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## Syllabus for

TMA975 - Real analysis

Owner: TTFYA
8,0 Credits (ECTS 12)
Grading: TH - Five, Four, Three, Not passed
Level: B
Department: 11 - MATHEMATICAL SCIENCES

Teaching language: Swedish

 Course module Credit distribution Examination dates Sp1 Sp2 Sp3 Sp4 No Sp 0197 Examination 4,0 c Grading: TH 4,0 c 11 Dec 2004 pm V, 02 Apr 2005 am V, 19 Aug 2005 am V 0297 Examination 4,0 c Grading: TH 4,0 c 14 Mar 2005 am V, 14 Jan 2005 am V, 24 Aug 2005 pm V

#### In programs

TTFYA ENGINEERING PHYSICS, Year 1 (compulsory)

#### Examiner:

Docent  Kjell Holmåker

#### Eligibility:

For single subject courses within Chalmers programmes the same eligibility requirements apply, as to the programme(s) that the course is part of.

#### Aim

The course provides basic knowledge of
the fundamental theories within mathematical analysis.

#### Content

Part A:
Ordinary differential equations: linear equations of the first order, separable equations, linear differential equations of arbitrary order with constant coefficients, systems of equations, some special types such as Euler's differential equation. Mathematical models giving rise to differential equations. Numerical solution of differential equations.
Taylor's formula, computation of limits, l'Hospital's rules.
Difference equations. Sequences, series, power series, convergence criteria, solution of differential equations by means of power series. Uniform convergence of function sequences and function series.
The vector space Rn, polar and spherical coordinates, some topological concepts.

Part B:
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, arc length. Line integrals, Green's formula in the plane, potentials and exact differential forms.
Sufaces 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.
Functional determinants, implicit functions. Extremal problems for functions of several variables, Lagrange's multiplier rule.

#### Literature

A. Persson, L.-C. Böiers: Analys i en variabel, Studentlitteratur, Lund.
A. Persson, L.-C. Böiers: Analys i flera variabler, Studentlitteratur, Lund.
Övningar till Analys i en variabel, Matematiska institutionen, Lunds tekniska högskola.
Övningar till Analys i flera variabler, Matematiska institutionen, Lunds tekniska högskola.
F. Eriksson, E. Larsson, G. Wahde: Matematisk analys med tillämpningar, del 3.

OTHER LITERATURE
L. Råde, B. Westergren: BETA - Mathematics Handbook, Studentlitteratur, Lund.
E. Pärt-Enander, A. Sjöberg: Användarhandledning för Matlab 6, Uppsala universitet.

#### Examination

Two written examinations.

Page manager Published: Mon 28 Nov 2016.