105A Forum 2005

Welcome to the 105A forum.

Thursday 22nd of September 2005 09:50:10 AM
this class rocks

THANKYOU. THANKYOU VERY MUCH...

CLASS, ANY SUGGESTIONS?

YES

WEIRD THAT THE OHMS SYMBOLS ARE MISSING. EVERYTHING IS IN OHMS.

HMMM. THANKS, I'LL LOOK INTO IT.

OY, THAT'S A PROBLEM, HAVING TO COPY N QUESTIONS ONTO THE ASSIGNMENT RATHER THAN JUST THE PROBLEM NUMBERS. BUT, I'M SUCH A NICE GUY. CHECK THE WEBSITE IN AN HOUR OR SO....

IT'S UP TO YOU WHAT YOU WANT TO HAND IN-- YOU NEED TO ENSURE THE SOLUTIONS ARE PROPERLY EXPLAINED. IF YOU FEEL A PIZZA HUT NAPKIN WILL HELP YOUR GRADE, HAND IT IN. WRAPPED AROUND A PIZZA, PREFERABLY (THE TA GETS HUNGRY).

THANKS FOR FINDING THE TYPO. I'LL TRY TO CATCH THEM, BUT CAN'T PROMISE I'LL BE SUCCESSFUL. LET ME KNOW IF YOU FIND MORE.

ACTUALLY, I'M HAPPY TO GET IT RIGHT IN ONE PLACE AT LEAST. THESE MISTAKES OCCUR IN THE TEXT BECAUSE THE TEXT WAS INCORRECTLY TRANSCRIBED FROM THE CD BY THE PUBLISHER, AND I DIDN'T CATCH THE ERRORS DURING THE PROOFING PROCESS. AS A RESULT THE CD HAS FEWER ERRORS THAN THE PRINTED BOOK. I DO CHECK FOR TYPOS CAREFULLY, BUT IT'S NOT EASY TO FIND THEM ALL. IT IS HELPFUL TO GET STUDENT FEEDBACK BECAUSE YOU SEE THINGS MY BRAIN SKIPS OVER. FEEL FREE TO RETURN THE BOOK IF YOU LIKE. YOU MAY WANT TO TAKE A LOOK AT THE TEXTBOOK WEBSITE (SEE THE LINK ON THE COURSE HOMEPAGE) WHERE THERE IS A LIST OF FOUND TYPOS. IT MIGHT PROVE USEFUL IN FUTURE WORK.

THE PROBLEM IS NOT REVISED, ONLY THE HINT. THE INITIAL CONDITIONS, IF THAT IS WHAT YOU WERE REFERRING TO, REMAIN UNCHANGED.

LATE HOMEWORK IS PENALIZED 10% PER DAY LATE, UP TO THE CUT-OFF DATE AFTER WHICH IT IS NOT ACCEPTED. IF IT COMES IN BETWEEN 5-6 PM ON FIRDAY, IT IS PENALIZED 5%

Best is to type it in. Otherwise, it depends on if the work is a stand-alone written problem, or supporting material in the notebook. If it is the former, it will be graded late according to when it arrives in our possession. If the latter, it may (depending on the judgement of the grader) be graded according to when the notebook arrives. If you can't submit such work on time, and don't want to type it into the notebook,you could scan it and email the scanned material.

IT WOULD BE NICE IF WE HAD THE LUXURY OF PUTTING OFF ASSIGNMENTS FOR A WEEK, BUT THAT WASTES TIME. IT IS EXPECTED THAT STUDENTS WILL HAVE TROUBLE FULLY ABOSRBING THE MATERIAL IF THEY ARE SEEING IT FOR THE FIRST TIME IN LECTURE. FIRST AND FOREMOST, YOU ARE SUPPOSED TO READ AHEAD. (READ SECS. 2.1-2.3 FOR NEXT WEEK). SECOND, THE ASSIGNMENTS THEMSELVES ARE ONE OF THE BEST WAYS TO LEARN THE MATERIAL. YOU MAY THINK YOU KNOW A SUV=BJECT, BUT YOU DON'T REALLY KNOW IT UNLESS YOU CAN DO THE PROBLEMS. IN MY OPINION IT IS BEST TO WORK ON THE PROBLEMS AT THE SAME TIME AS YOU ARE THINKING ABOUT THE MATERIAL IN THE CLASSES, NOT WAITING A WEEK.

I AGREE A SOLUTIONS MANUAL WOULD BE NICE. I'M WORKING ON ONE.

THE TA REMAINS IN THE LAB UNITL 5 PM BECAUSE HE HAS HIS OFFICE HOUR THERE, FROM 4-5 PM.

I will post last years midterm and solutions. Do you really want to spend three hours on the midterm? Past years have not done that and they have done OK.

ANY METHOD IS OK

THE PLOTSTYLE COMMAND ONLY WORKS IF THE ADD-ON GRAPHICS LIBRARY IS LOADED, AS IN THE FIRST LINE OF CELL 2.1. THEOREM 2.1 IS CORRECT AS STATED. THE FOURIER SERIES EQUALS F(T) FOR ALL T WHEN AN INFINITE NUMBER OF TERMS ARE KEPT. THIS IS AS OPPOSED TO FOURIER SERIES OF FUNCTIONS THAT DO NOT SATISFY THM. 2.1. HOWEVER, YOUR STATEMENT IS ALSO TRUE. IT NECESSITATES A DISCUSSION EARLY ON OF UNIFORM CONVERGENCE, WHICH I WANTED TO PUT OFF UNTIL THE SEC. ON UNIFORM AND NONUNIFORM CONVERGENCE. (ACTUALLY, IF YOU THINK ABOUT IT LONG ENOUGH, YOU CAN CONVINCE YOURSELF THAT THE TWO STATEMENTS OF THE THEOREM ARE THE SAME.)THANKS FOR POINTING OUT THE N=S M TYPO -- THAT'S ONE I WAS ALREADY AWARE OF, BUT HAVE NOT POSTED YET.

FIXED

NO, DO BOTH A AND B

UP TO SEC. 2.1 INCLUSIVE

WHAT FOLLOWS IS THE TEXT OF THE EMAIL I SENT TO ALL STUDENTS ON OCT. 20. The midterm for physics 105A is at 2:30PM -4 PM on Friday Oct. 28. For those of you with surnames starting with letters in the range A - HEDBERG inclusive, the test is in UH 6126. For the other students, the exam is in APM B337 & B349.

IT DOWNLOADS AND OPENS OK FOR ME.

YES DO ALL THE INTEGRAL WORK BY HAND, EXCEPT IN PROBLEM 18B. OF COURSE YOU ARE ALWAYS WELCOME TO CHECK YOUR RESULTS USING MATHEMATICA.

THIS YEAR 105B WILL BE OFFERED IN THE SPRING RATHER THAN THE WINTER

DUE MONDAY, AT 4 PM. TURN IN WRITTEN WORK AT MY OFFICE MH 3102 (DROP BOX OUTSIDE THE DOOR)

THANKS FOR THE USEFUL COMMENTS. SELECTED SOLUTIONS WILL EVENTUALLY BE POSTED ON THE BOOK WEBSITE, AS AN ONLINE SOLUTIONS MANUAL. NOT EVERY SPECIAL CASE CAN BE HANDLED WITH AN EXAMPLE IN THE BOOK ITSELF, OR THE BOOK WOULD BE MUCH TOO LONG. HOWEVER, YOUR COMMENTS ABOUT DEGENERACY ARE WELL FOUNDED AND I WILL EVENTUALLY INCLUDE EXAMPLES ON THE WEBSITE. NOTE THAT FINDING INDEPENDENT HOMOGENEOUS SOLUTIONS FOR THE CASE OF DEGENERACY IS COVERED IN SEC. 1.6.2.

THOSE ARE EXCELLENT TEXTS, AND I AGREE THEY ARE USEFUL ADJUNCTS TO THE COURSE. I ALSO LIKE POWERS, BOUNDARY VALUE PROBLEMS. ARFKEN MAY BE A LITTLE ADVANCED FOR AN UNDERGRAD. CLASS. EIGENFUNCTION EXPANSIONS ARE ALSO COVERED IN THIS BOOK, BUT NOT UNTIL CHAP. 4. (NOTE THAT EIGENFUNCTION EXPANSIONS ARE NOT NECESSARILY THE BEST WAY TO REPRESENT A GREEN'S FUNCTION, AKTHOUGH THEY HAVE THEUR USES. REPRESENTATION IN TERMS OF HOMOGENEOUS SOLUTIONS IS SOMETIMES PREFERABLE, AS IS A NUMERICAL REPRESENTATION (NOT COVERED IN ANY OF THE ABOVE). I THINK IT'S BEST TO COVER SEVERAL DIFFERENT APPROACHES. ANOTHER GOOD REFERENCE FOR ANALYTIC DESCRIPTION OF GREEN'S FUNCTIONS IS JACKSON, CLASSICAL ELECTRODYNAMICS.

I graded problem 1, the TA graded problem 2. If you have complaints about the grading, first see the TA to try to resolve it, and if you are not satisfied, come see me. For the midterm, just come see me directly on problem 1, since I graded it.

THE NEXT HOMEWORK WILL BE DUE NEXT MONDAY, SO YOU HAVE A WEEK BETWEEN ASSIGNMENTS.

I'M UNSURE MYSELF. IN PRINCIPLE, THERE IS NO DIFFERENCE AS LONG AS THERE IS CONSISTENCY. I'M MERELY POSTING THE COMMENTS AS THEY COME. I'M UNDER THE IMPRESSION THAT SOME PROPLE MAY HAVE ISSUES WITH THE WAY THE GRADING HAS BEEN DONE, HENCE MY COMMENT ABOVE.

DON'T USE MATYRIX INVERSION, DO THE PROBLEM ANALYTICALLY USING HOMOGENEOUS SOLUTIONS AS IN 2.4.2

NO, G IS NONZERO. THE ODE IS THIRD ORDER SO THREE BOUNDARY CONDITIONS ARE NECESSARY. NO TYPOS. EXCEPT, WHEN YOU PLOT, PLOT G(X,1/2) NOT G(X-X0).

I MEAN G(T,1/2). IT IS ARBITRARY, BUT YOU NEED TO CHOOSE SOME VALUE OR OTHER FOR X0 IN ORDER TO MAKE A PLOT. YOU CAN TRY X0 = 0 TOO IF YOU LIKE.

2.3.85 is only correct for ODEs with constant coefficients. Derive the green's function from first principles, using the jump condition. Re[] works on all complex functions.

WHAT YOU FOUND WAS A PARTICULAR SOLUTION. NOW ADD IN A HOMOGENEOUS SOLUTION TO THE ODE WITH COEFFICIENTS CHOSEN TO MATCH THE ICS/BCS

USE THE VERSION ON THE CD (WITH FORCE F(X) =X ADDED)

WELL, THAT'S QUITE A MOUTHFUL! OK, AS TO COMMENT 1, YES, FUNCTIONS CAN THEMSELVES BE ARGUMENTS OF OTHER FUNCTIONS (THESE OTHER FUNCTIONS ARE THEN 'FUNCTIONALS', FUNCTIONS OF FUNCTIONS. MATHEMATIUCA CAN DEAL WITH FUNCTIONALS VERY NICELY.). I WOULD HAVE WRITTEN YOUR INTEGRATION FUNCTIONAL, SLIGHTLY DIFFERENTLY, AS

FcSin[f_, {a_, b_}] := Integrate[2/L Sin[n Pi x/L] f[x], {x, a, b}]

. THEN IS ONE DEFINES A FUNCTION G[R_] = R^2; AND CALLS FcSin[G,{1,2}], the integrals are done over G.

AS TO THE USE OF UNDERSCORES, REMEMBER THAT WITHOUT THE UNDERSCORE THE FUNCTION IS DEFINED ONLY FOR THAT SPECIFIC VALUE OF THE ARGUMENT. IF YOU DEFINE J[R], AND THEN ASK FOR J[X] SOMEWHERE ELSE, YOU WILL FIND IT IS NOT DEFINED.HOWEVER, IF YOU DEFINE J[R_], IT IS DEFINED FOR ANY R. YOUR EXAMPLE ONLY WORKS BECAUSE YOUR INTEGRATION VARIABLE WAS r. IF YOU HAD WRITTEN

g[X_] = Integrate[j[X] Log[X],X]

, THE FUNCTION WOULD NOT HAVE WORKED UNLESS j WAS DEFINED WITH AN UNDERSCORE AS j[R_] = R^2.

THE MINUS SIGN ALREADFY APPEARS IN EQ. (3.1.29). THE SOURCE TERM HAS A MINUS SIGN IIN FRONT OF IT BECAUSE WE TOOK IT OVER TO THE RHS OF THE EQUATION.

I DON'T KNOW IF ITS THE ONLY WAY TO MAKE AN EXTERNAL PACKAGE, BUT I CAN TELL YOU THAT ALL EXTERNAL "ADD-ON" PACKAGES USE THE BEGINPACKAGE ENDPACKAGE SYNTAX. THE EQUIVALENT TO INCLUDE OR REQUIRE IN MATHEMATICA IS GET, OR <<. THIS READS IN A SPECIFIED EXTERNAL FILE OR SET OF FILES. IT'S WHAT WE USE TO LOAD ADD-ON PACKAGES.

WHEN INITIAL CONDITIONS DO NOT MATCH THE BOUNDARY CONDITIONS, A GIBBS PHENOMENON OCCURS. THE LARGE-WAVENUMBER OSCILLATION QUICKLY DECAYS AWAY IN THE HEAT EQUATION SOLUTION.

HERE'S AN EXAMPLE. SAY THE BOUNDARY CONDITIONS ARE T=1 AT X=0 AND T=2 AT X=L, BUT THE INITIAL CONDITIONS ARE T=1 + 2X/L. THE BOUNDARY CONDITION MATCHES THE INITIAL CONDITION AT X=0, BUT NOT AT X=L. A GIBBS PHENOMENON ARISES IN THE SOLUTION AT X=L. THE MORE TERMS ONE KEEPS IN THE FOURIER SUM, THE NARROWER THE REGION OVER WHICH THE OSCILLATIONS OCCUR.

IT DOESN"T MATTER.

FIRST, HAPPY THANKSGIVING. SECOND, CHECK TO SEE IF A SOLUTION EXISTS! (SEE EQ. 3.2.3)

THE CYLINDER IS BOTH LONG AND GROUNDED.

YOU ARE RIGHT THAT M5 DOES NOT SOLVE POISSON'S EQUATION. YOU CAN CHECK YOUR SOLUTION BY MAKING SURE THAT IT SOLVES LAPLACE'S EQUATION TERM-BY TERM (BY SUBSTITUTION) AND BY ENSURING THAT YOUR SOLUTION MATCHES THE BOUNDARY CONDITIONS.

YES, THERE IS A TYPO IN THE EQUATION AFTER 3.2.14. THERE SHOULD BE AN EXTRA FACTOR OF 2/A. FORTUNATELY, THE INTEGRAL IN THE EQUATION IS ZERO, SO ALL THE AN'S ARE ZERO ANYWAY, FOR N>0

YES

THE M=0 HARMONICS ARE INDEPENDENT OF PHI. THE BOUNDARY CONDITIONS ARE ALSO INDEPENDENT OF PHI, SO ONLY THE M=0 HARMONICS ENTER THE SOLUTION.
also, aside from l=0,1 the figures for the other l values in cell 3.21 look like wave-fitting around a circle. (n lambda = 2 pi r) this reminds me of fitting de broglie waves around a circle to (somewhat more rigorously) arrive at the bohr quantum condition. is there any significant relation in this coincidence? (the waves also look like closed orbit diagrams from Goldstein).

SPHERICAL HARMONICS FORM A COMPLETE AND ORTHOGONAL SET OF FUNCTIONS ON THE SURFACE OF A SPHERE. SO YES, THEY ARE LIKE WAVES ON A SPHERE. SINCE THEY ARE ALSO EIGENFUNCTIONS OF THE ANGULAR PART OF THE LAPLACIAN OPERATOR, THEY APPEAR IN MANY CONTEXTS INCLUDING SHRODINGER'S EQUATION.

TAKE IT AS DZ/DT = 0 INITIALLY.

PHI(A,THETA) IS NONZERO ON THE SECTOR -- THERE THE POTENTIAL IS ONE VOLT. PHI IS ZERO OUTSIDE THE SECTOR. SO PHI(A,THETA) = V0(THETA) WHERE V0(THETA) IS SOME FUNCTION OF THETA. YOU NEED TO DESCRIBE THAT FUNCTION AS A FOURIER SERIES, AND USE THESE COEFFICIENTS IN YOUR LAPLACE EQUATION SOLUTION.

THE HEIGHT OF THE SURFACE IS THE VALUE OF THE POTENTIAL. YOU CAN SEE THE POTENTIAL VARIES ROUGHLY LINEARLY FROM ONE POLE TO THE OTHER AS YOU MOVE ALONG THE AXIS OF THE SPHERE.