MATH-4210, MATHEMATICAL ANALYSIS II

# MATHEMATICAL ANALYSIS II

Time: Tuesday, Friday, 12:00 to 1:50 PM
Room: Darrin 232
Instructor: Gregor Kovacic
Office: 420 Amos Eaton
Phone: 276-6908
E-mail: kovacg at rpi dot edu

• This document contains the list of topics to be presented in this course, a list of textbooks, and some comments and recommendations about these textbooks.
• Click on each topic title to download the notes for that topic.
• To find some interesting papers on topics related to this class, click here.
• To check out the grading and homework rules, click here.
• To see the test dates and topics, click here.
• To find files of the homework assignments, click here.
• You will find statements of attendance policy, academic integrity, how to compute your current grade, grade appeals, etc. here.

## Topics

Click on each topic title to download the notes for that topic.

Numerical and Functional Series: Taylor's formula and Taylor series, Lagrange's and Cauchy's remainder, Taylor expansion of elementary functions, indefinite expressions and L'Hospital rule, numerical series, Cauchy's criterion, absolute and conditional convergence, addition and multiplication of series, functional sequences and series, pointwise and uniform convergence, Weierstrass test, integration and differentiation of functional series, power series and radius of convergence, complex exponentials. There will only be a homework given on this topic this year, because it was taught in Analysis I.

Metric Space Topology: Metric spaces, open and closed sets, Cauchy sequences and completeness, separable spaces, compact and connected sets, compactness and limit points of infinite subsets and sequences, Heine-Borel and Bolzano-Weierstrass theorems, continuous images and preimages of various types of sets, equicontinuity and compactness in the space of continuous functions, Arzela-Ascolli theorem, normed and Banach spaces, contraction mapping theorem, existence and uniqueness of solutions to ordinary differential equations.

Trigonometric Series: Periodic functions, orthogonal and orthonormal sets of functions, orthogonality of trigonometric functions, Fourier series and their convergence for piecewise continuous functions, differentiation and integration of Fourier series, Gibbs' phenomenon, Fourier series and mean-square convergence, solutions of classic partial differential equations by Fourier series.

Approximation of Continuous Functions: Uniform approximation by polynomials, Weierstrass theorem and separability of the space of continuous functions on a compact interval, approximation of derivatives, Stone-Weierstrass theorem.

Functions of Several Variables: Review of linear algebra, directional derivatives, partial derivatives and total differential, gradient, chain rule, equality of mixed partial derivatives, Taylor series in several dimensions, mean value theorem, extrema, inverse and implicit function theorems, multi-dimensional surfaces and their representations, conditional extrema and Lagrange multipliers.

Multi-Variable Integration: Riemann integral in several dimensions, integrable functions, Fubini's theorem, integrals with parameters, composite mappings, partitions of unity, change of variables, improper multiple integrals, Fourier integral.

Integration on Manifolds: Differential forms and their derivatives, Poincare lemma, Stokes' theorem for a rectangle, manifolds and charts, orientation and boundary, Stokes' theorem on manifolds, line integrals, surface integrals, volume integrals, classical vector analysis, Green's formula, Gauss' and Stokes' theorems, applications in electromagnetism.
Click here to find an alternative, more intuitive presentation of integrals of differential forms.

The daily build of notes is deposited here.

For my notes on Mathematical Analysis I, look here.

## Textbooks

The following textbooks contain (some of the) material presented in this course:

T. M. Apostol, Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra, Wiley.
T. M. Apostol, Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications, Wiley.
T. M. Apostol, Mathematical Analysis, Second Edition, Addison-Wesley.
V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag.
A. Browder, Mathematical Analysis: An Introduction, Springer-Verlag.
R. C. Buck, Advanced Calculus, Waveland.
R. Courant, Differential and Integral Calculus, 2 Vols., Springer-Verlag.
R. Courant and F. John, Introduction to Calculus and Analysis, 2 Vols., Springer-Verlag.
B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry - Methods and Applications : Part I: The Geometry of Surfaces, Transformation Groups, and Fields, Springer-Verlag.
B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry. Methods and Applications : Part 2: The Geometry and Topology of Manifolds , Springer-Verlag.
H. Flanders, Differential Forms with Applications to the Physical Sciences, Dover.
W. Fleming, Functions of Several Variables, Springer-Verlag.
J. D. Jackson, Classical Electrodynamics, Wiley.
O. D. Kellogg, Foundations of Potential Theory, Dover.
S. Lang, Undergraduate Analysis, Springer-Verlag.
R. Larson, R. P. Hostetler, and B. H. Edwards, Calculus: Early Transcendental Functions, Houghton-Mifflin.
D. Lovelock and H. Rund, Tensors, Differential Forms, and Variational Principles, Dover.
J. E. Marsden and M. J. Hoffman, Elementary Classical Analysis, W. H. Freeman.
J. C. Maxwell, Treatise on Electricity and Magnetism, Dover.
J. Munkres, Analysis on Manifolds, Westview.
M. Rosenlicht, Introduction to Analysis, Dover.
W. Rudin, Principles of Mathematical Analysis, McGraw-Hill.
G. E. Shilov, Elementary Real and Complex Analysis, Dover.
M. Spivak, Calculus, Publish or Perish.
M. Spivak, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus, Harper-Collins.
J. Stewart, Calculus: Early Transcendental Functions, 2 Vols., 5th Ed., Brooks-Cole.
R. S. Strichartz, The Way of Analysis, Jones and Bartlett.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge.
V. R. Zorich, Mathematical Analysis I, Springer-Verlag.
V. R. Zorich, Mathematical Analysis II, Springer-Verlag.

The following Schaum Outline Series textbooks contain exercises relevant to this course:

M. R. Spiegel, Schaum's Outline of Fourier Analysis with Applications to Boundary Value Problems, McGraw-Hill.
M. R. Spiegel, Schaum's Outline of Vector Analysis, McGraw-Hill.
R. C. Wrede and M. R. Spiegel, Schaum's Outline of Advanced Calculus, Second Edition, McGraw-Hill.