the range of understanding by autonomously solving problems of the corresponding exercise course.

According to the pandemic situation the procedure of the course may vary.

No lectures on 23 to 25 May 2023; please use the videos provided under course materials.

• Course assessment is done by written exam.
• usually 3 examination dates per semestre (roughly beginning, mid, end of term, cf. TISS)
• Registration for the exam is possible via TISS (consider the deadlines for registration or cancellation carefully); only registered students can take the exam (to be simply on the waiting list is not sufficient). Regarding examinations in front of a board of examiners, it is not possible to register in TISS. Instead the examinee has to notify, within the registration period, the main examiner as well as the dean's office of their intent to sit the exam.
• For organisational questions regarding the exams (registration, cancellation, announcement of sickness etc.) please refer to our secretary Ms Marietta Meszlenyi.
• There will be given 100 minutes (1 h 40 min) to complete the exam.
• The whole content of the lecture is subject to examination (not only the topics covered by the exercise sessions) [The term "content" describes the content of this lecture as given in the summer term of 2023; it does not apply and is perhaps unrelated to other content presented (by possibly other lecturers) in other semesters. It is the duty of the candidate to make sure they are aware of possible changes (if applicable) compared to previous semesters. Exams concerning the content of previous years (within the legal regulations) can be arranged on an individual basis.].
  As a general rule there will be 5 problems, each allowing to earn 16 out of 80 points total. There will be practical questions dealing with modelling or calculations similarly to the exercise sessions (usually 3/5), as well as theoretical questions concerning the definitions, concepts and theorems (including their proofs) covered in the lecture (usually 2/5). In general, the following marks are assigned for x out of 80 points: x<40: mark 5 (insufficient, fail), 40≤x<50: mark 4 (sufficient, pass), 50≤x<60: mark 3 (satisfactory), 60≤x<70: mark 2 (good), 70≤x≤80: mark 1 (very good). General rules regarding the distribution of questions and marks may be subject to exceptions (changes usually occur in favour of the examinees)! The content of the problems is randomly chosen from the topics of the lecture.
• Authorized resources:
  1) Collection of mathematical formulae: the collection "Mathematik für Chemiker" that is available at Institut 104 or a collection containing the endorsement "für die allgemeinbildenden höheren Schulen zur Abfassung der schriftlichen Reifeprüfung" authored by (a possibly proper subset of) {Götz, Bürger, Kraft, Unfried, Haschkovitz} published by öbv [other collections are explicity forbidden]; except for highlighting no additional notes in the collection of formulae are permitted.
  2) Calculators: calculators are admitted to perform the basic numerical operations (such as addition, subtraction, multiplication, division, square roots) and to calculate values of fundamental functions (trigonometric operations, exponentials and their inverses); calculators allowing for one of the following operations are not approved: numerical or symbolic integration, numerical or symbolic derivation, automated drawing or generation of value tables of graphs of real functions, matrix operations, integrated computer algebra system, programmable calculators, integrated regression
  FOR A LIST OF FORBIDDED / ALLOWED CALCULATORS, please see the German version of this site.
• In case of doubt, it is the responsibility of the candidate to check the acceptability of the means he or she intends to use in advance.
  Student identity cards are to be presented upon the exam to permit identity verification.
• After confirmed registration, unexcused absence will be sactioned within the legal regulations. Announce withdrawal due to health conditions to the person supervising the exam before the exam starts, i.e., before you will be handed out the exam paper. Otherwise, after the exam has started, illness cannot be recognised as an excuse to resign from the examination and the test will be marked (resulting in a failed attempt as the case may be).
• Whenever possible the exam will take place in a lecture hall.
• Necessary technical equipment in case of an online-exam: Computer with video camera, speakers and microphone; reliable internet connection; access to TUWEL.
  Detailed information regarding on-line exams will be made available (in TISS, or via message to the registered participants) should the necessity arise. On-line examinations will usually be split up into a written and an oral part.
• Oral inquiry in order to authenticate the written exam is possible.

Further reading: 

• H. Zachmann & A. Jüngel, "Mathematik für Chemiker", Wiley-VCH, Weinheim, 2007.
• J. Michael Fried," Mathematik für Ingenieure I und II -Für Dummies", Wiley-VCH, Weinheim, 2010 and 2013.
• H.-G. Roos & H. Schwetlick, "Numerische Mathematik. Das Grundwissen für jedermann." Mathematik für Ingenieure und Naturwissenschaftler. 220 pages. Teubner, Stuttgart/Leipzig, 1999
  ISBN 3-519-00221-3
• M. Drmota & B. Gittenberger & G. Karigl & A. Panholzer, "Mathematik für Informatik" Berliner Studienreihe zur Mathematik. Volume 17. 438 pages. Heldermann, Lemgo, 2008
  ISBN 3-88538-117-4
• A. Kielbasinski & H. Schwetlick, "Numerische lineare Algebra. Eine computerorienierte Einführung." Mathematik für Naturwissenschaft und Technik 18. 472 pages, Deutscher Verlag der Wissenschaften, Berlin, 1988.
  ISBN 3-87144-999-7
• Helmut Länger, "Grundlagen der Analysis und Linearen Algebra", TU-MV Media Verlag, Wien, 2018. ISBN 978-3-903024-83-0
• H. Heuser, "Lehrbuch der Analysis Teil 1" Mathematische Leitfäden. 17th edition. XI+632 pages. Vieweg+Teubner Verlag, Wiesbaden, 2009
  ISBN 978-3-8348-0777-9
• H. Heuser, "Lehrbuch der Analysis Teil 2" Mathematische Leitfäden. 13th edition. 737 pages. Vieweg+Teubner Verlag, Wiesbaden, 2004
  ISBN 978-3-663-01407-2

Additional material (multimedia resources):

• Grant Sanderson (3Blue1Brown), "Essence of linear algebra".
• Grant Sanderson (3Blue1Brown), "Essence of calculus".
• Khan Academy, "Multivariable calculus".

It is recommended to complete "Mathematics for Chemists I".