Edwin's loggbok, hösten 2006

Edwin's loogbok ht 2005

KS= G. Sparr och A. Sparr, Kontinuerliga system, Studentlitteratur, Lund (2000)
ÖB= Sparr och Sparr, Övningsbok till Kontinuerliga system, Studentlitteratur, Lund (2000)
T = Kapitel 1 i Tensoranalys
R=
A. Ramgard, Vectoranalys

Hemtalen finns här.

OBS If you write email to me I will answer usually in english since this is faster for me. However, it is prefectly fine if you write swedish. For the same reason I write recommendations for homework in English.

Sammanfattning av föreläsningarna

Vad hände egentligen på vilken föreläsning!?

• Lektion 1, 30/8: Introduktion, kursens mål och PM. KS Kap. 1.1, 1.2 t.o.m. 1.2.2. Hemtal 1 finns här.
• Lektion 2, 1/9: KS Kap.  1.2.3-1.2.1, 1.3, 1.4.1, 1.6 till Ex. 1.3, 1.7  NOTE: I made two minor corrections in "hemtal 1"COMMENT: Chapter 1 in our course book ([KS] Kap. 1) is very important, and I very much recommed that you study it repeatedly (read not only once but reread parts of it later when we will discuss examples). One key to successfully formulate mathematical models is to know how to adapt some basic physical laws to a specific situation and know how to use mathematics to formulate it is a precise way. This Chapter discusses the physics background to many of the examples which we will later consider in more detail. "RÄKNESTUGA" on Tuesday, Sept 5, 15.15h (until 17.15h or so): repetition of important topics from DIFF&TRANS (the room will be announded in time here)
• Lektion 3, 4/9: KS Kap. 1.5 (energi hos en sträng), 1.8 (existens och entydighet av lösningen till ett problem), 1.10 (linjära differentialoperatorer och superpositionsprincipen),  tolkning av sinusserie som utveckling av en vektor i en bas: inledning till H.1, H.2 (pdf fil); 3.2.1 (lösning av ett endimensionell homogen värmeledningsproblem med Fouriers method). NOTE: I suggest you study Kap. 3.2.1 in all detail. Many problems we will discuss later will be solved by a variant of the method explained there.
• RS 5/9: I suggested to solve the following problems (pdf file). In 2(a)-(c)  you are asked to recall a chapter from diff&trans which we will soon need: how to solve simple linear inhomogeneous ODEs. I this pdf  file I also descibe in an example how to use MAPLE to check a your own computations: I very much recommend to do this.
• Lektion 4, 5/9: lika med Exempel 3.1 i KS, Kap. 3.2.1; samanfattning Kap. 3.2.3 och 3.2.3 (lösning av vågekv. och Laplaces ekv. med Fouriers method); fysikalisk tolkning av produktlösningar (sid. 76-81 in KS);   inledning till Kap. 3.3 (inhomogena linjära problem); 3.3.1 B (inhomogen värmeledningsekv.). COMMENT: In the lecture I only gave an outline of how to solve problems like in Example 3.2 (sid. 77) and 3.5 (sid. 82). I suggest you try to solve these problems in detail by yourself and only check with the solution in the book if you get stuck or to check your solution.
• Lektion 5, 6/9: Tensorer [T] Kap. 1.
• Lektion 6, 11/9: Tensorer [T] Kap. 1. forts.
• Lektion 7, 13/9: Hur man flyttar inhomogeniteter i randvillkoren till PDE; Helmhotz ekvationen; egenvärden och egenfunktioner; Hilbertrum ([KS] Kap. 3.3.1 och Kap. H; pdf fil för Lektion 3 som finns ovan; delvis repetition av DIFF&TRANS kursen). Hemtal 2 finns här.
• Lektion 8, 14/9: Speciella funktioner ([KS], Kap. S).
• Lektion 9, 20/9: Symmetriska operatorer och spektralsatsen; blandade begynnelse- och randvärdesproblem i en, två och tre dimensioner som kan lösas exakt ([KS], Kap. H och Kap. 3). Exempel 3.13 och 3.14 ([KS], sid. 106-109).
• Lektion 10, 20/9: NOTE:  since I realized that the original course plan was somewhat tight and I did not manage to discuss everything which I intended to discuss about Chapters 3, H and S in the course book. I thus still used this lecture to discuss these chapters. Until further notice the course plan in the following will thus be shifted by one. The same applies to the "Övningar": Ö9 will still be on Chapter 3.  Symmetriska operatorer och spektralsatsen;  Sturm-Liouvilleoperatorer;  blandade begynnelse- och randvärdesproblem i en, två och tre dimensioner som kan lösas exakt ([KS], Kap. H och Kap. 3). Exempel 3.16, 3.19 ([KS] sid. 111-113, 118-119).
•  Lektion 11, 26/9:  Laplaces ekv. för cirkelskivan (Exempel 3.17 och 3.19 i [KS]).  Fouriertransformen , distributioner (repetition);  PDE problem för  obegränsade områden: värmeledning i oändlig stav; Dirichlets problem för ett halvplan  ([KS] Kap. D och 4.1).  In the lecture I only summarized the main results from distribution theory which we need in a somewhat pragmatic way: Dirac delta and Hearviside function; derivation of distributions; solution of equations like xf(x)=0; Fourier transformation. I very much recommend you study  Chapter D in our course book in some detail  to get a better feel for the subject.
• Lektion 12, 27/9:   Greenfunktioner (pdf.fil  och [KS] Kap. 5).  In my lecture I put emphasis on the concept and physical intuition behind Green's functions - the notes in the pdf summarize most of what I discussed. The examples in the course book and "övningar" will (hopefully) demonstrate the usefulness of the method.
• Lektion 13, 2/10:  Greensfunltioner för  Poissonekv. med Dirichletrandvillkor: sammanfattning (Kap. 5.4 i [KS]). Vågutbredning: d'Alemberts formel och fysikalisk tolkning; harmononiska vågor; dämpning och dispersion: telegrafekv., grupp- vs. fashastighet; sfäriska vågor;  Kirchhoffs formel och retarderade potentialer ([KS] Kap. 7).
• Lektion 14, 3/10:  Variationsräkning (Kap. 8 i [KS] och pdf.fil). The section on variational calculus in the book is rather brief, and I thus wrote this notes that you have a more detailed discussion of the theory back ground behind the exercises on variational calculus in the "övningar".  Note that I typed in the notes rather quickly and I thus expect that there are typos.
• Lektion 15, 9/10: Variationsräkning (sammanfattning). Diskreta modeller: inledning till obligatoriska hemuppgift  ([KS] Kap. 2). Sammanfattning av kursen. Below I made available a simple MAPLE program solving the discretized hear equation in 1D, as discussed in Kap. 2 of [KS]. You can use this program as a starting point when you work on the solution of the "obligatory hemuppgift" coming out tomorrow. Note that my program below is rather basic.