Aktuella meddelanden /Messages.
Välkommen till kursen/Welcome to the course.
Exam
2017-Jan-2 with solution
Schemat för
kursen hittar du via länken till webTimeEdit på sidans topp./The schedule is found on the web in the previous
directory.
2016-11-03. The grading of the Exam 2017-10-22 will be finished on
Tuesday Nov. 8. (It might take a while
to get your official results on LADOK.)
You can get your exams back and ask questions on the grading on
Wednesday Nov 9. 12:00-12:30,
MVH12 (MV building H, first floor).
2016-10-26. Exam 2016-Oct-22 with solution
2016-10-10. Home work 3 rewritten
(the earlier exercise on Laplace transform was too difficult). Deadline
Thursday 13 Oct.
2016-10-06. We’ll have one hour
lecture on Laplace Tranform on Monday Oct 10 (to
catch up the schedule – we’ve gone through a bit more material on Fourier T.
than usual)
2016-09-08. The course representatives are
Anton Johansson, erikantonjohansson@live.com
Philip
Nilsson, Philip.nil@gmail.com
Nazli
Raufi, nazli.raufi@gmail.com
Lärare /Teacher
Kursansvarig och Övningsledare / Examinator: Genkai Zhang
Email: genkai@chalmers.se,
office: MVH5023.
Kurslitteratur /Litterature
Text book: G. Folland, Fourier Analysis and Its
Applications, Amer. Math. Soc., 1992
(A list of corrections to Folland's
book (in Swedish), and another one in English)
Exercises
[F]: Exercise from Folland's book, part of them will be solved during the
exercise classes. See the Planning below
[E]: Additional Exercises.
[S]: Supplementary (and elementary)
exercises. See the Planning below.
Check-list: A
description of the main concepts and theorems along with list of required proofs of some major theorems.
Program
Planning
Lectures on Tuesdays/Thursdays, exercise
classes Mondays/Wednesdays. (Tuesday Aug. 30, (part of) Wednesday 31 and Monday
Sept. 19 will be instead lectures). To benefit from the exercise classes you
should prepare at least a few of the suggested problems before the class.
Föreläsningar/Lectures.
Totally 8 weeks, week 1 = Monday Aug. 29- Sunday Sept. 4, week 2 = Monday Sept
5 --11, and so on.
|
Dag/Day |
Avsnitt/Chapter |
Innehåll/Content |
|
Week 1. 29/8- Tues. 30/8 |
1.1-1.3 |
Introduction/Motivation. Wave equations/heat equations. Method of Separation of Variables for Solving PDE. |
|
Wedn. & Thurs 31/08-1/9 |
1.3, 2.1 |
Separation of varibles. Fourier series |
|
W2. 5/9- Tues. |
2.2-2.4 |
Inversion/Convergence Theorem |
|
Thurs. |
2.3-2.4 |
Convergence. Further Properties of Fourier Series (FS): Differentiation and Integration of F.S. |
|
W3. 12/9- Tu. |
2.5-2.6 |
Application of FS to Heat and Wave Equations, Gibbs phenomenon |
|
Th. |
3.1-3.4 |
Hilbert spaces, L2 theory of Fourier series |
|
W4. 19/9 Mon. |
3.2-3.4 |
More on Hilbert spaces and L^2 spaces. Completeness. Parseval theorem. |
|
Tu |
3.5 |
Sturm-Liouville problems |
|
Th. |
4.1 |
Inhomogeneous
boundary value problems |
|
W5. 26/9 Tu |
7.1-7.2 |
Fourier Transform (FT) |
|
Th |
7.1-7.2 |
Properties of FT. Riemann-Lebesgue Lemma. Inversion and Plancherel formula. |
|
W6. 3/10 Tu |
8.1-8.2 |
Laplace Transform (LT) |
|
Th |
7.3 pp. 229-231. 8.3 pp 273-279 |
Applications of FT and LT. Diff. Eqs on the real line/half line. |
|
W7.10/10 Tu |
4.4 |
Laplace equations, Harmonic functions. Dirichlet problems. |
|
Th |
4.4 |
Poisson formula for solutions of Dirichlet problem on the disc. |
|
W8.17/10 Mon.
Tu |
Repetition |
Repetition. Solving earlier examination problems. |
|
Exam. Sat. 8:30-12:30 |
|
|
Rekommenderade
övningsuppgifter/Recommended Exercises
[F] refers to Folland's book and [E]
to the additional exercises.
(We shall solve some of the exercise below for the demonstration on the
blackboard. Please try yourself
before the class, and try those unsolved
ones afterwards.)
|
Dag/Day |
Uppgifter/Exercises |
|
W1 Wedn |
[E]
1,2, [F] 1.1.1, 1.1.5, 1.1.6, 1.2.4, 1.3.1,
1.3.4, 1.3.7 |
|
W2 Mon. |
[F] Table 1: 2, 8, 20, [E]3, 4 |
|
Wedn |
|
|
[F]
2.2.2, 2.2.4, 2.4.3, 2.4.7,
2.4.11, [E] 5, 6, 86 |
|
|
W3 Mon |
F]
2.3.2, 2.3.3, 2.3.5, [E] 8-10,
31-32. |
|
Wedn |
F.
2.3.2, 2.3.5,
[E] 13, 15 |
|
W4 Mon.
Wedn |
[F] 2.5.1, 2.5.5, ] 3.3.10, 3.4.6, 3.4.7,
[E]16-18, 36-37 |
|
W5 Mon |
[F]
3.5.1, 3.5.2, [E] 51-53 |
|
Wedn |
[F]
4.2.3, 4.2.5, 4.3.6,
[E]25-28 |
|
W6 Mon |
[F]
7.1.1, 7.2.2,
7.2.3, 7.2.12, [E]54ace,
56, 58. |
|
Wedn |
[F]
7.2.13ab, 7.3.1, [E] 60cd, 64, 67, 76. |
|
W7 Mon |
[E] E 74-75, 78-79, 81 |
|
Wedn. |
[F] 4.4.1, 4.4.2, 4.4.6, [E]22, 30. |
|
W8 Mon |
[S] Supplementary and elementary
exercises:
(The following exercises
are more elementary than those in the text book and are to help you get
started.)
Part I (Fourier Series).
Part II (L^2-theory, convergence, Sturm-Liouvill).
Part III (Differential equations).
Part IV (Fourier and Laplace transforms).
Supplementary texts on (1) Applications
of Laplace transform on Filter, (2) Laplace transform, (3) Gibbs phenomenon by Hjalmar
Rosengren
Further recommended textbooks:
1. A. D. Wunsch, Complex variables. Pearson, Addison
Wesley.
There is an introduction of complex function
theory from the very beginning of algebra and geometry of complex numbers. Also
many example
of Laplace transforms are computed using residue calculus.
2. G. Arfken and H. Weber, Mathematical methods for
physicists. Academic Press.
Many physics problems are presented motivating the mathematical tools such as
Laplace and Fourier transforms and Fourier series.
(Auxillary
links to Math. Softwares: Matlab/Mathematica/Maple)
(En del av studenter kan redan göra avancerade Matlab
beräkningar. Ni som inte har lärt sig Matlab kan
försöka kolla material nedan. Mathematica och Maple
är också bra verktyg, man kan till ex rita Fourierserier
och ha en intuitiv uppfattning för konvergensen. /Some of the students
have already learnt Matlab. Otherwise below is some list of references.)
Referenslitteratur:
- Material
(utvecklat av MV) som ger en kortfattad introduktion till Matlab
- Holly More,
MATLAB for Engineers
Ger en introduktion till Matlab och kräver inledningsvis ingen matrisalgebra. Är utmärkt för självstudier. - Per Jönsson, MATLAB-beräkningar
inom teknik och naturvetenskap
Kräver kunskaper i Matrisalgebra. Innehåller lite mer avancerade övningar och modelleringsuppgifter. Är utmärkt som referenslitteratur/uppslagsbok.
Kurskrav
Kursens mål finns angivna i kursplanen.
Duggor
Homework with bonus points.
Three sets of home-work-exercises for hand-in, on weeks 3, 5, 7. You may work in teams (of max. 3 team members) and submit
you team’s work.
Max 2 bonus points will be given for the home-works. The bonus can be added
to reach G (Pass) but not VG (Very Good)
Examination
Examination
Written examination with total 24 points. One can get max 2 bonus
points for the oral and written presentation of the assigned exercises, the
bonus
can also be added to gain the grade G (Pass) but not VG (Very Good).
Grade:
MMG710 (GU students), grades are G (12-17 points) and VG (18-24 points).
TMA362 (Chalmers students), grades are 3 (12-14 points), 4 (15-17 points) and 5
(18-24 points).
Rutiner kring
tentamina
General Guide/Routines on Examination
(Please ask me or your fellow students if you need help for the Swedish text).
I tentamensscheman
anges alla tentor för respektive period.
Vid tentamen ska du kunna uppvisa giltig legitimation.
Du kan läsa i Chalmers
studentportal om vilka regler som gäller kring att tentera på Chalmers, men
observera att du som går på GU ska anmäla dig till tentan via GU:s studentportal.
För att se ditt resultat gå till Ladok via inloggning
i Studentportalen (GU).
Granskning vid ordinarie tentamen:
Då det är praktiskt möjligt ordnas ett separat granskningstillfälle av
tentamen. Tidpunkt för detta meddelas på kurshemsidan. Den som inte kan delta
vid granskningen kan efter granskningstillfället hämta och granska sin tenta på Matematiska vetenskapers studieexpedition, se
information om öppettider. Kontrollera att Du har fått rätt betyg och att
poängsumman stämmer. Eventuella klagomål på rättningen ska lämnas skriftligt på
expeditionen, där det finns en blankett till hjälp.
Vid omtentamen:
Tentorna granskas och hämtas ut på Matematiska
vetenskapers studieexpedition, se
information om öppettider. Eventuella klagomål på rättningen ska lämnas
skriftligt på expeditionen, där det finns en blankett till hjälp.
Kursutvärdering
I början av kursen bör minst två studentrepresentanter utses för att
tillsammans med lärarna genomföra kursutvärderingen. På kursens aktivitet i GUL
(inloggning
via Studentportalen) finns en enkät som används vid utvärderingen.
Utvärderingen sker genom samtal mellan lärare och studentrepresentanter under
kursens gång samt vid ett möte efter kursens slut då enkätresultatet diskuteras
och rapport skrivs på speciell blankett.
Gamla tentor