Mathematics 203-204 - Basic Analysis I-II

Fall Semester 2009


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:
  • Office Hours: Monday and Friday 11:30am-2pm and by appointment.
  • Time and Place for Mathematics 203, Fall Semester 2009

  • Mondays, Wednesday and Fridays 10:20am-11:10am, Physics 119
  • Syllabus For Mathematics 203

  • 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
  • Syllabus For Mathematics 204

  • 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.

  • Grading

    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 Wednesday, September 30. PDF

  • Comments

    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.

    Math 203 Notes

  • Sets, relations and functions PDF Revised August 28, 2009
  • Initial segments, well ordering and the axiom of choice PDF Revised August 31, 2009
  • Binary operations and some basic algebraic structures PDF Revised July 10,1999
  • The natural numbers and arithmetic PDF Revised September 3, 2009
  • Introduction to the theory of infinite sets PDF Revised September 9, 2009
  • The integers and the rational numbers PDF Revised August 28, 2009
  • The real numbers PDF Revised September 11, 2009
  • Topology PDF Revised September 21, 2009
  • Metric spaces PDF Revised September 27, 2009
  • An extremely useful abstract closure principle PDF Revised September 30, 2009
  • Uniform convergence PDF Revised September 30, 2009
  • Summation PDF Revised October 9, 2009
  • Differentiation on the line PDF Revised October 12, 2009
  • Power series PDF Revised October 24, 2005
  • The complex exponential function PDF Revised October 30, 2008
  • Algebras of sets, multirectangles and integration of simple functions. PDF Revised October 26, 2009
  • The Riemann and Lebesgue integrals. PDF Revised November 16, 2009
  • The Theorems of Fubini and Tonelli. PDF Revised November 18, 2009
  • The inequalities of Hölder and Minkowski; convolution and other stuff. PDF Revised December 4, 2009
  • Hermitian inner product spaces PDF Revised October 31, 2005
  • Hilbert space PDF Revised November 6, 2003
  • Fourier series PDF Revised December 4, 2009
  • Fourier inversion on the line PDF Revised December 4, 2009
  • Homework problems for Math 203, Fall Semester 2009

  • Homework One PDF Due August 31, 2009
  • Homework Two PDF Due September 7, 2009
  • Homework Three PDF Due September 14, 2009; revised September 11,2009!!
  • Homework Four PDF Due September 21; revised September 16, 2009!!
  • Homework Five PDF Due September 28, 2009.
  • Homework Six PDF Due October 9, 2009.
  • Homework Seven PDF Due October 16, 2009.
  • Homework Eight PDF Due October 23, 2009.
  • Homework Nine PDF Due November 9, 2009.
  • Homework Ten PDF Due November 23, 2009.
  • Final Problem Set PDF Due Friday, December 11, 2009 at 5pm.
  • Math 204 Notes

  • 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
  • Some References

  • Functions 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

  • Return to: Duke University * Mathematics Department
    Last modified: March 20, 2009