Edwin's loggbok, våren 2007

Edwin's loogbok ht 2006

KS= G. Sparr och A. Sparr, Kontinuerliga system, Studentlitteratur, Lund (2000)
ÖB= Sparr och Sparr, Övningsbok till Kontinuerliga system, Studentlitteratur, Lund (2000)
[HS1]=Notes on Hilbert space theory
[VAR]=Notes on variational calculus

Old (somewhat more difficult) homework problems with solutions are  available  here.

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: 17/1. Introduktion, kursens mål och PM. KS Kap. 1.1, 1.2 t.o.m. 1.2.1. Delvis 1.6.1 och 1.6.2.
• Lektion 2: 18/1. [KS] 1.7, 1.2.2-1.2.3, 1.3, 1.4.1, 1.6 (delvis). 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.
• Lektion 3: 25/1. [KS] 1.4.1, 1.4.5, 1.4.4. (delvis), 1.5, lite om 1.8, 1.10(!), 3.2.1. Recommended reading: 3.1 and [HS1]. COMMENT:  As  discussed in the lecture, a key property of the PDE and RV problems which we treat in this couse is linearity.  I  thus strongly recommend you study [KS], 1.10 carefully.
• RÄKNESTUGA 30/1: I pointed out the importance of [KS], 1.10 for the course: linear differential equations and how to exploit linearity to solve such equations. You used this method before, and to recall I reviewed in this RS how linearity is used to solve linear second order ODE with constant coefficients, i.e., find the general solution f(t) of af''(t)+bf'(t)+cf(t)=0 (linear homogen eq.) and, more generally, af''(t)+bf'(t)+cf(t)=F(t) (linear inhomogen problem; F some given function). I recalled the rule "f_inhom=f_hom+f_part" (general inhom. solution = general homogen solution + one particular inhomogen solution). I tried to emphasis that it is useful to look at the theory and examples of such ODEs again,  trying to be more aware of the key role of linearity: we will use a very similar strategy to solve linear ODE problems.
• I also handed out a problem sheet: I suggest you use these to test yourself: try to do these problems yourself and only look at the solution when done. If you feel these problems are hard practice on similar problems from the teaching material you got in your ODE courses.
•  Lektion 4: 1/2: [KS] 1.10 (linjära differentialoperatorer och superpositionsprincipen); 3.1.2 (Fourierserier); 3.2.1 (värmeledning i en dimension) och 3.2.3 (vågekv. i en dimension); [HS1] (inledning till Hilbertrumteori); H.1 och H.2 (vektorrum, pre-Hilbertrum: teori och exempel).   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.  Appendix  H in our course book [KS] summarizes the mathematical theory of Hilbert spaces. I very much recommend you study that carefully (in my lectures I can only  give motivation and an overview).
• Lektion 5: 8/2: Operatorer i Hilbertrum; symmetriska operatorer;  spektralsatsen; Sturm-Liouville satsen: [KS] H.8, H.9,  H.10,  H.11. Recommended reading: I suggest you study all of appendix H in [KS] carefully. I also recommend to read Sect. 2 in [KS]: it gives a better intuitiv understanding about how symmetric matrices approximate symmetric operators and of Fourier's method.
• Lektion 6: 15/2: Problem i två dimensionoer som kan lösas exakt i polära koordinater; Besselfunktioner: Exempel 3.14 i [KS]; speciella funktioner: S.2 och S.3 i [KS];  exempel 3.18 och 3.19 i [KS].
• Lektion 7: 22/2: Problem som kan lösas exakt i sfäriska koordinater; sfäriska Besselfunktioner, Legendrepolynomer och klotfunktioner: Exempel 3.16 och Bilaga S.4, S.5 i [KS]. Variationsräkning: Kap. 8.1 i [KS] och [VAR](=pdf-notes on variational calculus).  NOTE: A good way to get more familiar with the special functions we discussed is to plot them using MATLAB. A program showing a few examples of how to do this can be found can be found here (bessel.m). THERE WILL BE A EXTRA LECTURE ON 23/2, 8-10, in FB53!!!
• EXTRA Lection: 23/2: Applications of special functions: examples of problems which can be solved exactly in polar and spherical coordiates. Inhomogeneous problems: I will discuss how the spectral theorem is used to also solve inhomogeneous problems.
• Lektion 7: 01/3: Variationsräkning och tillämpningar: Kap. 8 och [VAR]. NOTE: The list of variational problems misses "simple" examples (where the solution does not require longer computations), and a few examples do not have solutions. I try to improve that during the next week. I WAS ASKED TO SPREAD INFORMATION A SCHEDULE CHANGE IN AM36 (KÄRNFYSIK) KURS MODERN FYSIK.
• Lektion 8: 08/3:  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 recommend that you study  Chapter D in our course book in some detail  to get a better feel for the subject.
• Lektion 9: 15/3:   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 10: 20/3: Kap. 3.2 i [KS] ( dimensionlös form av en PDE). In the rest of the lecture I gave a summary of the course, similar the following pdf file).  A suggested solution for KS2 is available hereNOTE: There will be a "räknestuga"/"exempelräkning" (majority of students will decide) next monday, 14-16h, in FB53 (I reserved the room until 17h).