Time: Monday, Thursday, 2:00 to 3:50 PM
Room: Carnegie 205
Instructor: Gregor Kovacic
Office: 419 Amos Eaton
E-mail: kovacg at rpi dot edu
Office Hours: Click here.
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 brief review of this topic this year, because I taught it 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.
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 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.
I have ordered the book Strichartz, because Mathematical Analysis I was taught from it. It has a very intuitive approach and presents important results from a relatively practical point of view. If you want to buy another general book on mathematical analysis, get the book by Rosenlicht from amazon.com, because it is a cheap Dover, and both precise and concise. I have also ordered the book by Spivak on Calculus on Manifolds. This is an honest and not too complicated book on rigorous multi-variable calculus, and I will follow the material if not all the details in this book. I recommend buying it to everybody. The book by Munkres is similar. My favorites are the books by Courant and John. They are genuine classics, and are unsurpassed in conveying the true understanding of mathematical analysis. Very often, I will follow the details of the material from these books. The reason I did not order them is because they are very expensive. If you want to buy them, it may be cheaper to get them used from amazon.com or abebooks.com. The books by Courant alone are older and more calculusy versions of the books by Courant and John. The books by Shilov and Zorich are translations of Russian books, and are also very intuitive, connected to physics, and user friendly. The book by Zorich is very extensive, and covers many results. It is a great reference book. The book by Rudin has great exercise problems, and I will assign many of them in the homework. It is often used as the standard Mathematical Analysis text. The book by Whittaker and Watson is an old classic, and will give you a glimpse of what modern analysis was about a hundred years ago. You will see that not so much has changed, and that the problems in the book are still challenging after all these years. The book by Flanders addresses differential forms from the physicist's poin of view. While a bit less rigorous than the rest, it has great, nontrivial examples and applications. Most other books not mentioned explicitly are some of the better standard mathematical Analysis textbooks. Of the books on related topics, the one by Lovelock and Rund presents a connection to tensor calculus and calculus of variations, as do the geometry books by Dubrovin & Co. which were written by three Russian greats. They have few proofs but great connections to and deep understanding of geometry and both classical and modern physics. The book by Arnold contains a brief and intuitive discussion of differential forms which it uses to derive important results in classical mechanics. The book by Kellogg contains a rigorous, old-fashioned proof of the Stokes' theorem from before the time of differential forms and partitions of unity. The book by Jackson is the standard graduate text on electricity and magnetism, which was one of the first applications of vector calculus and Stokes'-type theorems. The original source of this field is the classic by Maxwell. Finally, we used either the book by Larson & Co. or the book by Stewart in our Calculus sequence, depending on when you took Calculus. I strongly recommend either for reviewing elementary material.
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