After successful completion of the course, students are able to solve polynomial interpolation problems (using Newton's or Lagrange's method), approximate roots of real functions in one variable using Newton's algorithm and to calculate approximations of solutions for ordinary differential equations of first order via Runge-Kutta-type methods. They can check and compare the order of convergence for different such algorithms for the respective problems.

They are able to perform basic vector calculations such as addition, scaling, scalar and cross product, projections, finding the angle between two vectors. They are able to differentiate between linear and non-linear maps, and they can represent linear maps between finite dimensional vector spaces by matrices. They have mastered the matrix-vector calculus, and they can decide whether vectors are linearly (in)dependent, a generating (spanning) system or a basis of a vector space. In simple cases they can determine the dimension of a vector space. They are able to solve systems of linear equations using Gaussian elimination; they can determine whether such a system is (uniquely) solvable or not. They can explain the connection between the solution method for systems of linear equations and linear differential equations. They can compute the determinant of a square matrix (using Gaussian elimination or Laplace's theorem). They are able to determine whether a matrix is regular in they are able to compute an inverse. They can correctly modify the matrix representation of a linear map when the bases change. They are able to explain the geometric meaning of an eigenvector and an eigenvalue and they can obtain such computationally. They are able to diagonalise square matrices if possible.

The participants are able to determine whether a function in multiple variables is partially differentiable and they can compute such a derivative,give a geometric interpretation, and verify its continuity. They can compute the derivative of a function in serveral variables via its Jacobian and use it to give a linear approximation of the function. They are able to compute the gradient, divergence and curl of a scalar / vector field. They can check whether a vector field has a scalar potential and they are able to compute one such. They can compute the derivative of compound functions via the chain rule using a matrix multiplication. Moreover, they can locally determine the derivative of implicitly given functions. They can furthermore compute a directional derivative of a scalar function, and they are able to interpret it. They are able to compute higher order derivatives by successive differentiation and they are able to perform a Taylor expansion of a scalar field up to a certain order. They can find critical points of a scalar field and check those for local extremal points using sufficient and necessary conditions. In easy cases they can also determine global extrema and give evidence of the global nature of the extremal point. Using Lagrange multipliers they are able to solve extremal problems with equational side conditions. They are moreover able to compute simple integrals over n-cubes and normal domains. They can also compute line integrals of vector and scalar fields, and they can explain the connection between independence of the line integral from the chosen path and the existence of a scalar potential for a vector field. Given a parametrization they are able to compute volumina of curved spacial domains (curves, surfaces etc).

Content: numerical methods, linear algebra, analysis of functions in several variables and examples from chemistry. Applications of the presented methods to the mathematical modelling of chemical problems will be given.

Presentation of general theoretical arguments, discussion of concrete problem cases, showcasing of (computational) solutions for concrete questions.

Dates (currently not relevant):

• on Tuesday, Wednesday and Thursday (8:10 to 9:00 a.m.) in GM2 (Radinger lecture hall)
• starting from 3 March 2020
• last date: 29 May 2020

Dates (during the corona induced situation):

• on Tuesday, Wednesday (8:10 to 9:00 a.m.), and Thursday (9:00 to 9:50 a.m.) via Zoom
• starting from 28 April 2020
• last date: 29 May 2020

Examination (information without guarantee):

• examination on 05 June 2020 concerns the content lectured in summer 2019 !
• first possible date (2020S): 03 July 2020
• alternative date (2020S): 04 September 2020,
• also within the subsequent semesters (cf. TISS), usually 3 dates (roughly beginning, mid, end of term)
• registration through TISS (consider the deadlines for registration or cancellation); only registered students are allowed to participate in the exam (to be only on the waiting list is insufficient)
• further information available under "Examination Modalities"

Exercise sessions:

• Introductory meeting ("Vorbesprechung") on Wednesday, 4 March 2019, 9:15-11:00 a.m. in GM 2 (Radinger lecture hall), directly following the lecture
• Registration for the exercises is to be done via TISS (after the introductory meeting).
• Beginning from 18 March for odd groups, from 25 March for even groups; (all groups fortnightly)
• Wednesdays (9:15-11:15 a.m.)
• exact dates and further details: see TUWEL
• problems to be solved: see TUWEL (will appear at least one week before the exercise session)
• information regarding exercise sessions during the corona pandemic: see TUWEL
  
  
• Announce (and upload) prepared problem solutions timely via TUWEL.
• latest timely submission (no exceptions):
  ##############################
  #      every Tuesday, 9:55 p.m.       #
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• necessary condition for a positive result:
• at least 60% of all problems have to be checked (29 > 6×8*0.6) and
• the presentation by the board is positive (on average)
• A bad average assessment of the oral presentation implies a bad final mark (independently of a possibly high percentage of prepared problem solutions).
• A low percentage of prepared problems (e.g. only slightly above the minimum of 60%) entails a bad (at best mediocre) final result, even in the case of a very good oral performance;
• however, a good to excellent oral presentation will relax the minimum requirements regarding the percentage of solved problems to achieve a certain final mark.
• There is no explicit written / oral examination for the exercise sessions (even if TISS is using this kind of terminology for the type of course assessment). Evaluation of the candiate's qualifications is done throughout the whole course during the exercise sessions.

• Course assessment is done by written exam.
• 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)
• 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 2020; 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 he/she is 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! 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} [other collections are explicity forbidden]; 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, drawing of graphs of real functions, matrix operations, integrated computer algebra system, programmable calculators
• 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.
• If (after confirmed registration) at the date of the examination the exam cannot be taken due to illness, a medical report is to be presented. Unexcused absence will be sactioned within the legal regulations. Announce your condition to the person supervising the exam before the exam starts, i.e. before you will be handed out the exam paper. Otherwise, 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).

• "Mathematische Grundlagen für Chemiker", Volumes I, II and III. These books are available as e-books (TU Studienbibliothek).

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

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".