Course syllabus for Multivariable analysis

Course syllabus adopted 2024-02-05 by Head of Programme (or corresponding).

Overview

  • Swedish nameFlervariabelanalys
  • CodeMVE600
  • Credits7.5 Credits
  • OwnerTKTEM
  • Education cycleFirst-cycle
  • Main field of studyMathematics, Engineering Physics
  • DepartmentMATHEMATICAL SCIENCES
  • GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail

Course round 1

  • Teaching language Swedish
  • Application code 59123
  • Maximum participants50
  • Open for exchange studentsNo
  • Only students with the course round in the programme overview.

Credit distribution

0119 Examination 6 c
Grading: TH		0 c0 c6 c0 c0 c0 c
0219 Intermediate test 1.5 c
Grading: UG		0 c0 c1.5 c0 c0 c0 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 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

Linear algebra MVE670 and Real analysis TMA976 or equivalent courses.

Aim

The course will provide familiarity with the most basic theories in mathematical analysis in several variables and shed light on their applications in physics and technology.
Specific aim for the 1.5 credit part: Provide a brief introduction to numerical analysis.

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

The goal is to provide the students with the necessary mathematical tools in multivariable calculus and 3-dimensional vector analysis for subsequent courses in the Engineering Physics and Technical Mathematics programs. Among the most important learning outcomes are the following:
  • To understand the basic concepts of multivariable differential calculus, such as: partial derivative, differentiability, linearization, gradient, implicit and inverse function theorems.
  • To be able to apply the chain rule to changes of variables in PDE.
  • To be able to find and classify the stationary points of a multivariable function and apply this knowledge to the solution of optimization problems.
  • To understand the definition of Riemann integral in arbitrary dimension.
  • To be able to apply some basic techniques when computing multiple integrals, such as: inspection/symmetry, Fubini's theorem, change of variables, level surfaces.
  • To be able to handle different parametrizations of curves and surfaces in 3-space, and understand the meaning of and be able to compute line and surface integrals.
  • To understand Green's theorem in the plane, plus Gauss' and Stokes' theorems in 3-space and apply these to the computation of line and flux integrals.
  • To acquire some basic knowledge of how the concepts of the course arise in physics, especially in mechanics and electromagnetism.
  • To be able to differentiate under the integral sign.
The course also includes a module of 1.5 credits, which gives a brief introduction to numerical analysis. After completing the course, the student should independently be able to solve simple problems in the field and explain the theory.

Content

Functions of several variables. Partial derivatives, differentiability, the chain rule, directional derivative, gradient, level sets, tangent planes. Taylor's formula for functions of several variables, characterization of stationary points. Double integrals, iterated integration, change of variables, triple integrals, generalized integrals. Space curves. Line integrals, Green's formula in the plane, potentials and exact differential forms. Surfaces in R3, surface area, surface integrals, divergence and curl, Gauss' and Stokes' theorems. Some physical problems leading to partial differential equations. Partial differential equations of the first order. Differentiating through the integral. Functional determinants, inverse functions theorem, implicit functions. Extremal problems for functions of several variables, Lagrange's multiplier rule.
For the 1,5 hp module: Erroranalysis and floating point arithmetics.Numerical solution of linear and non-linear equations and systems.
Simple numerical methods for computations of derivatives and integrals.

Organisation

Lectures and exercises.

Literature

A. Persson, L.-C. Böiers: Analys i flera variabler, Studentlitteratur, Lund. Övningar till Analys i flera variabler, Institutionen för matematik, Lunds tekniska högskola. OTHER LITERATURE L. Råde, B. Westergren: BETA - Mathematics Handbook, Studentlitteratur, Lund.

Examination including compulsory elements

A written examination.
A mandatory test.
Mandatory black-board presentation.
Bonus point rewarding tests.

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 on educational support due to disability.