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Graduate courses

Departments' graduate courses for PhD-students.


Syllabus for

Academic year
TMV225 - Differential calculus and algebraic equations  
Differentialkalkyl och algebraiska ekvationer
Syllabus adopted 2020-02-10 by Head of Programme (or corresponding)
Owner: TKMAS
7,5 Credits
Grading: TH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Education cycle: First-cycle
Major subject: Mathematics

Teaching language: Swedish
Application code: 55112
Open for exchange students: No
Only students with the course round in the programme plan

Module   Credit distribution   Examination dates
Sp1 Sp2 Sp3 Sp4 Summer course No Sp
0108 Examination 7,5 c Grading: TH   7,5 c   27 Oct 2020 am J   07 Jan 2021 am J,  19 Aug 2021 am J  

In programs



Anders Logg

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General entry requirements for bachelor's level (first cycle)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Specific entry requirements

The same as for the programme that owns the course.
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Course specific prerequisites



The purpose of the course is to strengthen, deepen and develop the
knowledge in secondary school mathematics and to thereby give a solid
ground for further studies in mathematics. The course will also
introduce the habit of using numerical computation in mathematics.

Learning outcomes (after completion of the course the student should be able to)

  • apply basic notation for set theory;
  • apply basic notation for mathematical logic;
  • define the real numbers;
  • define and apply the concept convergent sequence;
  • define and apply the concept Cauchy sequence;
  • define and apply the concepts open and closed set;
  • reason about computer representation of real numbers (floating point numbers);
  • define and apply the function concept;
  • define and apply basic concepts for functions, including surjectivity, injectivity, bijectivity, inverse function, graph, restriction, function algebra and composition;
  • apply basic properties of the elementary functions, including polynomials, rational functions, the power function, the exponential, the natural logarithm, the trigonometric functions and the inverse trigonometric functions;
  • define and apply the concepts limit, continuity, uniform continuity and Lipschitz continuity;
  • determine limits symbolically by using standard limits;
  • compute limits numerically by extrapolation;
  • define and apply the concepts derivative and differentiability;
  • determine derivatives of the elementary functions;
  • define and apply the differentiation rules for sum, difference, product, quotient, composition (the chain rule) and inverse function;
  • determine extreme values;
  • define and apply the mean value theorem;
  • define and apply the concept linearization;
  • define and apply numerical derivative;
  • define and apply Taylor polynomials and Maclaurin polynomials;
  • define and apply ordo notation;
  • determine limits by using Taylor expansion and ordo notation;
  • define the concepts series, convergence, absolute convergence and conditional convergence;
  • determine convergence for series by using convergence tests and standard series (geometric series and p series);
  • define the concepts power series, center and radius of convergence;
  • determine the center and radius of convergence of a power series;
  • define the concepts Taylor series and analytic function;
  • define the concepts equation, root and fixed point;
  • rewrite equations in fixed point form;
  • compute the solution of equations by using the bisection algorithm;
  • analyze the bisection algorithm (Bolzano's theorem);
  • compute the solution of equations by using the fixed point algorithm;
  • analyze the fixed point algorithm (Banach's fixed point theorem);
  • compute the solution of equations by using Newton's metod;
  • define the concept order of convergence and know the order of convergence for the bisection algorithm, the fixed point algorithm and Newton's method;
  • implement and apply the above mathematical concepts and algorithms to solve complex problems by simple computer programs.


The course is about real numbers, functions, derivatives and algebraic equations. Equal weight is given to the three pillars mathematical theory, analytic techniques and numerical methods.

Set theory, mathematical logic, number systems, sequences, convergence, Cauchy sequence, real numbers.

Functions, function algebra, the elementary functions.

Limit, continuity, uniform continuity, Lipschitz continuity, symbolic and numerical computation of limits.

Derivative, derivatives of the elementary functions, differentiation rules, extreme values, the mean value theorem, linearization, numerical derivative.

Taylor polyomial, series, limits by ordo notation, series, power series, Taylor series.

Equation, root, fixed point, bisection algorithm, fixed point algorithm, Newton's method, order of convergence.

Implementation and application of the above mathematical concepts and algorithms.


Instruction is given in lectures and classes. More detailed information will be given on the course web page before start of the course.


S. Larsson, A. Logg, A. Målqvist, Analys och linjär algebra del I: Differentialkalkyl och algebraiska ekvationer

Examination including compulsory elements

More detailed information of the examination will be given on the course web page before start of the course.
Examples of assessments are:
-selected exercises are to be presented to the teacher orally or in writing during the course,
-other documentation of how the student's knowledge develops,
-project work, individually or in group,
-written or oral exam during and/or at the end of the course.
-problems/exercises are to be solved with a computer and presented in writing and/or at the computer.

Page manager Published: Thu 04 Feb 2021.