Spring 2007 Math 425 Homework from Spivak's _Calculus on Manifolds_
(due each MONDAY in class as indicated at the left margin - the !ed
problems are for you to ponder - these assignments are always subject
to change as necessary):
[Please read indicated sections even if no problems are assigned]
[Supplemental reading or problems from Marsden's & Tromba's _Vector
Calculus_ is indicaated by M&T....]
[Also, please have a look at my colleague Roman Federov's problems:
http://www.math.umass.edu/~fedorov/425/]
2/5 [Warm up with some geometric linear algrebra problems:]
An (infinite) chessboard can be thought of as the integer points
in the plane: in other words, a possible chess move is of the
form mH + nV where H and V are horizontal and vertical moves
by a single square, and where m and n are (positive or negative)
integers.
Analyze the various moves of different chess pieces. For example,
can a knight move to any position? How few knight moves would it
take to make a basic H or V move?
Bishops move diagonally, and thus (unlike the other pieces)
they always stay on their same color. These moves are of the
form mH + nV where m+n is even; alternatively, they are of
the form aA + bB where A = H + V and B = -H + V are basic
diagonal moves. Explain why these descriptions of bishops'
moves are equivalent (see next problem for one way to go
about this).
* * *
The atoms in salt (NaCl) are located at the points of the integer
(cubic) lattice in 3-space:
lI + mJ + nK where l, m and n are integers, and
where I, J and K are translation vectors along the axes.
If we place a Na atom at the origin O, we have seen that
there are two descriptions for the sites of the Na atoms:
(*) lI + mJ + nK where l+m+n is an EVEN integer,
(**) aA + bB + cC where a, b, and c are integers, and
where A = I + J, B = J + K, C = I + K are diagonal vectors
on the face of the cube.
We want to see that these two descriptions of the Na sites
(also known as the face-centered cubic, or FCC, lattice) are
equivalent. To do this, find a formula
F(a, b, c) = (l, m, n)
which gives the location in description (*) of a point V =
aA+ bB + cC in description (**). Also find an inverse formula
G(l, m, n) = (a, b, c)
which lets us recover the coefficients in (**) from those in (*).
* * *
Consider a tetrahedron in R^3 with vertices A, B, C, D. Show that
the (outer) face normal vectors (weighted by their respective areas)
sum to the zero vector. (Hint: express areas using cross products.)
Generalize this to any polyhedron in R^3. (Hint: dissect into
tetrahedra.)
What generalizations can you make to R^n? (Recall the case n=1 is
trivial, and n=2 was done in class, so we are interested in n>3.)
(Hint: use determinants.)
* * *
[please read the Preface and Chapter 1 of Spivak]
2/12 Spivak: 1-1, 1-2, 1-3, 1-4!, 1-5, 1-6, 1-7, 1-8!, 1-9
1-10, 1-11!, 1-12! (Hint: think transpose), 1-13
M&T: 1.5
2/19 Spivak: 1-14, 1-15, 1-16, 1-19, 1-23, 1-24, 1-25, 1-27, 1-28
M&T: 2.2
2/26 Spivak: 2-1, 2-2, 2-6, 2-7!, 2-8! 2-9!, 2-10h,a,b,j,e!, 2-13!,
2-14!, 2-15!, 2-16
M&T: 2.3, 2.5, 2.6
3/5 Spivak: 2-17h,a,e, 2-18, 2-20a,c,e, 2-21!, 2-22!, 2-23, 2-25!,
2-29, 2-32, 2-34!, 2-35!
M&T: 3.1, 3.2, 3.3
3/12 Spivak: 2-36!, 2-37!, 2-38, 2-40!
Space of circular arcs problem (as discussed in class): please
work out the linear approximation of (s,c) in terms of (R,\theta)
at other locations besides the semicircle (\theta = \pi - suggestions
are near \theta = \pi/k where k = 2, 3...)
M&T: 3.4, 3.5
[MIDTERM EXAM in class on Monday 4/2 - please think about
the following two "topical" functions:
A) (x,y) -> (e^x cos y, e^x sin y): R^2 -> R^2
B) (x,y,z) -> 1 - x^2 - y^2 - z^2: R^3 -> R^1
- think about all the things I could ask you about these...!]
4/9 Recall that a subset S of \R^n has (n-dimensional Lebesgue) measure
zero if it can be covered by open rectangles R_1, R_2,..., R_i,...
such that for an real e > 0, \sum m(R_i) < e. (Here m(R_i) is
n-volume of the rectangle. i.e. the product (b_1-a_1)...(b_n-a_n)
if R_i = (a_1, b_1)\times...\times(a_n, b_n).
Show that the subset \Q^n of \R^n consisting of all vectors with
rational coordinates has measure zero.
[Hint: first do the case n=1, recalling that \Q is countable.]
Also show that the boudary of a rectangle R = [a_1, b_1]\times...
\times[a_n, b_n] has measure zero (and consquently, we could have
alternatively used closed rectangles in defining measure zero).
[Hint: first show more generally that an (n-1)-plane in \R^n has
measure zero.]
Spivak: 3-1, 3-5, 3-6, 3-7!, 3-8, 3-10, 3-14, 3-18!
4/16 Spivak: 3-27 [there's a printing error: explain why the missing
upper limits on the right are both b], 3-28, 3-35, 3-36, 3-40, 3-41
4/23 Spivak: 4-1, 4-2, 4-3, 4-8, 4-19, 4-20!, 4-21
4/30 Spivak: 4-23, 4-24, 4-25, 4-26, 4-27, 4-28!, 4-31!, 4-33!, 4-34!
5/7 Spivak: 5-1, 5-5, 5-6, 5-9, 5-15!, 5-16, 5-17
5/14 Spivak: 5-18!, 5-20!, 5-22!, 5-25, 5-26, 5-31, 5-32, 5-33....
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Here is your FINAL EXAM, on which I expect you to work ALONE and which
I would like to you to complete on or before Saturday 26 May 2007.
You may place the exam under my office door (LGRT 1435G) IN A SEALED
ENVELOPE, or you may mail it to me (postmarked on or before Saturday
26 May 2007)
Rob Kusner 17 May 2007
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1. Even though polar coordinates (r,t) on the plane are not both
globally defined functions (for example, the angle "function" t is
multivalued and undefined at the origin) on the plane R^2, the
corresponding 1-forms are well defined (at least on the punctured
plane R^2 \ {(0,0)}), and can be expressed using cartesian coordinates
(x,y).
a. Using the relations x = r cos t, and y = r sin t, find functions
A, B, C and D which express the 1-forms
dr = A(x,y) dx + B(x,y) dy
and
dt = C(x,y) dx + D(x,y) dy
in terms of dx and dy.
b. On what open subsets are A, B, C, D continuous? ... differentiable?
c. Show that the integral of dr around any closed loop in the
punctured plane R^2 \ {(0,0)} is zero.
d. Compute the integral of dt around the (counter-clockwise oriented)
unit circle c parametrized by c(s) = (cos s, sin s).
e. Suppose b is any loop which winds once (counter-clockwise) around
the origin. Show that the integral of dt around b is the same as for
the unit circle c. (Hint: show d(dt)=0 and use Stokes' Theorem.)
f. Generalize part e to a loop which winds n times around the
origin. (For example, the circle above winds n=1 times, while the
clockwise circle b(s) = (cos s, - sin s) winds n=-1 times, and a loop
which doesn't surround the origin winds n=0 times.)
2. Let W=dx*dy*dz (I will let * stand for wedge product for this
exam) denote the volume 3-form on R^3, let S be surface (such as a
plane or sphere or whatever) in R^3 with unit normal vector field N,
and let w = W(N,_,_) be the 2-form that results from inserting N in
the first of the three slots of W.
a. In case S is the xy-plane {z=0} with upward unit normal N = e_3,
show that w = W(N,_,_) = dx*dy, namely the area 2-form on S.
b. In case S is the graph of a function f(x,y) over this plane, find a
forumla for the upward unit normal in terms of grad(f)=(f_x, f_y), and
show that w = W(N,_,_) = (1+|grad(f)|^2)^{1/2} dx*dy is again the
area form for the graph.
c. When S is the unit sphere around the origin with outer unit normal
N, show w = W(N,_,_) = x dy*dz - y dx*dz + z dx*dy, and argue
geometrically that is again the area form on S (in fact, your argument
should work for any "nice" surface with unit normal N).
d. Using the relation x^2+y^2+z^2=1 to express each of the terms in
the 2-form w on the appropriate hemisphere and corresponding unit disk
(for example z dx*dy = (1 - x^2 - y^2)^{1/2} dx*dy for the upper
hemisphere over the unit disk in the xy-plane) integrate w over S.
(The 6 integrals should sum to area(S) = 4\pi).
e. Thinking of w = x dy*dz - y dx*dz + z dx*dy as a 2-form on R^3,
compute the 3-form dw on R^3.
f. Apply Stokes' Theorem to the unit ball B with boundary S (the unit
sphere) and conclude
vol(B) = area(S)/3 = 4\pi/3
Generalize this argument to show that the n-volume of the unit ball
B in R^n and the (n-1)-area of its boundary (n-1)-sphere satisfy
the relation
n vol(B) = area(S).
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Do your best with this - good luck!!!
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Have a great summer & here's hoping to do math with you in the future!