**Time:** Tuesday and Friday, 10:00 to 11:50 AM.

**Room:** Carnegie 201

**Instructor:** Gregor
Kovacic

**Office:** 419 Amos Eaton

**Phone:** 276-6908

**E-mail:** kovacg at rpi dot edu

**Office Hours:** Click here.

**Number Systems:**
Integers, countable infinity, induction, rationals, irrationals, the ordered ring of real numbers, suprema and infima,
the least upper bound and Archimedean properties, decimal representation, uncountable infinity of real numbers,
basic inequalities, complex numbers, roots of unity.

**Topology of the Real Axis
and the Complex Plane:** Basic set theory, Cartesian producs, Axiom of Choice, open and closed
intervals, open and closed balls, open and closed sets, cluster points, interiors, closures, boundaries, bounded sets, connected sets,
open subsets of the real line, Cantor set.

**Limits:** Functions, domains, ranges,
onto and one-to-one functions, inverses, infinite sequences and their limits, convergence and divergence,
Cauchy sequences, bounded sequences and cluster points, monotonic sequences, operations on limits, some special limits,
limsup and liminf, completeness of real numbers, construction of the real numbers, limits of functions,
left-hand and right-hand limits.

**Continuity:**
Continuity at a point, limits and continuity, continuous functions, examples of continuous functions, operations on
continuous functions, continuity of composite functions, continuous preimages of open and
closed sets, compactness, Bolzano-Weierstrass and Heine-Borel theorems, uniform continuity,
continuous images of compact intervals, Lipschits and Holder continuity,
discontinuities, monotonic functions and their inverses.

**Differentiation:**
Derivative at a point, slope of the tangent, differentiable functions, derivatives of elementary functions,
differentiability and continuity, calculus of derivatives, derivatives of composite functions and the chain rule,
mean value theorem, derivatives and monotonicity,
derivatives of inverse functions, higher derivatives, maxima and minima of functions, lack of differentiability at a point,
intermediate value theorem, L'Hospital's rule.

**Rieman Integral:**
Riemann and Darboux sums, partitions and refinements, the Riemann integral, integrals of continuous functions,
operations on integrals, integrals of the absolute value, integrals over adjacent intervals, mean value theorem,
antiderivatives and the fundamental theorem of calculus, change of variable,
integration by parts, integrability of piecewise-continuous functions, a nonintegrable function, logarithm and exponential functions,
hyperbolic functions and their inverses, integration methods for rational functions, improper integrals, gamma function.

**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, Weierstrass Approximation Theorem.

Class notes from the Fall semester of 2013 are deposited here.

Here is another set of notes, written up by Joshua Sauppe.

T. M. Apostol, * Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra*, Wiley.

T. M. Apostol, * Mathematical Analysis, Second Edition*, Addison-Wesley.

A. Browder, * Mathematical Analysis: An Introduction*, Springer-Verlag.

R. C. Buck, * Advanced Calculus*, Waveland.

R. Courant, * Differential and Integral Calculus, Vol. 1*, Springer-Verlag.

R. Courant and F. John, * Introduction to Calculus and Analysis, Vol. 1*, Springer-Verlag.

S. Lang, * Undergraduate Analysis*, Springer-Verlag.

R. Larson, R. P. Hostetler, and B. H. Edwards, * Calculus: Early Transcendental Functions*, Houghton-Mifflin.

J. E. Marsden and M. J. Hoffman, * Elementary Classical Analysis*, W. H. Freeman.

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.

J. Stewart, * Calculus: Early Transcendental Functions, 2 Vols., 5th Ed.*, Brooks-Cole.

R. S. Strichartz, * The Way of Analysis*, Jones and Bartlett.

V. R. Zorich, * Mathematical Analysis I*, Springer-Verlag.

I have ordered the book by Strichartz, because it has a very intuitive approach and presents important results from a relatively practical point of view. If you want to have two books, buy the one by Rosenlicht or the one by Shilov, because they are cheap. My favorite is the book by Courant and John. It is a genuine classic, and is unsurpassed in conveying the true understanding of mathematical analysis. Very often, I will follow the material from this book. The reason I did not order it is because it is expensive. If you want to buy it, it may be cheaper to get it used from amazon.com or abebooks.com. The book by Courant alone is older and a bit more calculusy version of the book 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 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. Most of the other books not mentioned explicitly are some of the better standard mathematical Analysis textbooks. 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.