Course syllabus for Multivariable calculus and partial differential equations

Course syllabus adopted 2026-02-19 by Head of Programme (or corresponding).

Overview

  • Swedish nameFlervariabelanalys och partiella differentialekvationer
  • CodeMVE255
  • 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 44114
  • Open for exchange studentsNo
  • Only students with the course round in the programme overview.

Credit distribution

0108 Examination 7.5 c
Grading: TH		 7.5 c

In programmes

Examiner

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 course
Applicants 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 multivariable calculus, with a clear connection to analytical and numerical methods for solving partial differential equations (PDEs).

The course develops the student’s 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 integral calculus), linear algebra, and introductory programming in Python.

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

  • Explain and apply basic concepts in the analysis of functions of several variables, such as sets, sequences, and convergence in $R^n$, limits and different types of continuity, and formulate and use the concepts of derivative, gradient, Jacobian and Hessian matrices, chain rule, and Taylor’s theorem to analyse and approximate such functions, for example by linearisation and numerical differentiation.
  • Formulate problems in several variables as systems of equations and optimisation problems, use the gradient and directional derivatives to find stationary points, classify these using the Hessian matrix, and apply Newton’s method and basic methods for optimisation on compact sets and under constraints.
  • Formulate and compute double and triple integrals, use Fubini’s theorem to separate integrals, perform change of variables with the Jacobian determinant (for example in polar, cylindrical, and spherical coordinates), and apply double and triple integrals to determine mean values, centres of mass, moments of inertia, and improper integrals.
  • Formulate and compute line and surface integrals of scalar and vector fields, use gradient, curl, and divergence, apply Green’s theorem, Stokes’ theorem, and Gauss’ theorem (divergence theorem) in simple situations, and analyse conservative and incompressible fields with interpretation in terms of work, circulation, and flux.
  • Formulate and use basic concepts for discretisation and approximation in several dimensions, such as triangulations and tetrahedral meshes, basis functions and spaces of piecewise polynomial functions, and barycentric coordinates, and apply these to interpolation, projection, and numerical integration (quadrature).
  • Formulate basic partial differential equations with associated boundary and initial conditions as models for stationary and time-dependent phenomena, give an overview of strong and weak formulations, and apply the finite element method to linear problems, including derivation of the weak form, stiffness and mass matrices, assembly, and the use of simple time-stepping methods.
  • 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.

Content

Differential calculus in several variables: Sets, sequences, and convergence in $R^n$; limits and continuity (including uniform and Lipschitz continuity); differentiability, derivative, partial derivatives, gradient, Jacobian and Hessian matrices; chain rule; Mean Value Theorem and Taylor’s theorem; linearisation and numerical differentiation.

Equation solving and optimisation: Newton’s method for systems; gradient and directional derivatives; stationary points and classification via the Hessian matrix; optimisation on compact sets and under constraints (Lagrange multiplier method); overview of the inverse and implicit function theorems.

Integral calculus in several variables: Double and triple integrals; Fubini’s theorem; change of variables and the Jacobian determinant (polar, cylindrical, spherical coordinates); mean values, centres of mass, and moments of inertia; improper integrals.

Line and surface integrals: Line and surface integrals; work, circulation, flux; gradient, curl, and divergence; Green’s theorem, Stokes’ theorem, and Gauss’ theorem; conservative and incompressible fields.

Discretisation and approximation in several dimensions: Triangulations/tetrahedral meshes; basis functions and spaces of piecewise polynomial functions; barycentric coordinates; interpolation, projection, and quadrature.

Numerical solution of PDEs: Basic concepts for PDEs; strong and weak formulations; boundary and initial conditions; basic models (Laplace/Poisson, heat conduction, advection–diffusion); finite element method for linear problems, weak form, stiffness and mass matrices, assembly and sparse structure; time stepping (explicit/implicit Euler, midpoint method); nonlinear problems and Newton’s method; visualisation.

Applications and programming: Python as a computational tool for PDEs; implementation, testing, and verification; for example, advection–diffusion, the Schrödinger equation, and training neural networks as an application of multivariable calculus and the gradient method.

Organisation

Teaching consists of lectures and exercise classes in smaller groups. More detailed information is provided on the course web page before the start of the course.

Literature

S. Larsson, A. Logg, A. Målqvist, MATEMATISK ANALYS & LINJÄR ALGEBRA (IV): Flervariabelanalys och partiella differentialekvationer

Examination including compulsory elements

Assessment is by a written examination. Assignments or short quizzes that may give bonus points towards the examination may be included.

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.