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2.1 Misc. notes

  1. it is OVER !! finals finished.
  2. math 121A. review HW 11 done 11 PM wed. review of HW 12...finished 3:20 AM. (4 hrs per HW to go over!). getting ready to start on HW 13...6 AM, almost finished HW 13 most of the rest I know, about calculus of variations, studied that before, but need to finish HW13 (may be one more hr). Then start on the HW 10 (Fourier series). will do after wake up. now going to sleep. 1 pm Thursday.. finishing HW 13 now.... 3:00 PM finished HW 13. This contained important stuff. 8 hrs study only today
  3. now midnight Thursday. 1 hr study. went over some problem from midterm2, and Lagrange equations physics problem derivations. need to finish review of midterm 2, then back to HW review. it is 4 AM now, read notes and finished midterm, starting on HW 9, getting tired, will not  be able to review everything before finals, need to try to concentrate on last stuff only... 4:40 AM finished HW 10 (Fourier series, easy stuff), now starting on HW 9...5:20 AM Friday, finished HW 9. HW 8 is on Laurent series, so important... 6:40 AM, ok finished. going to eat something and sleep and wake up for the exam.
  4. finals for math 121B over. I made 3 very stupid mistakes., can’t believe I did those. blow away 3 fairly easy questions I could have full credit on. but I think I can pull a B in the course. keep fingers crossed.
  5. Practice more chapter 7 Fourier series tricks (odd/even) stuff
  6. Make sure I remember \(ds^{2}\) in all coordinates
  7. learn better how to evaluate this:\(\left (-\frac {1}{x}\frac {d}{dx}\right ) ^{n}\left (\frac {\sin x}{x}\right ) \)
  8. HW’s for 121B went over since midterm exam: HW5  chapter 12, HW 6 done, HW 7 stop here. Saturday night.., HW 11 done, HW 12 working on..finished. Now studying probability distribution, last HW
  9. write down the same space for the 2 die, with the sum, some problems use it.

questions:

  1. Why did we use series method to find solution to Legendre ODE, but used generalized series method to find solution to Bessel ODE? how to know when to use which? Answer: if ODE has something like \(\left (1-x\right ) y^{\prime \prime }\), then at \(x=0\) we’ll have problem, then use the generalized power series)
  2. The Legendre ODE is solved using series method, assuming \(l\) is an integer. We get one solution which is Legendre function of first kind \(p_{l}\relax (x) \). What if \(l\) is not an integer? A: Legendre \(P_{l}\) is only defined for integer \(l\)? YES? No, there are tables for non-integer, but these cases are not important.
  3. What if we get a legendre ODE and we want to find solution for \(x>1\) ? Since legendre functions are only defined for \(x\) less than one (to have convergence). Physics example? usually \(x\) is the cosine of an angle so it is \(\leq \) 1.
  4. What if \(l\) is not an integer in the legendre ODE? how to get a solution? this is special cases, not important, look up handbooks.
  5. problem I solved in HW#6, chapter 12, 16.3. check my solution. I claimed that the second solution is \(N_{p}\) but since I found \(P\) NOT to be an integer, hence the second solution is one containing log and not a combination of \(J_{-p}\).  When I solved it in mathematica, I get this solution (notice complex number?), could this second solution be converted to \(\log \) function? answer: OK, the solution I did will turn out to have log in it if I put p=integer and use L’hospital’s rule to evaluate.

    pict

    pict

    pict

  6. When solving for equation 16.1 on page 516, we seem to only take the positive root for the variables, why? see for example page 516. \(b=2\) but it is really \(b=\pm 2\) answer: OK, any of these will give a good solution, just pick one.
  7. on page 528, can I just set \(n=0\) always to solve for the indical equation as shown in the example? is it better to solve this using the \(\sum \) directly as shown in the example instead of setting up a table? table seems more clear, but the example method seems shorter.
  8. How to solve chapter 16, 4.1 part (c) using Bayes rule? I write: Let A=event first chair is empty, let B=event second chair is empty. We need to find \(P\left (AB\right ) =P\relax (A) P_{A}\relax (B) =(\frac {1}{10})(\frac {1}{9})=\frac {1}{90}\), but the answer should be \(\frac {1}{45}\), what is it I am doing wrong? wrong. \(P(A)=1/5\,\ not\ 1/10.\)
  9. for problem HW 12, chapter 16, 4.8, part b. It says given 2 cards drawn from deck, if you know one is an ace, what it the chance the BOTH are an ace? I know how to solve by the book. but why can I not say the following: since we KNOW that one card is an ace, then the chance that both cards are an ace is just the chance the second card being an ace (since we know the first is an ace). So this should give \(\frac {3}{51}\)
  10. random variable is defined as a function on the sample space. however, it is multivalued. for example, if x= sum of 2 die throw, then more than one event can give the same random variable. is this OK? I thought a function must be single valued? answer: I am wrong. it is NOT multivalued.
  11. check that my solution for chapter 16, 5.1 MATH 121B is correct, I have solution on paper. this is the last HW

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