MATH 308 HOMEWORK ASSIGNMENTS, SPRING '08
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
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 WEDNESDAY, JAN. 23
Section 3. 4, pp. 99-100 # 5, 6, 7, 8
Section 3. 5, p. 106, #27, 28
In class, we sketched the first part of the proof that the medians of any triangle
meet at a point C called the centroid of the triangle and C is two-thirds of
the way along each median from the vertex to the midpoint of the opposite
side. Generalize this to a corresponding statement about the centroid C
of a tetrahedron with C being the intersection point of the 4 medians (line
segment from a vertex to the centroid of the opposite triangular face) of the
tetrahedron and C located three-fourths of the way along each median from the
starting vertex for the median. The numbers ฝ for midpoints
(centroids) one-dimensional line segments, 2/3 for centroids of
two-dimensional triangles, พ for centroids of three-dimensional tetrahedra suggest a pattern with n/(n+1) being the
appropriate ratio for centroids of n-dimensional tetrahedral involving n+1
vertices with opposite faces being (n-1)-dimensional tetrahedral. For
extra credit, prove that this is correct. Hint: With P0, P1,…Pn being the vertices, make sense out of the assertion that
C = 1/(n+1)(P0+P1+…+Pn)= common value of Pj +n/(n+1)((centroid of the face
opposite Pj) – Pj)
for each j.
In
this group 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 #2, DUE WEDNESDAY, JAN. 30
First read Sections 3.7 and 3.10 in the text,
then do the problems indicated below:
1. Section 3.7, pp. 131-2 #25, #29, #35
Hints: As mentioned in the text, there
are two kinds of orthogonal matrices, i.e. n x n matrices A whose columns are
an orthonormal basis for the Euclidean space R^n equipped with the usual Cartesian distance function and
usual dot product. The determinant of A must be +1 or -1 with det A =+1 signifying that the columns of A have the same
orientation as the standard basis while det A = -1
signifies that the columns have the opposite orientation from that of the
standard basis. The first kind are called
rotation matrices and the second kind reflection matrices. So the first
thing to do is to check whether or not the matrix is orthogonal by checking to
see if the columns are mutually perpendicular unit vectors; if so, the second
thing to do is to calculate the determinant. For n=3, when A is a
rotation matrix, there is always a non-zero vector X = (3 by 1 column matrix)
with AX=X and the entries of X can be found by solving 3 simultaneous
equations; then A rotates about the axis in R^3 determined by X and
it’s not hard to figure out the angle of rotation. When n=3 and A
is a reflection matrix, there is always a vector X with AX = - X; again
the entries of X are found by solving simultaneous equations. In this
case, the easiest way to picture the action of A on
vectors is by a 2-stage process: first rotate by some angle about the
axis determined by X, second reflect through the plane perpendicular to
X. The n=2 case is similar but easier; A just rotates when det A =1 while A rotates and reflects through a line
determined by a solution of AX=X when det A=-1.
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^4 which can be adjoined to your subspace vectors to yield an orthonormal basis of R^4. To get the fourth vector, you can solve equations
(ugh!), continue to use Gram-Schmidt with some arbitrarily chosen fourth
vector like (1,0, 0, 0) (more ugh! since Gram-Schmidt quickly gets tedious), or
devise an analog of cross-products for 4-vectors involving the
“cross-product” of 3 non-zero vectors to get a vector perpendicular
to each of the three along with a recipe involving 4 3-by-3 determinants to
explicitly compute the components of the vector.
3. 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 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 #3 DUE WEDNESDAY, FEB. 6
Browse through Chapter 4. Most of it reviews things about partial
derivatives you learned in Math 233. We’ll dwell only on manipulations
of differentials, various versions of the chain rule, and the implicit function
theorem. Don’t worry about several variable power series or the
intricacies of techniques for several variable max/min problems. Do the
following problems:
Section 4.4, p. 198, #3, #7, #14, #16
Section 4.5, p. 201, #1, #2
Section 4.6, p. 203, #2, #6
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ASSIGNMENT #4 DUE MONDAY, FEB. 18 (TURN IN ON
FRIDAY IF YOU WISH)
Section 4.6, p. 203, #4, #7
Section 4.7, p. 210, #2, #10, #14, #28
Section 5.3, p. 255 #1
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ASSIGNMENT #5 DUE
Wednesday, FEB. 27
Section 5.4, pp. 267-269 #20
Section 5.5 pp.272-273 #2 (follow the approach in Example 1 on p.272), 4, 7, 10, 11 (you can find the centroid by the Euclidean geometry method we discussed earlier and avoid doing any integrals), 12 (sadly, here you can’t avoid doing integrals)
Section 5.6 pp.273-275 #11
General Comment: Always IGNORE any suggestion in a problem to do a computer plot UNLESS
you’re adept at a package like MATLAB and love to use it.
ASSIGNMENT #6 DUE WEDNESDAY, MAR. 5
Section
6. 10, pp. 323-4, #2, #3,
#5, #6, #8, #12, #15
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ASSIGNMENT #7 DUE WEDNESDAY, MAR. 19
In class today, we
discussed how to calculate the electric field produced either by electric
charges with constant line density or by charges on a cylinder with constant
area density.In both cases with the z-axis being
either the line or axis of symmetry of the cylinder,E is expressed in cylindrical coordinates by (c/r )
(unit vector e_r) and one can use Gauss’s Lawto figure out c. The text uses this approach to compute
E for charges on a flat plate with constant area density. The
electric field produced by charges on several lines or surfaces is just the sum
of the fields produced for the individual lines or surfaces:
#12
from Section 6.8 (the problem held over from this week’s assignment)
Compute
E for charges on a sphere with constant area density. For this,
you’ll need to use the formula for div E in spherical coordinates,
arguing by symmetry that E must be a function of r times e_r where now r is the spherical coordinate, ie. distance from the center of
the sphere to the point where E is being evaluated. Why is E = 0
inside the sphere?
Use
2 to compute E for charges on two concentric spheres, each with constant charge
area density. What about N concentric spheres?
Use
the plate computation in the text to compute E for charges with constant
density on two parallel plates. Warning: You have to consider 3
regions, the region between the plates and the two regions outside the plates.
Modify #12 to compute E for constant density charges
both on a line and on a cylinder having the line as its axis of symmetry.
ASSIGNMENT #8 DUE WEDNESDAY, MARCH 26
DO THE FOLLOWING PROBLEMS FROM CHAPTER 6:
SECTION 8 (pp. 308-9) #10,13.15,16,17
SECTION 9
(p.314) #4,8, 12
ASSIGNMENT #9 DUE FRIDAY, APRIL 4
DO THE FOLLOWING:
(i) Problems 6,7,10, 15, and 17a,b,c from Section 11 of
Chapter 6 in
the textbook. Although this is the section on Stokes’
Theorem, you’ll probably want to use the divergence theorem for #10 and
perhaps for parts of #17.
(ii) Use the formulas for the
curl in cylindrical and spherical coordinates to find, in two special cases, a
vector field F for which E = curl(F). See the handout for details. (Problems 1 and 2 at the end of the handout).
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