INSTRUCTOR'S POLICY:  TRY HARD TO HAND IN ASSIGNMENTS IN CLASS ON THE INDICATED DUE DATE AND, IF YOU'RE UNABLE TO ATTEND, GIVE YOUR COMPLETED ASSIGNMENT TO A FRIEND TO HAND IN.  PAPERS SLIPPED UNDER MY OFFICE DOOR WITHIN AN HOUR AFTER CLASS ON THE DUE DATE HAVE FAIRLY GOOD ODDS  OF GETTING INTO THE GRADER'S HANDS AND BEING GRADED.  IN THE EVENT OF A SPECIAL PROBLEM (ILLNESS, ETC.) CONTACT ME BY E-MAIL. UNLESS I'VE MADE A SPECIAL ALLOWANCE FOR SOME REASON, PAPERS RECEIVED AFTER I'VE GONE HOME ON THE DUE DATE WILL NOT BE GRADED AND HENCE WON'T RECEIVE ANY CREDIT.
                IT'S PERFECTLY O.K. TO ASK FOR A HINT DURING OFFICE HOURS.  I'LL USUALLY BE QUITE "GENEROUS" WITH HINTS, MAY COME VERY CLOSE TO COMPLETING THE PROBLEM.  REMEMBER THAT IT'S ALSO PERFECTLY O.K. TO DISCUSS PROBLEMS WITH OTHERS AS LONG AS YOU INDIVIDUALLY WRITE OUT SOLUTIONS AND PENCIL IN A NOTE IDENTIFYING THE OTHERS ON THE TOP OF YOUR PAPER (SEE ACADEMIC INTEGRITY SECTION).   THERE'S NO DEDUCTION AT ALL FOR SUCH IDENTIFYING NOTES BUT THERE MIGHT BE A SUBSTANTIAL DEDUCTION WHEN IT'S CLEAR THERE'S BEEN COLLABORATION BUT NO NOTES  APPEAR.
 

ASSIGNMENT #1.  DUE MONDAY, JANUARY 30
                               Section 1. 14, p. 35 (3rd Edition) #4, 5

                              Section 1.15, p. 40     #7, 9

                                Section 2.5, p. 54      #55, 64

                                 Section 2.6, p. 57     # 7, 13

                               Section 2.7 , p. 59    #8, 15

                               Section 2.9, p.63      #18, 25

ASSIGNMENT #2.  DUE WEDNESDAY, FEB 1

                    Section 2. 16, p. 78   #7a, #11

                            Section 2. 17, p. 80   #5, 8, 18, 21

ASSIGNMENT #3  DUE WEDNESDAY, FEB. 8

                            Section 3. 4,  pp. 99-100  #3, 4, 5, 6, 7, 8

                            Section 3. 5, p. 106, #27, 28

        In both of these groups of problems, the results mentioned are general ones true for any Euclidean space.
So don't presume anything about the dimension of the background space (it's irrelevant)..  While it's possible to set up a coordinate system relevant to the setting mentioned, this very likely will just bog down your arguments with extra baggage.  Instead, try to do everything just
using the properties of vectors (both bound and free) on Euclidean spaces and the associated inner product as discussed in class.
Bringing in angles (other than right angles) is another red herring, i.e. it's not wrong to do so but it won't help and may hinder.
 

ASSIGNMENT #4 DUE WEDNESDAY, FEB. 15

        The problems below are selected from the later sections of Chapter 3.  They involve two sets of ideas:
(i) Transformations on a Euclidean space preserving the distance between points are called isometries (taken from a Greek word meaning same distance).  Those that fix some point give rise to linear transformations on the free vectors which not only don't change the lengths of vectors but also don't change the inner products of vectors; in particular, when two vectors are orthogonal, their images are orthogonal, and for this reason such transformations are called orthogonal.  For finite dimensional spaces, when we assign rectangular coordinates to a space, orthogonal transformations correspond to multiplying a column of coordinates by a matrix A whose transpose is the same as its inverse and such matrices are said to be orthogonal.  Easy properties of matrices show that any orthogonal matrix has determinant +1 or -1; those that have determinant +1 just involve rotations in planes, those that have determinant -1 combine a reflection with a rotation.
The upshot is that having determined that a 3 by 3 matrix A is orthogonal, the next thing to do is compute the determinant of A.  If det  A = -1, there will be a unit vector v whose coordinate vector X (expressed as a column) satisfies AX = -X.  Solve this system of equations to find X.  Then the transformation corresponding to A will reflect v through the plane through the origin perpendicular to v and will rotate members of this plane about the line through v (rotation axis).  By choosing an orthonormal basis for the plane, you can determine the rotation angle.  If det A=1, there will always be a unit vector v fixed by the transformation:  solve AX=X to find its coordinates.  Then, as before, the plane perpendicular to v will be rotated about the line through v.  For 2 by 2 matrices, the situation is easier; when det A = 1,
you can just read off the rotation angle from the matrix and when det A= -1, you can either solve AX = -X to find a vector v reflected through the line perpendicular to it or solve AX=X to find the line of vectors fixed by the transformation with those perpendicular to the fixed line being reflected through it.

(ii)  Gram-Schmidt process for construction of orthonormal bases in an inner product space (as discussed in class and gone over in the textbook at the very end of Chapter 3).

                    Section 3.7, pp. 131-2  #25, #29, #35

                    Section 3.10, p. 147  #4a and #4c.  In each case, first use Gram-Schmidt to find an orthonormal basis for the subspace of R^4 consisting of linear combinations of the given vectors, then find a fourth vector in R^n which can be adjoined to your subspace vectors to yield an orthonormal basis of R^4.

                    Problem 6.   Consider the space V of all Hermitian symmetric 3 x 3 complex matrices.  As mentioned in the text, a complex matrix A is Hermitian symmetric if the the transpose of A is the same as the complex conjugate of A, or, equivalently, the conjugate transpose A* of A is equal to A.  For A and B in V, show that <A , B> = trace (AB) defines a real inner product on V.  [Hint:  For A in V, use the Hermitian symmetry condition to check that the trace of the square is A is the sum of the squares of the magnitudes of the entries in A].  Deduce that V is a 9-dimensional real vector space by exhibiting an orthonormal basis for V with 9 elements.  Don't make heavy work of this:  start with the diagonal matrices in A and get the first three members of your orthonormal basis using the "easiest" possible diagonal matrices.  Then go on to off-diagonal matrices and again use only easy ones with lots of zero entries.

                    Problem 7.  On pp. 182-183, the text uses the Gram-Schmidt process to construct the first 4 Legendre polynomials, i.e. those with degree 3 or less. Continue the process to construct e_4 from f_4 and e_5 from f_5 (here the 4 and 5 should be subscripts but the package I'm using won't do subscripts).  This ought to be enough to convince you that Gram-Schmidt by hand is no fun.  As mentioned in the text, there are round-about ways to get an orthonormal basis for the Legendre polynomials without grinding through Gram-Schmidt.

ASSIGNMENT #5  DUE WEDNESDAY, MARCH 1

Section 7.9, pages 378-379, #4-#11.   Of these problems, #4 is just a matter of unraveling the definitions given, #5-#9 are just a matter of looking up the reference given for a Fourier series and then applying the Plancherel theorem to evaluate the given infinite series, #10 is a consequence of the Plancherel theorem and the ability to expand out any real inner product <f,g> as 1/2<f+g,f+g>-1/2<f-g,f-g>.
But #11 is stated in a very confusing way and there is a typo rendering the statement in 11(b) meaningless.  What's going on is that if a certain function f is symmetric about x=L, one can talk about 3 sorts of Fourier expansions for f:
            (i)  Extend f to be an odd 4L-periodic function and use a sine series
            (ii) Extend f to be an even 4L-periodic function and use a cosine series
            (iii) Take the given symmetry to extend f to be an even 2L-periodic function and again use a cosine series.
The idea of #11 is to unravel the relationships between these 3 Fourier series.  Try your hand at this and don't pay a lot of attention to the socalled hints in the text.

ASSIGNMENT #6 DUE WEDNESDAY, MARCH 10

This assignment was handed out in class.  It involved using d'Alembert's formula to draw pictures of the wave ampltitude function u(x,t) for various t's and various initial conditions f(x) = u(x,0),  g(x) = partial derivative with respect to t of u at (x,0).

ASSIGNMENT #7  DUE MONDAY, MARCH 27

          Section 4.2, p. 192, #2, #4

               Section 4.4, p. 198, #3, #7, #16

               Section 4.5, p. 201, #2

               Section 4.6, p. 203, #4, #7

               Section 4.7, p. 210, #2, #10, #14, #28
 

ASSIGNMENT #8 DUE MONDAY, APRIL 3

            Section 5.3, p. 255 #1

            Section 5.4, pp. 267-269  #3(a), 6, 8, 17, 20

            Section 5.5 pp.272-273  #2, 7, 10

            Section 5.6 pp.273-275 #11
 

ASSIGNMENT #9 DUE WEDNESDAY, APRIL 12

            Section 6. 6, p.295, #10, #11, #13

            Section 6.8, pp.306-7, #7a,c , #9, #12, #15, #17, #18
 

ASSIGNMENT #10 DUE WEDNESDAY, APRIL 19 (OR FRIDAY, APRIL 21)

          Section 6.7, pp. 298-299, #3, #8, #9, #12, #18, #20

             Section 6.9, p. 314, #3, #5, #7, #12
 

ASSIGNMENT #11 DUE FRIDAY,  APRIL 28 (OR MONDAY, APRIL 30)

            Section 6. 10, pp. 323-4, #3, #6, #8, #15

              Section 6. 11, pp. 334-5, #4, #5, #7, #8, #15

In most of these problems, the idea is to avoid doing a nasty integral either by using the Divergence Theorem or Stokes' Theorem to do a much easier integral or by using one of these theorems to deduce that the nasty integral is 0.  For many of the 6.11 problems, numerical answers are in the back of the book.  But don't just write down these answers.  Show how they arise.