There will be three texts; the main one will be my own typeset notes which will appear as links on this page; the other two will be
Introduction to Analysis, by Maxwell Rosenlicht, Jr., Dover Publications, New York.
This is a soft cover reprint of an earlier hard-cover edition. It's real cheap, say $11.95!
Advanced Calculus of Several Variables, by C.H. Edwards, Jr., Dover Publications, New York, 1995.
This is a soft cover reprint of the 1973 hard-cover edition. It's real cheap, say $15.95!
William K. Allard, Professor of Mathematics
Office: 029A Physics Building Phone: (919) 660-2861 Fax: (919) 660-2821 E-mail: wka@math.duke.edu Office Hours: Tuesday and Thursday 10am-Noon and by appointment.
Tuesdays and Thursdays 2:50-4:05pm, Physics 235
Naive but nontrivial set theory including uncountable sets and the axiom of choice Construction of the real numbers from the natural numbers Topological spaces Metric spaces The topology of Euclidean space Infinite series The complex exponential function The Riemann and Lebesgue integral in Euclidean space Fourier series and integrals
Tangency and differentiation Higher derivatives and Taylor's Theorem The contraction mapping principle The inverse function theorem The implicit function theorem and functional dependence Existence, uniqueness and smooth dependence on parameters for systems of ordinary differential equations Vector fields Exterior algebra Mapping formulae for multiple integrals Submanifolds of Euclidean space Exterior differential calculus Stokes' formula The Brouwer fixed point theorem DeRham cohomology. Introduction to Riemannian geometry and the tensor calculus with applications to physics Since this program is probably too ambitious, some of these topics will likely be omitted. High on the list of my priorities is Stokes' Theorem. So we will be sure to cover that material which is necessary to prove Stokes' Theorem. Using Stokes' theorem we will prove the Brouwer Fixed Point Theorem.
Your grade will be based on the weekly assignments. Believe me, they will keep you busy! There will also be two in class tests on terminology.
Definitions to memorize for test on October 7. PDF
This material is rather challenging. I am happy to report that in spite of the challenging nature of the material many of our undergraduates have been equal to the task of dealing well with it. You should not take this course unless you have had prior experience with proving things and with mathematical abstraction. We don't have higher level honors courses but if we did these courses would be among them. It is also absolutely necessary for more advanced work in mathematics and can come in handy in other branches of science and engineering.
In previous calculus courses you learned to differentiate and integrate functions of one or more real variable and to apply these techniques to solve problems of various sorts. You probably didn't pay may attention to mathematical rigor. In fact, when you studied things like integration formulae in several variables you probably didn't pay much attention to intellectual rigor either. I assume that part of the reason you are taking this course is to get all that stuff straight.
Sets, relations and functions PDF Revised September 9, 2008. Initial segments, well ordering and the axiom of choice PDF Revised September 9, 2008 Binary operations and some basic algebraic structures PDF Revised July 10,1999 The natural numbers and arithmetic PDF Revised September 9, 2008 Introduction to the theory of infinite sets PDF Revised September 9, 2008 The integers and the rational numbers PDF Revised September 1, 2008 The real numbers PDF Revised September 15, 2008 Topology PDF Revised September 22, 2008 Metric spaces PDF Revised September 29, 2008 An extremely useful abstract closure principle PDF Revised October 4, 2005 Uniform convergence PDF Revised October 25, 2005 Summation PDF Revised September 12, 2008 Differentiation on the line PDF Revised October 18, 2002 Power series PDF Revised October 24, 2005 The complex exponential function PDF Revised October 30, 2005 Algebras of sets, multirectangles and integration of simple functions. PDF Revised November 6, 2005 The Riemann and Lebesgue integrals: Part One PDF Revised November 16, 2005 The Riemann and Lebesgue integrals: Part Two PDF Revised December 6, 2005 Hermitian inner product spaces PDF Revised October 31, 2005 Hilbert space PDF Revised November 6, 2003 Fourier series PDF Revised December 6, 2005 Fourier inversion on the line PDF Revised December 2,2002
Homework One PDF Due September 4, 2008 Homework Two PDF Due September 11, 2008 Homework Three PDF Due September 18, 2008 Homework Four PDF Due September 25, 2008 Homework Five PDF Due October 2, 2008 Homework Six PDF Due October 16, 2008
Some basic linear algebra PDF Matrices PDF Introduction to multilinear algebra and norms PDF Revised 1/12/2004 Differentiation PDF The symmetry of higher differentials PDF Tangency PDF Inversion PDF The contraction mapping principle PDF The inverse function theorem PDF Two examples illustrating the inverse function theorem PDF The implicit function theorem; functional dependence PDF The algebra of alternating multilinear functions PDF How elementary linear maps change areas PDF Partitions of unity PDF The singular set gets mapped to a set of measure zero PDF The change of variables formula for multiple integrals PDF Submanifolds of Euclidean space PDF Inscribing simplices to compute the area of a submanifold of Euclidean space PDF Integration of a scalar function over a submanifold of Euclidean space PDF The coarea formula PDF Differential forms PDF Orientation PDF Integration of differential forms PDF The Poincare lemma PDF Stokes' theorem PDF The solid angle form PDF The Brouwer fixed point theorem PDF The Gauss-Bonnet formula for surfaces in three space PDF Vector fields; curl and divergence PDF
Return to: Duke University * Mathematics DepartmentFunctions of several variables by Wendell H. Fleming Real and complex analysis by Walter Rudin Advanced calculus by R. Creighton Buck Advanced calculus by Angus E. Taylor A primer of modern analysis by Kennan T. Smith