MIT
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18.085 Spring 2010



Tomorrow (Wednesday's) exam will cover Fourier series, a discrete convolution computation, circulant matrices, and a Fourier integral.



2010: We will follow the 2008 outline or indeed the video lectures fairly closely perhaps with a slight more emphasis on computing. For 2010 exams will be in class, open book and open notes, not evenings, no computers. Homeworks will be due on Fridays in spring 2010.
The exam dates will take place on the classes numbered 13, 26, and 39. The dates are Wednesday March 3, Friday April 9, and Wednesday May 12.
Strang's Videos (terrific)

Quizzes and Solutions for Spring 2010


Quiz 1 (pdf) and Solutions (pdf) .
Quiz 2 (pdf) and Solutions (pdf)
Quiz 3 (pdf) and Solutions (pdf)

HOMEWORK for Spring 2010


#11 (Final HW) Due Friday April 30 (Solutions to HW 11)
4.3: 6,7,9,20
4.4: 1,2,4,5,6,7
4.5: 3,4,5,7,11

#10 Due Friday April 23 (Solutions to HW 10)
Note: we may not cover this material in class, but it's very useful and similar in many ways to material we saw in class.
4.2: 1,2,3,7,8,10,20,21
Hint for problem 8: Write down T4(x) and combine until you get x^4.

By popular request (matlab to watch Fourier Series for square wave):

x=pi*[-1:.01:1]; for n=3:5:51, k=(1:2:n)'; plot(x, sum(diag(1./k)*sin(k*x)));title(['n=' num2str(n)]); pause(0.1); end

#9 (Solutions to HW 9)
Section 4.1: 2,3,4,5,7,8,9
Hints: 4: assume f(x) is general, not any one particular function
Hints: 9: Note the taylor series for 2*log(1+z) is 2(z-z^2/2+...). It's not that imaginary part of the taylor series is 2 log(1+z). Guess this could have been written clearer, but if you knew the series, you'd be ok. The real problem is a bit of a trig identity.

#8 (Solutions to HW 8) Due April 12 1pm
Section 3.4: 1,2,3,12,13
Section 3.5: 8,9,10

#7 (Solutions to #7) Due April 2 1pm
Section 3.2: 6. 8, 9. 10
Section 3.3: 1,2,3,4,5,6, 9
Hint on 3.2,6: I think we must remember that w=u" to think clearly here. Perhaps also it is good to realize that we will need all the functions to cancel out to get higher derivatives 0.
Hint on 3.3,9c: I think s(x,y)=arctan(y/x) might be a good stream function

#6 (Solutions to #6) Due March 19 1pm 3.1:3,4,7,10,12,13,15

#5 (Solutions to #5) Due March 12 (Friday) 1pm
2.7:4,5,8,9,10,13 (Note 12 is a copy of 8)

#4 (Solutions to #4) Due March 8 (monday), 1pm 12. (My survey of students was that extra time would be nice given the exam, but the homeworks are useful for mastering the material).
2.2: 7
2.3: 3,4,10
Problem 2.2:7 requires understanding that lambda=exp(i*theta). If you expand lambda as a power series of h, it looks like the power series of exp(i*h) for a few terms. The problem asks you to recognize where they differ.
Problem 2.3:3 is long but I hope very instructive. The U matrix in the Householder code has the information to multiply the Householder's in reverse order.
I got the exact answer using matlab's symbolic toolbox and then rounded only at the end. I got
    1.000000000705105e+000
   -2.964328419073236e-007
   -7.683186917434977e+000
   -2.206297290004058e-004
    9.840487549806722e+000
   -9.324611932899178e-003
   -5.011975877477831e+000
   -4.685413770053830e-002
    1.417778585705657e+000
    2.287193842956169e-002
   -3.135724049839490e-001
    7.196408559132141e-002

If you have the tollbox you can type
AAA=sym(0:49).'/49;
AAA=(AAA*ones(1,12) ).^ (ones(50,1)*(0:11));
bbb= 0:sym(8/100):4-sym(8/100);
bbb=cos(bbb).';
digits(100);
xxx=double(vpa(AAA.'*AAA)\vpa(AAA.'*bbb));

2.4: 2,5,6,7,9

#3 (Solutions to #3) for Friday February 26
1.6: 15, 16, 17, 21, 23
2.1: 4,5,6
2.2: 5
Recommended but do not hand in: Do the MATLAB homework in Problem set 3 by scrolling all the way down to "First MATLAB Homework in 18.085" from the previous semester.

#2 (Solutions to #2) for Friday February 19
1.3: 5,6,7,13
1.4: 2,6,12,15
1.5: 2,10,11,12
1.6: 12

#1 (Solutions to #1) for Friday February 12 (Please submit in class, or by 1pm near 2-343) (Please write your name extra extra neatly)
1.1: 1,7,10,12
Hint on 10: The idea is to check that UU' truly is H and then invert using the rule inv(A*B)=inv(B)*inv(A).
1.2: 3,4,7,18