Materials to the seminar "Theoretisches Minimum (160203)"

• Announcement of the seminar with description of aims and further details can be found here. The name of the seminar is taken from famous Landau Theoretical Minimum (LTM). You can look the recent programme of LTM exams at Moscow Landau Institute for Theoretical Physics here. You see that our seminar is a preparation to the first step of LTM. Very brief info about LTM you can find e.g. here.


• 22.04.2009
  First, about organization: we shall have common seminar every two weeks on Wednesdays (Mi) at 11:00 in NB6/173 (next seminar on 06.05.2009) where we shall discuss one particular mathematical method and you will get your homework. Your homework we discuss a week after the common seminar. It should be personal discussion, meaning that you will show the way you solve the problems in personal discussion. The solution you can discuss with Maxim (NB6/123), Alena (NB6/125) or Alexei (NB6/135). For the moment we suggest the following "time windows" for such discussions: Di. 14:00-16:00, Mi. 10:00-12:00, 14:00-17:00 and Do. 9:00-11:00. We are thinking about possibility to make the second common seminar for those who can not attend the seminar on Mi at 11:00, please send your wishes to Maxim per e-mail.
  
  On the first common seminar we discussed the elementary integrals and each student received a list of 100 numbers, each corresponds to an integral in the list which you can download here. Please bring the list of numbers you were given to the personal discussion next week. If somebody would like to get the list for homework please contact Maxim.
  
  Good luck with solving the first 100 integrals!!


• 28.04.2009
  Several people already successfully solved first 100 integrals. Till Mi. 06.05 these people are given a set of 10 more complicated integrals from the list which you can download here. Please return the solution in written form till Mi. 06.05. If you have difficulties with some of integrals or you find some elegant solution please come to us and we discuss! MAA
  
  


• 06.05.2009
  Today we discussed important clas of integrals which can be guaranteed computed - that are intergrals of a rational function . Such integrals are computed with help of Partialbruchzerlegung . Very helpful method for fast calculation of integrals of rational functions is the Ostrogradsky method.
  We also discussed that many seemingly very complicated types of integrals can be computed with clever substitutions. For example, with Euler substitutions or with Weierstrass substitution . However, to see which substitution to perform one needs to calculate many, many integrals!
  You can download the new set of integrals here. You solve 33 integrals from this list and next week come to Maxim (Alena and Alexei will be away) to present your solutions at blackboard!
  
  
• 14.05.2009
  Our next common seminar is on Mi. 20.05 at 11:00 (NB6/173). Here you find 7 "Kopfbrechende" integrals which I ask you to try to solve till 20.05 and give me back in written form. The first 3 are more or less standard but one need to know one method (which was known to Newton), I suggest you invent this method youself (or find it in books). Another 4 integrals could be very difficult but they can be computed!
  
  From the previous list of integrals I did not manage to compute integral Nr. 68, those, who "geknackt" it please tell me the solution! For the integral Nr. 40 I could not find an elegant way to solve, I did it only through standard method finding all roots. Who knows an elegant way, send me e-mail!



• 20.05.2009
  At common seminar we discussed 3 topics: 1) integrals depending on a parameter 2) multidimensional integrals and 3) curve integrals. The set of problems is here. This set consists of 33 integrals, 11 for each discussed topic. The first 11 integrals can be computed using tricks with differentiation or integration over a parameter. In this set you will compute famous integrals of Gauss, Frullani, Dirichlet, Fresnel and Laplace. These integrals are frequently used in physics and we would be happy if you will invent methods to compute them! Also in the list of integrals we offer problems with multidimensional and curve integrals.
  
  As usually please solve these integrals at home and next week we discuss your solutions individually at blackboard.
  
  As an advise for literature we recomend books by V.I. Smirnov and G. Fichtenholz. These gentlemen wrote classical courses of analysis. Read info about them on Wiki. I checked that the physics library has books by V.I. Smirnov and books of G. Fichtenholz can be found in library in NA.
  
  
• 22.05.2009
  The integrals from the list of 14.05 belong to the class of binomial integrals. I am glad that some people managed to solve them! The integral Nr. 68 from the list of 06.05 can not be computed in elementary functions! Please consult Jan Sieverding, he proved that!
  
  
  
• 25.05.2009
  We updated the list of integrals from 20.05! There were several misprints!
  
  
• 10.06.2009
  Today on the common seminar we started with differential equations. We considered the simplest types of differential equations: 1) DE with separable variables 2) homogeneous (scale invariant) DE 3) Linear DE and as special cases two famous types of DEs: Bernoulli equations and Riccati equations . You can download first 77 DEs to be solved by you and presented to us at the blackboard next week.
  
  
• 19.06.2009
  More complicated differential equations you can download here . Please return the solutions in written form after next common seminar on Mi. 24.06.
  
  
• 24.06.2009
  On todays seminar we discussed linear differential equations and different methods of their solution - for example method of variation of constants. The set of 55 differential equations are here . Please note that for the differential equation of the second order sometimes one of the solutions is given, one has to find the second solution with e.g. Liouville-Ostrogradsky method. If the solution is not given try to look for it in the form of a polynomial, or exponential form. Good luck!
  
  
• 08.07.2009
  Today we had the last common seminar in the framework of "Theoretisches Minimum" :( The last topic was about the stability of the solutions of the differential equations. We discussed Lyapunov theorem which is the central theorem in the theory of the stability of dynamical systems. We touched also very interesting topic from linear algebra - Hurwitz criterium for the negativity of roots of the algebraic equations. The problems to this topic are here . As usualy you present your solution on the blackboard next week.
  
  I am thinking whether it makes sense to do "Theretisches Minimum II" in future. To make decision I need a response from you! You can either use the faculty evaluation system (passwords you can pick up in my office) or you can directly write to me what do you think.
  
  Thank you very much for your interest. For me it was very interesting and useful!