Math 334, Accelerated (Honors) Advanced Calculus, Fall, 2016
This is the Math 334 homepage. Consult it from time to time to find
useful information for the course. I will include links to the syllabus and
other course information.
The following are links to current course information.
- Sample problems for the final.
- More on parametrized surfaces.
- Sample problems for the second midterm.
- Limsup by Will.
- Sample problems for the first midterm.
- Mason Kamb's conjecture about
the triangle inequality
- Analytic Inequalities
- Landau's book Foundations of Analysis is an excellent discussion of the
creation of real numbers.
- Implicit Function Theorem.
- Young's theorem on
mixed partials.
- Exotic examples of critical points: examples I and II.
- Yet another discussion of least squares.
- More on the chain rule.
- Lim sup, A discussion of lim sup.
- An excellent resource Real Mathemamtical Analysis by Charles Pugh.
- Baire Category theorem
- From Hausdorff's Set Theory:
"The continuous image of a closed linear segment -- say of the interval
T=[0,1] of the real number system is called a continuous curve"; ...
In the very next sentence: "But we shall refer to continuous curves as
interval-images since, as we shall see, they need have little resemblance
to our intuitive notion of a curve."
- Intermediate Analysis by John Olmsted is a good
reference.
- A quote from Abel Cauchy is mad, and there is no way of
being on good terms with him, although at present he is the only man who
knows how mathematics should be treated. What he does is excellent,
but very confused . .
- Real Analysis is a text by John Olmsted that I highly recommend. You may
download a copy. If has many good exercises and detailed discussion of subtle
points in analysis, but beware that the notation may differ from the
notation I will use.
- George Green
- Definition of cosine.
- A Surface area counterexample.
- A counterexample to
Tonelli's theorem.
- Dave Duncan's thesis and a discussion of Fubini's theorem.
- Quote from Jacobi. Dirichlet alone, not I, nor Cauchy,
nor Gauss knows what a completely rigorous mathematical proof is. Rather we
learn it first from him. When Gauss says that he has proved something, it is
very clear; when Cauchy says it, one can wager as much pro as con; when
Dirichlet says it, it is certain ...
Quoted in G Schubring,
Zur Modernisierung des Studiums der Mathematik in Berlin, 1820-1840.--
- Integrability of sup and inf
- Fundamental theorem of algebra
- Links to
336_16 papers
336_15 papers
336_14 papers,
336_13 papers,
336_12 papers, and
336_11 papers.
- A discussion of
Tiknonov regularization and least squares.
- Dirichlet-(Heine?)-Lebesgue
proof of the uniform continuity
of a continuous function on a compact set.
- The connected sets of \( \mathbb{R} \) are
intervals.
- More on norms.
- A proof of the
Heine-Borel Theorem. (a distillation of
various proofs)
- More details on
sequential continuity.
- History of Heine-Borel Theorem
- Topologist's sine curve
- A proof of Lebesgue's theorem on Riemann integration.
- A discussion of norms.
- The class email address is math334a_au16@uw.edu.
- Hyperbolic functions.
- The Cauchy-Binet formula and areas
of submanifolds.
- Areas of hypersurfaces and
parallelotopes.
- Federer's exposition of Geometric Measure Theory. His very dense book is 676 pages long.
- Volume of the n-ball.
- Fubini's theorem
- A set is Jordan-measurable if and only if the outer area of its boundary is 0.
- A geometric proof that (sin x)/x ->
1 as x->0.
- Differentiability
- Thomae's function.
- Best book ever written on inequalities: Inequalities by Hardy, Littlewood, and Polya.
- Jensen's Integral Inequality.
- Hadamard's Inequality
for determinants of matrices and its application to measures.
- Jensen's inequality.
- The Arithmetic mean - geometric mean inequality and consequences.
-
Derivation of
least squares.
- Scanned first exercises from Folland)
- Cauchy's inequality
- The book Principles of Mathematical Analysis by
Walter Rudin has a construction of the real numbers (as Dedekind
cuts) from the rational numbers. It is on reserve in the math
library. The reals can also be constructed from the rationals
using Cauchy sequences by a general process known as
completion that applies to any metric space.
- Basic Real Analysis by Tony Knapp is a good reference.
- A quote from Felix Klein (famous German mathematician)
Everyone knows what a curve is, until he has studied
enough mathematics to become confused through the countless
number of possible exceptions.
-
When I see a bird that walks like a duck and swims like a duck and
quacks like a duck, I call that bird a duck. -- Poet James Whitcomb Riley.
- A quote from Henri Poincare
Mathematics is the art of giving the same name to different things.
[As opposed to the quotation: Poetry is the art of giving different
names to the same thing].
- Some printings of Folland have an error on problem 6b,
page 125. It should read grad F3(a)=0. (There should be a
subscript 3 on F.)
- SAGE website.
- For those of you who have either of the first two
printings of the text the old
errata link is appropriate.
- The
On-Line Encyclopedia of Integer
Sequences is a
great resource.
- Mathworld
link.
- History of Mathematics Archive
- Errata for Folland's text. It is updated regularly. You should send email to folland@math.washington.edu if you spot any errors not already listed
- In problem number 7, section 1.3, consider
f(x) to be defined only for x>0. Also assume the integers
p and q are positive.
- Syllabus
morrow@math.washington.edu