Math 335, Accelerated (Honors) Advanced Calculus, Winter, 2017
This is the Math 335 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.
 Inner products discusses
inner products on the same vector space.
 Sample problems for the second midterm.
 The LevySteinitz rearrangement theorem.
 Inequalities cheat sheet.
 Divergent Series by Hardy.
 Sample problems for the first midterm.
 A history of the
concept of a differential form.
 Dixon's proof that
\( \int_0^{\infty} \frac{\sin x}{x} = \frac{\pi}{2} \).
 History of
vector analysis.
 My two favorite references on trigonometric series: A Treatise on Trigonometric Series by Nina Bari and Trigonometric Series by Antoni Zygmund.
 How to compute the Laplace transform of
\( \frac{\sin x }{x}\).
 Integrability of uniform limit of Riemann integrable functions.
 In 1943 Einstein answered a little girl who had difficulties in school with mathematics.
"Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.
Best regards
Professor Albert Einstein."
 There is no perfect test for
convergence. (Taken from Knopp.)
 Will on quotient spaces and differential forms.
 Quote of Galileo: "Philosophy is written in this grand book, which stands continually open before our eyes (I say the 'Universe'), but can not be understood without first learning to comprehend the language and know the characters as it is written. It is written in mathematical language, and its characters are triangles, circles and other geometric figures, without which it is impossible to humanly understand a word; without these one is wandering in a dark labyrinth."
 Schwarz's example (with Schwarz misspelled).
 More on surface area
 The CauchyBinet formula and areas
of submanifolds.
 Areas of hypersurfaces and
parallelotopes.
 3.1415926... How I wish I
could calculate pi easily ...
 Five Short Stories about the Cardinal Series. Take a look at this interesting article.
 Abel's theorem on continuity
of power series and a similar theorem for Laplace transforms.
 Compactness by David Jekel
 Proof and applications of the bounded convergence theorem.
 You might find this note on
hyperbolic functions useful.
 Several articles on surface area surface 1, surface 2, surface 3 ,
elementary, directional norms by Toralballa.
 An article on
A Geometric Theory of Surface Area
 The Baire category theorem and related
ideas. This document will resemble a blog and will evolve over time. The
Baire category theorem has an elementary proof but it has many sophisticated
consequenses.
 The best book on infinite series is Theory and Application of Infinite Series by Konrad Knopp.
 G.H. Hardy's book Orders of Infinity might answer some
of your questions about comparitive growth of terms in a series.
 Rearranging alternating harmonic series
 A Primer of Real Functions by Ralph Boas is one of my favorite
books on introductory analysis. I read it as an undergrad.
 Riemann's rearrangement theorem
 I'd like to recommend a book that you can download for free:
Elementary Real Analysis by Thomson, Bruckner, and Bruckner. You
might like to take a look at the other books on their website
Classical Real Anallysis.

Suppose
\(
\sum c_n e^{inx}
\)
converges for all x to a function f that is
Riemann integrable (or just Lebesgue integrable). Then the \(c_n\) are
the
Fourier coefficients of f (and hence the series is a Fourier series). This is
a theorem of du Bois Reymond and de la Valee Poussin.
 The probability integral.
 Artin, Emil. The Gamma Function. New York,
NY: Holt, Rinehart and Winston, 1964.
 Arzela's theorem on dominated
convergence.
 Some stuff on series.
 A hint for problem 14 in section 6.4. Prove that for small \(x\)
\[
\frac{1}{x} \log(1+x) = 1  \frac{x}{2} + \frac{x^2}{3} + \dots,
\]
and hence that
\[
(1+x)^{\frac{1}{x}} = e(1\frac{x}{2} +\frac{7}{12}x^2 +\dots).
\]
Hence prove that
\[
\frac{e(1+\frac{1}{n})^n}{1/n} \to \frac{e}{2} \text{ as } n\to \infty.
\]
 page 1 page 2, and page 3 of my proof that
\[
u(x) = \int \frac{\rho (x+y)}{y}dy\]
solves
\[\nabla ^2 u(x) =  4 \pi \rho(x)
\]
 Bump functions
 Please volunteer to help me with Mathday. There will be
1600 high school students on campus and I will need at least 75 volunteers
to help escort then around and do other routine duties that require no
mathematical knowledge. Here is a link to my mathday website. Mathday is Monday, March 20,
2017  the Monday of Spring Break.
 Kernels, a discussion of solving
constant coefficient odes with kernels.
 Closed and exact vector fields
 Proof of the RiemannLebesgue Lemma.
 Abel's theorem on Fourier series.
 Uniform version of Abel's test
 A summary of facts about power series.
 A short note on compactness.
 A note on double series.
 A note on Abel's test on series.
 Dirichlet's test
 A general pointwise convergence theorem.
 How Newton made his famous discoveries: By always thinking about them. I keep the subject constantly before me and wait til the first dawnings open little by little into full light.
 Binomial series.
 A sentence from a letter by C. G. J. Jacobi to
A. von Humboldt,
If Gauss says he has proved something,
it seems very probable to me; if Cauchy
says so, it is about as likely as not; if
Dirichlet says so, it is certain.
 Infinite sums, some notes on summing
possible infinite sets of numbers.
 A simple discussion of the fundamental solution of a constant coefficient linear differential equation.
 A primer on differential equations.
 A direct, simple, proof of a convergence theorem for
improper integrals.
 A simple, quick, introduction to differential forms is
Differential Forms by Harley Flanders which is available as a Dover
reprint.
 Change of variables
formula for spherical coordinates.
 Spherical Coordinates
 Here's a proof of the CauchyBinet formula, which has a
nice application to give a formula for the measure of a
parameterized manifold in R^{n}.
 A proof of the Poincare Lemma.
 Closed and Exact differential forms.

The Marquis and the Land Agent
by G. N. Watson, The Mathematical Gazette, Vol. 17, No. 222 (Feb., 1933), pp. 517
 I will not cover Raabe's test and I will not ask you to
work any problems using it.
 Starter books on manifolds and Stokes's theorem: Loomis
and Sternberg, Advanced Calculus; Hubbard and Hubbard,
Vector Calculus, Linear Algebra, and Differential Forms;
Flanders, Differential Forms.
 There is an error in the answer to problem 2b in
section 5.8. The answer should be (xz^{2}/2,
xyzz^{2}/2x^{2}/2, 0)+grad(f)
 For problem number 3 in section 5.8, assume that
Laplacian(f)=div(H) has a solution. You don't need to justify this.
 The 1854 Smith Prize Exam at Cambridge University that Stokes
wrote can be found in the Michigan online
library. The Smith Exams are in the last volume and the 1854
exam is on page 320, and it is problem number 8. Apparently William Thomson (Lord Kelvin)
stated the result to Stokes in a letter in 1850. James
Clerk Maxwell won the Smith Prize in 1854 and Gabriel Stokes
himself won it in 1841 and Thomson in 1845. Other winners are
Arthur Cayley (1842),
G.H. Hardy (1901), Arthur Eddington (1907), Alan Turing
(1936). A history of the
prize.
 History of Stokes' theorem.
 Make sure you check Jerry Folland's website for misprints.
 Syllabus(pdf)
morrow@math.washington.edu