MATH 4111 HOMEWORK ASSIGNMENTS, FALL '07
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 OR TWO AFTER CLASS ON THE DUE DATE
HAVE 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,
PAPERS RECEIVED AFTER I'VE PASSED ON ALL OTHER PAPERS TO THE GRADER (USUALLY
DONE BY MID AFTERNOON ON THE DUE DATE) WILL NOT BE GRADED AND WILL NOT RECEIVE ANY CREDIT. DON'T PUT ANY
PAPERS IN MY MAILSLOT SINCE I CHECK IT ONLY SPORADICALLY.
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 YOUR OWN 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.
AS
MENTIONED IN THE ACADEMIC INTEGRITY STATEMENT FOR THIS COURSE, THE BIG
“NO-NO” IS SUBMISSION OF SOLUTIONS FROM AN “ASK THE
EXPERT” WEBSITE; THIS IS PLAGIARISM AND MAY RESULT IN AN ASSIGMENT OF A
FAILING COURSE GRADE BY THE ACADEMIC INTEGRITY COMMITTEE.
ASSIGNMENT #1. DUE TUESDAY, SEPTEMBER 11
(i) Read Chapter 1 in Rosenlicht's book and get started on reading Chapter 2.
(ii) Do the following
problems from Chapter 1 (pages 12-13): 4 and 5 (all parts), 7(b),
(c), (e), (f) , 8, and 9.
(iii) Use mathematical
induction to prove that, for each positive integer n, the sum of the third
powers (cubes) of the first n positive integers is [n(n+1)/2]^2. Thus,
for n = 3, 1 +8+27=36=6^2 with 6 = 3(4)/2. Done efficiently, the
induction proof takes only a few lines.
(iv) Use mathematical
induction to prove the binomial theorem (look it up if you’ve forgotten
the statement). Then go on to prove by
induction a trinomial theorem for expansion of the nth power of a sum a+b+c in terms of products of powers of a, b, and c with
certain “trinomial” coefficients all involving n! in the numerator
and appropriate products of three factorials in the denominators.
(v) For extra credit, hypothesize a statement of a general
multinomial theorem for expansion of the nth power of a sum of m variables and go on to prove your hypothesis either by
induction on m for each fixed n or induction on n for each fixed m (one way is
easier than the other!)
Example of an inductive proof: For each
positive integer n, we assert that
(Hn) 1 + 2 + ...+n = n(n+1)/2
[arithmetic progression formula]
The assertion (H1) is true since
1=1(2)/2. If (Hn) is true, then the sum
of the first n+1 integers being the sum of the first n plus n+1, we infer that
the sum of the first n+1 integers is equal to
n(n+1)/2 + (n+1) =(n+1)(n/2 +1) =
(n+1)(n+2)/2. Since n+2 = (n+1)+1, this means
(H(n+1)) is true.
By the principle of mathematical induction, (Hn) is true for every positive integer n.
[Using a few shorthand symbols to eliminate the words I used above, the same
proof could be written down in two lines.]
________________________________________________________________________________________________________________________
ASSIGNMENT
#2. DUE TUESDAY, SEPTEMBER 18
Do the following problems from Chapter 2 (pages 29-31): 4 (b) [try to avoid brute force
division/multiplication via algebraic trickery], 5 (c),(d) [be careful with
signs], 6, 7, 10(b),(c), 13, 14, 15, 16.
Feel free to ask in class for hints on some of these problems.
ASSIGNMENT #3 DUE THURSDAY, SEPTEMBER 27
NOTE: THE DUE DATE HAS BEEN
PUSHED UP TO THE 27TH BECAUSE THERE WON’T BE ANY HOMEWORK DUE
THE FOLLOWING WEEK IN VIEW OF THE EXAM TO BE GIVEN ON THURSDAY, OCT. 4. THE HOMEWORK PROBLEMS ON THIS ASSIGNMENT WILL
BE GOOD PRACTICE FOR THE EXAM ALTHOUGH THE EXAM PROBLEMS WON’T BE AS HARD
AS SOME OF THESE HOMEWORK PROBLEMS.
Do the following 4 problems from Chapter 3 (pages 61-63): 11, 19, 21,23. Also do the following 3 additional problems:
(5) Let f(x) = 1/(x+2) for x a real number not equal to -2. Obviously, f is
continuous on the set of real numbers not equal to -2. We can use f to inductively define sequences
by choosing a starting value x0 and defining xn+1 = f(xn).
Check that f has two equilibrium values (solutions of f(x) =x), one positive
and one negative. If x0 > -2, show that the sequence
converges to the positive equilibrium value.
As mentioned in #13, the key thing is to check
that the subsequences with n even and n odd are each monotonic, one monotone
increasing and the other monotone decreasing. For this, you'll want to
study f(f(x))= (x+2)/(2x+5).
For extra credit, discuss what happens when x0 < -2.
WARNING: this gets tricky so, if your time is limited, don't consider it.
Among other things, you have to worry about initial values x0
for which there is some value of n such that xn =-2 with xn+1 then not defined. For this, it’s easiest to calculate
the inverse function g of f, i.e. solve y=f(x) for x=g(y) and look at the sequence defined by g
with initial value -2.
(6) Check that the arc length metric on the unit sphere in R^3 is
equivalent to the chord length metric.
Thus, for each two points on the unit
sphere, there is a unique great circle passing through these two points and the
arc length metric is the arc length of the shorter of the two arcs of the great
circle while the chord length metric is the Euclidean distance between the two
points (= restriction to the unit sphere of the Euclidean/Cartesian metric on
R^3). Why is it enough to make your
comparisons for two points on the equator of the sphere? Note that this reduces you to a comparison of
arc lengths and chord lengths for circles in R^2 and you can use polar
coordinates to simplify the calculations.
(7) In class, we’ll eventually show
that, for 1<p<(infinity), there is a norm || . ||p on R^n defined by
||(x1 , x2, …,xn
||p = 1/p power of the sum of |xi|p
from i=1 to n.
For p not equal to q, show directly that || . ||p
and || . ||q are equivalent. If you can, use Lagrange multipliers to
determine the best constants relating these two norms, i.e., maximize/minimize
one norm subject to the constraint that the other norm has value 1.
ASSIGNMENT#4, DUE
THURSDAY, OCTOBER 11
1.
Using the
definitions of open, closed, interior, closure, boundary given in class for
subsets of a metric space, show the
following for any subset S of a metric space (E, D):
(i)
The closure of S
is closed (thus, every limit point of the closure of S belongs to the closure);
(ii)
The interior of S
is open (thus, any point in the interior of the interior of S is in the
interior of S);
(iii)
E is the disjoint
union of the interior of S, the interior of the complement of S, and the
boundary of S;
(iv)
The interior of S
is the complement of the closure of the complement of S;
(v)
The closure of S
is the disjoint union of the interior of S and the boundary of S (also the
non-disjoint union of S and the boundary of S);
(vi)
The boundary of S
(and its complement) is the intersection of the closure of S and the closure of
the complement of S.
Hints: Draw the picture we put on
the board in
class. Each of (i)-(vi) should only take a couple of lines.
2.
Let f(x) = x/(1+|x|). Check that
f maps the real line 1-1onto the open interval (-1,1)
with both f and its inverse function g(y)=y/(1-|y|) being continuous. Extend f to a function from the extended real
numbers (= union of R and {+infinity, -infinity}) onto [-1,1]
by defining f(infinity) =1, f(-infinity)=-1. Then check that d(x,y) =|f(x)-f(y)| defines a metric
on the extended real numbers in which the extended real numbers are the
completion of the restriction of d to R.
Justify the statement that (R, | |) is locally equivalent to (R,d) but not globally equivalent.
3.
Identify R^(n+1) with the Cartesian product of R^n
and R and identify each x in R^n with the pair (x,0)
in R^(n+1). Let ||.|| be the Euclidean
norm on either R^n or R^(n+1). Call q=(0,1) the
“north pole” in R^(n+1) and let S be the unit sphere about 0 in
R^(n+1), i.e. all points (u,t) for which ||u||^2
+t^2=1.
(i)
For x in R^n, check that F(x) =(2x/(1+||x||^2),
(||x||^2 -1)/(||x||^2 +1)) defines a point p in S\{q} with p on the line
joining x and q.
(ii)
Check that F is
1-1 from R^n onto S\{q} with
inverse G(u,t) =u/(1-t). G is called stereographic projection from S\{q} onto R^n.
(iii)
Show that d(x,y) = ||F(x)-F(y)|| defines a
metric on R^n which is locally equivalent to the
Euclidean metric but not globally equivalent.
In what sense is S the completion of (R^n, d)?
4.
Do #6, #10, 26, 30 on pp. 61-64.
______________________________________________________________________________
ASSIGNMENT #5 DUE TUESDAY, OCTOBER 23
Do the following problems from Chapter 4, pp. 90-95: 1(b), 1(d), 4,
9(b), 10(b), 14,15, 17, 21,22. Also prove the following variation on
#21: with E and E’ as in #21 and with S a subset of E whose closure K in
E is compact, a continuous function f from S into E’ extends to a
continuous function from K into E’ if and only if f is uniformly
continuous on S.
Comments and hints:
1(b)
is easy--use continuity of F(x) =x to get continuity of f. But 1(d) is
not easy--you need to separately investigate continuity at rational and
irrational points and use the properties of rational numbers we discussed
earlier.
The key point in #4 is the assumption that f
is onto--without this, the conclusions wouldn't follow.
#9(b) and 10(b) are "old chestnuts"
which likely appeared in your Calculus III textbook--just a line or two ought
to suffice.
#14 and #15 are "intuitively clear"
but you'll need to think through how to concisely justify the indicated
conclusions.
For #17, you
use the same trick as in 9(b), namely that when a and b are
positive, a - b is the product of the sum and difference of the postive square roots of a and b.
#21 and the variation on #21 aren’t particularly
hard but are very useful in applications.
The idea in both problems is to use uniform continuity of the original
function to get continuity of the extended function.
#22 may cause you trouble, especially if
your linear algebra background is weak.
But it’s very important in multivariable calculus.
________________________________________________________________________________
ASSIGNMENT #6. DUE TUESDAY, OCT. 30
Do the following problems from Chapter 4, pp. 90-95: 30, 32, 34, 37, 39,
41, 44, 45.
Also do the
following additional problem: In
class, we gave two examples of subsets of R^2 which are connected but not path
connected. If we take the Cartesian
product of one
of these sets with an interval, we get a subset of R^3 which is connected but
not path connected. Give an example of a
subset of R^3 which is connected, not path connected, and NOT the Cartesian
product of a subset of R^2 and an interval.
For these, you might want to generalize the graph of sin(1/x)
example by the graph of something like sin(1/(|x| +|y|) or alternative
generalize the spiral curve example with a spiral surface spiraling outward to
the unit Euclidean sphere in R^3.
For your amusement (?), you might
want to read over #31. The idea of a continuous
function on [0,1] having the solid cube in R^3 as its
image set is decidedly counter-intuitive.
Such space-filling curves are often used in geometry and topology for
counter-examples to various assertions and go a long way toward establishing
the principle that continuity is not enough to get nice geometric properties.
Comments and hints:
#30. When Rosenlicht
says "closed interval in E^2", he means "closed rectangle in
R^2", i.e. the Cartesian product S = I x J of two closed intervals I=[a,b] and J=[c,d]
in R. If f is a continuous R-valued function on S, then, for each y in J,
g(x) = f(x,y) is continuous
from I to R. Use the intermediate value theorem for each such g to show f
can't be one-to-one.
#32. Note that we are dealing with
a sequence of functions defined inductively by initial function =
square root of x, (n+1)st function at x = square
root of (x+(nth function at x)). This is the same kind of thing we
encountered earlier with dynamical systems problems so the same techniques
apply. Thus, if the sequence is going to converge at some x, the limit
F(x) must be >= 0 and satisfy the equation F(x)^2
= x + F(x). The point of the problem is to show that the sequence does
converge to F(x). Is the convergence uniform or only pointwise?
#34. These are a little
tricky. In each case, it's not hard to calculate pointwise
limits but determining whether the converge is or
isn't uniform takes a little work.
#37. It's easy to see that
the sum functions converge uniformly but not so clear what happens for the
product functions. First consider the case where the limit functions for
each of the two sequences are bounded and use the triangle inequality to see
that the product sequence converges uniformly to the product of the limits.
Then take E = R (real numbers) and look at examples of situations where one or
both of the limit functions isn't bounded.
#39. There are many ways to
go about constructing a sequence of functions with the stated properties. You
may want to try to find a pointwise convergent
sequence of continuous functions where the limit function is 1 at the points 1,
1/2, 1/3, 1/4,..., 1/n,... and 0 elsewhere. Think of functions whose graphs are sharply
peaked “tents” with a small base.
#41. This one is admittedly not
easy but it isn't as bad as you may initially think. It's important that
the functions are defined on a compact metric space. Try to come up with
an argument first showing that any point p lies in an open set where the
sequence converges uniformly, then use compactness to obtain E as the union of
finitely many such open sets and deduce uniform convergence on all of E.
#44 and #45. In contrast to the last
three problems, these two are very easy. Just unravel the
definitions. So, if you're having trouble with some of the earlier
problems, you may want to do these problems first.
________________________________________________
ASSIGNMENT #7.
DUE TUESDAY, NOV. 6. Do the following problems from Chapter 5, pp.
108-110: # 4, 6, 7, 12. Also do the following
problems from Chapter 9, pp. 212-213: #1,3,4.
Comments:
The four Chapter 5 problems plus #3 and #4 from Chapter 9 rest on the Mean
Value Theorem. The wording in the hint in #7 from Chapter 5 is very
confusing. What’s meant is that
the subset of R^2 defined in the hint is connected and the function defined on
the set is continuous so its image in the real line is connected, i.e. an
interval.
For #12 from Chap. 5, you'll need to use the Mean Value Theorem several
times for f and functions closely related to f. Unraveling the somewhat
confusing wording in the problem, for any a and b in the interval I on which f
is defined with a<b and any points c and x in (a,b),
there are 3 (x,y) points be considered: the
point (x,y_chord(x)) on the chord line through (a,f(a)) and (b,f(b)); the point (x,f(x) on the graph of f; and the point (x,y_tan(x)) on the line tangent to the graph of f
at (c,f(c)). Convexity of f means that
y_tan(x)
<= f(x) <= y_chord(x) for all a,b,c, and x. As
stated in the text,
Problem #12 asks you to
show that, when f happens to be twice differentiable at each point in I, then f
is convex on I if and only if the 2nd derivative of f is everywhere
non-negative. In fact, the hypothesis of
non-negative 2nd derivatives serves only to guarantee that f ′
is monotonically increasing;
using just the assumption that f is differentiable on I plus
using the mean value property for derivatives established in #7, show that f is
convex on I if and only if f ′ is increasing. All you'll need is the point-slope formula
for straight lines and the Mean Value Theorem.
For #1 in Chapter 9,
you'll want to first show that the indicated function is differentiable at
points (a,b) where both a and b are non-zero, then go
on to investigate whether or not the function is differentiable at (a,b) when either a or b or both are zero. Note that if f is
differentiable at (a,b),
then at (a,b) the differential df
must be the linear map taking (x-a,y-b) to A(x-a) +B(y-b) where A and B are the
partial derivatives of f at (a,b) with respect to x
and y. So, to investigate differentiability at (a,b), first see whether or not A
and B exist, then go on to see whether the error function goes to zero as (x-a,y-b)
approaches (0,0). We'll prove next week that if all partial derivatives
of a function exist and are continuous on some open set, then the function is
differentiable at each point of the open set. You can just quote this
result as a shortcut to proving that the error function goes to zero provided
you can show that the partial derivatives exist and are continuous in an open
ball around your chosen point.
Note that Rosenlicht
uses a highly non-standard notation for partial derivatives, namely f prime sub
i for the partial derivative ∂f/∂xi of f
with respect to the ith variable.
ASSIGNMENT #8 Due Tuesday, Nov. 27
Do the following problems from
Chapter 9, pp. 212-214: # # 9, 11, 13, 14, 16, 20.
Also read over Chapter 6 and start browsing through
Chapter 7.
Hints: On #9,
you can use without proof the differentiation under the integral formula of #6.
On #11, you may want to warm up with the hint in the
book for the cases n=2 and n=3.
Note that #13 generalizes our discussion in class of 2nd
order differences to 3rd order differences.
Aside from Rosenlicht’s
horrible notation for partial derivatives, #14 is a straightforward
computation.
For #16, recall that an n-by-n matrix A is symmetric
if A is equal to its transpose (equivalently, the i,j entry of A coincides with the j,i
entry for all i and j) and A is then said to be
positive definite if
<Ax, x> is > 0 for each non-zero column
vector x. Here < .
, .> stands
for the standard inner product (dot product) on R^n. For positive semi-definite we replace > by
>= .
Similarly, A is negative definite if <Ax,x> is < 0
for each non-zero x, relaxing < to <= for negative semi-definite.
To be honest, I dislike #20 since it entails a sufficient condition
for contraction mappings which is more restrictive than necessary and whose
proof entails accepting on faith an important theorem in Euclidean geometry. To demystify this statement, observe that
there are many candidates for useful norms on the space of n-by-n real
matrices. We showed in class that the
operator norm of a linear transformation R^n with matrix A
relative to the norm ||.||_(infinity) on R^n is the
maximum of the numbers rho_i =sum of the magnitudes
of the ith row entries of A for 1<= i <=n . Call this the row sum norm of A. We went on to
use the multi-variable mean value theorem to show that, if F is a continuously
differentiable map on U with U containing a closed convex subset S for which
F(S) is contained in S and, if there is a positive number c<1 for which the
row sum norm of the Jacobian matrix JF(x) of F at x is
less than or equal to c at each point x in S, then F is a contraction mapping on
S. In particular, this condition holds
if each entry of JF(x) has magnitude less than or equal to c/n
and we relied heavily on this fact in carrying out the proof via contraction
maps of the Implicit Function Theorem.
The Hilbert-Schmidt norm of a matrix A, commonly denoted by |||A|||, is
NOT an operator norm but is instead the Euclidean norm of A as a member of
R^{n^2}. Thus |||A||| is the square root of the sums
of the squares of all entries of A.
Problem #20 asks you to prove that, for F and S as above, F is a
contraction mapping on S if there is a positive number c < 1 for which |||JF(x)|||
<= c for each x in S. Do this as
follows: (i) Let
|| . || =||.||_2 be the Euclidean norm on R^n. Use the Cauchy-Schwartz inequality to show that ||(dF)_x (v)|| <=|||JF(x)|||
||v|| for every vector v.
(ii) Accept without proof the famous adage that “a
straight line is the shortest distance between two points”; more precisely, for
each pair of points c and d in R^n, ||d-c|| is less
than or equal to the arc length distance along any smooth path C joining c to d
(iii)
For any a and b
in S, apply (ii) to c=F(a), d=F(b),
and C = {p(t) =F(a + t(-a): 0<= t <= 1}, noting
that dp/dt (t) =application to b-a of the
differential of F at the point a + t(b-a)
for each t. Then use (i) to reach the desired conclusion.
In
representation theory,tensor
analysis, and their generalizations to infinite dimensional Euclidean spaces, the
Hilbert-Schmidt norm for linear transformations plays a very important role.
But, if we want to use the Euclidean norm || . || , the appropriate matrix
norm is the operator norm of (dF)_x associated with || . || and the
calculation in (i) shows that this operator norm is <= |||JF(x)||| for each x. It turns out this operator norm is equal to
the square root of the largest eigenvalue of the
product of JF(x) times its transpose while |||JF(x)||| is the square root of
the sum of these eigenvalue and could be much
larger. So |||JF(x)|||
could be > 1 and F still a contraction.
This isn’t the end of the story. It can be shown that a linear transformation
is a contraction (with 0 its unique fixed point) relative to SOME norm on R^n if and only if all eigenvalues
of the transformation have magnitude < 1.
So the “right approach” to determining necessary and
sufficient conditions for a non-linear transformation F to be a contraction has
to involve eigenvalues of (dF)_x having magnitude <= c for all x with c fixed and c<1. This result is of large importance in signal
processing via wavelets, data compression, and related topics. To summarize,
the row sum criterion for contractibility we developed in class is the easiest
one to apply, the necessary and sufficient criterion involves eigenvalue estimates (not easy), and the business about
Hilbert-Schmidt norms in #20 is just “blowing smoke” on an
important question.
_______________________________________________________________________________________________________________
ASSIGNMENT
#9, DUE THURSDAY, DEC. 6
Do the following
problems from Chapter 6, pp. 133-134: #8, 13, 16, 19, 20, 22. Also do the following
problems from Chapter 7, p. 161: #3, #4, #5.
Hints:
On #8, Chap. 6, you may want take a sequence of partitions with widths
going to 0 and try to show that the Riemann sums for these partitions using the
right endpoint rule form a Cauchy sequence and so do the Riemann sums using the
left endpoint rule with the limits for these two types of Riemann sums being
equal. Depending on the approach you
use, you may or may not need to show that every monotone function has at most a
countable number of discontinuities.
For #13, remember that the limit
of nth roots of any positive number as n goes to infinity is equal to 1.
#16 is very easy—just a
matter of reviewing definitions.
Do #19 by induction plus
integration by parts.
Among other things, #20 involves a proof that among all smooth curves
joining two points, the shortest arc length distance is the
straight line distance. To carry out
#20, you’ll need to use the Mean Value Theorem for EACH component.
In #22, log(x) means the natural
log of x or ln(x).
Recall that ln(x) is the integral of the
function 1/t from t=1 to t=x for each positive x. Using this plus a Riemann sum argument, #22 “drops
out”.