Course syllabus adopted 2026-02-19 by Head of Programme (or corresponding).
Overview
- Swedish nameIntegralkalkyl och ordinära differentialekvationer
- CodeTMV151
- Credits7.5 Credits
- OwnerTKMSK
- Education cycleFirst-cycle
- Main field of studyMathematics
- DepartmentMATHEMATICAL SCIENCES
- GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Course round 1
- Teaching language Swedish
- Application code 44119
- Open for exchange studentsNo
- Only students with the course round in the programme overview.
Credit distribution
Module | Sp1 | Sp2 | Sp3 | Sp4 | Summer | Not Sp | Examination dates |
|---|---|---|---|---|---|---|---|
| 0108 Examination 7.5 c Grading: TH | 7.5 c |
In programmes
Examiner
- Axel Målqvist
- Full Professor, Applied Mathematics and Statistics, Mathematical Sciences
Eligibility
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
Specific entry requirements
The same as for the programme that owns the courseApplicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements
Course specific prerequisites
Aim
The aim of the course is to study the theory and applications of integral calculus in mathematical analysis, with a clear connection to analytical and numerical methods for solving ordinary differential equations (ODEs).The course develops the students ability to use both analytical and numerical methods, and to use programming in Python as a natural tool for modelling and computation. By combining theory, computation, and applications, the course strengthens both theoretical understanding and practical problem-solving skills, and provides a solid basis for further studies in mathematics and engineering.
The student is expected to have prior knowledge corresponding to introductory courses in mathematical analysis (differential calculus) and introductory programming in Python.
Learning outcomes (after completion of the course the student should be able to)
- Explain and apply the concept of the Riemann integral for functions of one real variable, understand the interpretation of the integral as area, use the Fundamental Theorem of Calculus, and give an overview of improper integrals and associated notions of convergence.
- Formulate ordinary differential equations of first and higher order as mathematical models for time-dependent phenomena, give an overview of existence and uniqueness, and solve basic classes of ordinary differential equations using analytical methods.
- Formulate and analyse systems of first-order ordinary differential equations in vector and matrix form, and reduce higher-order equations to such systems.
- Understand and use the Laplace transform as a tool for analysing and solving linear initial value problems, use basic operational rules for the transform, and handle step functions and impulses.
- Formulate initial value and boundary value problems for ordinary differential equations, and choose, apply, and give a basic analysis of fundamental numerical methods for their solution, with respect to, for example, order of convergence, stability, and preservation of relevant quantities.
- Implement and use numerical algorithms in Python for computations within the scope of the course, and analyse, interpret, and communicate in writing modelling choices, computational results, and limitations in engineering applications.
- Use classical analytical methods and basic numerical methods to compute integrals, and use integrals to determine quantities such as arc length, area, volume, and mass-related quantities in simple models.
Content
Integral: Area as a limit; Riemann sums; definition of the Riemann integral; Mean Value Theorem for integrals; Fundamental Theorem of Calculus; improper integrals, absolute convergence, and the comparison test.
Computation of integrals: Change of variables (including inverse substitution); integration by parts; integration of rational functions via partial fractions; arc length, area, and volume; solids of revolution; numerical integration (midpoint, trapezoidal, Simpsons rule); error estimates and convergence.
Ordinary differential equations: Classification of differential equations (ordinary/partial, order); existence and uniqueness (Picards theorem); first-order ODEs (separable equations by separation of variables, linear equations with integrating factor); reduction from second-order to first-order ODEs; linear ODEs with constant coefficients; Eulers equation.
Systems of ordinary differential equations: Matrix and vector notation; systems of first-order equations with initial conditions; reduction of higher-order ODEs to systems of first-order equations; existence and uniqueness for systems; elementary functions defined via ODEs.
Laplace transform: Definition and linearity; scaling and exponential scaling; differentiation/integration; convolution; step function and Dirac delta (impulse); solution of initial value problems.
Numerical solution of ordinary differential equations: Initial value problems; explicit/implicit Euler methods and the midpoint method; order and stability analysis (stability region); energy-preserving properties; boundary value problems; the finite element method in one dimension (strong/weak form, simple assembly).
Applications and programming: Python as a computational tool for integration and ODEs; implementation, testing, and validation; for example, computation of mass and centre of mass, simple ecosystem models, and two- and three-body dynamics.
Organisation
Literature
Examination including compulsory elements
The course examiner may assess individual students in other ways than what is stated above if there are special reasons for doing so, for example if a student has a decision from Chalmers about disability study support.