Math 4567, Spr 2023 (secs 002), Prof. D.A.Hejhal This page will be used for general announcements and the homework listing. ----------------------------------------------------------------------- LAST UPDATE: 5-11-23 ----------------------------------------------------------------------- Final Exam Tally (out of 125; scores include any bonus) 125,121,114,113,112; 106,102,87,87; 82; ; (n=10) ----------------------------------------------------------------------- Mini-Test #3 tally (take-home) (n=10) 20,20,17,17; 17,17,17,16,14; 10 ----------------------------------------------------------------------- Midterm Exam Tally: 94,89,87,85,85,82,77,65,63,61 (n=10) ----------------------------------------------------------------------- MINI-TEST #2 on Wed, March 1, in-person. Grade Tally: 20,18,18, 15,15,15, 12,11,9, 4 (n=10) ----------------------------------------------------------------------- Tally for Mini-Test 1 (n=12) 20,12,12,12,12,11,11,10,10,8,7,5 ======================================================================= A list of sections that we have covered in Churchill & Brown during our lectures thus far. (I refer to the 8th edition.) Chap 1. Sec 1, 2 (note even/odd periodic extension), 3,4; skip 5; 6 (we used cases 3,3a), 7,8 (in sec 8, we did things using cases 1,2,3,3a). Chap 2. Sec 9 (f_R prime and f_L prime at x_0), 10 (we developed Bessel's inequality for general orthogonal fcns), 11 (see the Dirichlet kernel D_N on p.32 top), 12 (we covered Fourier's theorem on p.35 in a more general context, ie case 3a, and presented the proof in a pdf handout based on D_N and the Riemann-Lebesgue lemma [see p.165]), 13 (we stated the alternate form of Fourier's theorem on p.38 in the context of case 3a), 14, 15 (we based things on FSS(f) = FS(f_odd) and FCS(f) = FS(f_even), then utilizing the FS theorems on pp.35, 38), 16 (the fundamental lemma for type (abc) functions in case 3a), 17 (uniform convergence, the Weierstrass M-test, the 3 basic uniform convergence theorems on p.50 middle, and the Key FS Theorem for type (abc) functions on [a, a+2L]), 18 (we gave a brief discussion of Gibbs' effect based on studying the FSS of pi - x), 19 (differentiation for FS for type (abc) functions), 20 (we stated a better form of p.56 Theorem by considering case 3a and using Parseval's formula; we stressed that the base point x_0 is arbitrary and that the resulting series is always both uniformly and absolutely convergent). Chap 7. Sec 66 (gave Parseval's equation for cases 1,2,3a; compare p.207(8), 208(probs 2 - 4)); also in Sec 65 (discussed the equivalence of p.203(4), completeness, and Parseval's equation p.205(10) for perfectly general orthogonal functions on [a,b]; stressed the Pythagorean theorem throughout!). Chap 3. Sec 21, 22 (secs i & ii; got the heat equation for a 1-dimensional metal rod, both uniform and nonuniform density, also with internally generated heat), 26 (parts i & ii only); skipped 23-25 for now, also sec 27. Chap 4. Secs 32-36 and Separation of Variables. Sec 35 gives the very important Sturm-Liouville Theorem (needed to control the form of baby solutions); see, e.g., Sec 36 equations (9) thru (12). Chap 5. Intro on pp.113-114. Sec 39 (first half), 40 (first example). ======================================================================= HOMEWORK : "H" problems are the hand-in ones. All others are just recommended; do not hand the latter in! DUE DATES for the "H" problems will be sporadic and decided only after discussion with our HW grader. NB: unless otherwise indicated, all HW is to be handed in ON PAPER (*NOT* as a pdf emailed to me). assgnmt problems ----------------------------------------------------------- 1 "H" 1-5 #9 (using only trig functions, no complex numbers) "H" Using integ by parts, show that the FSS of f(x) = x(1 - x^2) on [0,1] is, in fact, the series given on p.382 of the book. Write out the formula for b_n clearly; show all steps in your integ by parts; write neatly. No use of a change-of-scale or complex numbers is allowed. "H" Show that EVERY inner product on vector space R^3 must have form = summation x_m times A_{mn} times y_n (where m, n range 1 to 3) for v = vector , w = vector . Here A_{mn} is some constant for each (m,n). Explain why A_{mn} = A_{nm}. Suggestion: use the i,j,k notation for v and w. Note: mn is a double-index, not m times n. ( Optional: if you happen to know, what type of matrix is [ A_{mn} ] ???? ) [end of mathematical part of Ass. 1; due Fri, Jan 27, 4:30pm] 2 REC: 1-5 #3,5,10 1-7 #1,5 1-8 #5,8,9 "H" 1-7 #2 (show all steps; use no complex numbers) "H" 1-7 in #2, using Theorem on p.38, find the value of this Fourier series when x = -20 to four decimal places "H" 1-7 #6 (DO NOT use prob 4 or 5, or any complex numbers; use only trigonometric identities instead) [end of Ass. 2; due date Fri, Feb 3 by *start* of lecture] 3 REC: Sec 15 #4 ; Sec 20 #1,2 "H" problems: Sec 15 #6 Sec 20 #5 (first Fourier series; use the integration theorem given in lecture and express your final graph on [-pi,pi] without any infinite series appearing) Sec 20 #5 (second Fourier series; use the same theorem & express your final graph on [-pi,pi] without any infinite series appearing) 4 "H" Problem: using Parseval's relation for Fourier Sine Series on [0,1] (state what it says first!) and a suitable FSS from the table on p.382, evaluate summation of 1/(m^6) for ODD m going 1 to +oo. See p.207(8) for Parseval's relation for a full Fourier series taken over [-pi,pi]. See also p.208 prob 4(c). 5 All are "H" problems. Section 39, problem 4. Use the PHI(x) method. Show any separation of variable procedure, but (to make matters shorter) you can quote the S-L Theorem in Sec 35. Do not quote anything from any other problem or from Example 2 (on pp. 116-117). Section 40, problem 4. Show all steps, but (to make matters shorter) you can quote the S-L theorem in Sec 35. Do not give any physical interpretation. Hint: use the stated answer to figure out the correct PHI(x). Wave equation with free ends. Using separation of variables, solve the wave equation u_{tt} = 16 u_{xx} on [0,1] subject to the conditions u_{x} (0,t) = 0, u_{x} (1,t) = 0, u(x,0) = x^2, u_{t} (x,0) = 4 + cos(3*pi*x). Show your separation and write your ansatz clearly. You can quote the S-L theorem in Sec 35. Suggestion: split into 2 problems as suggested by p.144 (middle). Note: you do NOT need to do anything like p.150(10); an "unreduced" form like p.150(9) or as in lecture is fine