# Math 334, Accelerated (Honors) Advanced Calculus, Fall, 2015

### 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. Implicit Function Theorem. Young's theorem on mixed partials. Sample problems for the second midterm. Exotic examples of critical points: examples I and II. Yet another discussion of least squares. More on the chain rule. There will be review sessions in PDL C36, from 2-4 on Saturday, Oct. 24, and Sunday, Oct. 25, for the first midterm. Sample problems for the first midterm. Lim sup, Will's discussion of lim sup. Will's Tuesday office hour is now 12:30-1:30. 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_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_au15@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. James Whitcome Riley, a poet. 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 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