Announcements

Lecture No	Date	Content	Corresponding Reading material
1	Feb. 14	Real Numbers: Field axioms, positivity or order axioms and some of its consequences, intervals.	Pages 1-4 of Fitzpatrick
2	Feb. 16	Natural Numbers: Inductive sets, a definition of the set of natural numbers, principle of mathematical induction, an application of proof by induction; Integers, rational numbers, and some of their algebraic properties; Theorem: x^2=2 has no rational solution	Pages 5-8 of Fitzpatrick
3	Feb. 21	Upper and lower bounds, bounded subsets of R, supremum and infimum, Completeness Axiom, Archimedean property, few theorems on the distribution of integers in R.	Pages 8-15 of Fitzpatrick
4	Feb. 23	Dense subsets of R, Theorem: Sets of rational and irrational numbers are dense in R; Absolute value and its basic properties, the triangular inequality; Some useful identities: Difference of n-th powers of two numbers, geometric sum formula, binomial expansion; Sequences: Convergence of sequences in R, Comparison theorem, limit of the sum	Pages 16-29 of Fitzpatrick
Quiz 1	Feb. 25
5	Feb. 28	Limit of the product, ratio, and linear combinations of two convergent sequences; Bounded sequences, sequential denseness, closed and open subsets of R	Pages 29-37 of Fitzpatrick
6	Mar. 02	Monotone Convergence Theorem and its application in proving that the harmonic series is divergent; The Nested Interval Theorem (proof left for self-study and will be given in a PS), Subsequence of a sequence, Existence theorem for a monotone subsequence of every real sequence, Sequential compactness, Bolzano-Weierstrass Theorem, Theorem: Every closed and bounded subset of R is sequentially compact (Study of the converse is left as exercise)	Pages 38-46 of Fitzpatrick
7	Mar. 07	Continuous functions, sum, scalar product, product, ratio, and composition of of functions and their continuity. The Extreme-Value Theorem.	Pages 53-61 of Fitzpatrick
8	Mar. 09	The Intermediate-Value Theorems. Proof of the existence of a real and positive solution of x^2=c for each positive real number c, Theorem: Image of an interval under a continuous function is an interval; Uniform continuity, examples	Pages 62-67 of Fitzpatrick
Quiz 2	Mar. 11
9	Mar. 14	Thm: A continuous function is uniformly continuous on a closed interval, epsilon-delta characterization of continuity and uniformly continuity. Monotone functions, Continuity of f(x)=x^r for any rational number r.	Pages 67-80 of Fitzpatrick
10	Mar. 16	Limit point of a subset of R, limit of a function, limit of sum, scalar product, product, ratio, and composition of functions, differentiability of a real-valued function of a real variable, differentiability implies continuity, derivative of sum, scalar product, product, and ratio of differentiable functions, derivative of x^n for n integer.	Pages 81-93 of Fitzpatrick
11	Mar. 21	Extra PS	-
Midterm Exam 1	Mar. 28
12	Mar. 30	Derivative of the inverse function, Chain Rule, differentiability of f(x)=x^r for any rational number r, local minimum and maximum of a function, Rolle's theoream, Mean-value theorem, Applications of Mean-Value Theorem: f is constant iff f '=0	Pages 96-105 of Fitzpatrick
Spring Break	Apr. 4-8
13	Apr. 11	Applications of Mean-Value Theorem: f is strictly increasing if f '>0, a is a local minimum (maximum) if f'(a)=0 and f''(a)>0 (f''(a)<0); Cauchy Mean-Value Theorem; Cauchy Sequences: Lemma: Every Cauchy sequence is bounded, Theorem: A real sequence converges iff it is a Cauchy sequence	Pages 105-111 & 228-230 of Fitzpatrick
14	Apr. 13	Real series and various tests of their convergence and absolute convergence. Sequences of functions and their pointwise and uniform convergence, Theorem: The limit of a uniformly convergent sequence of continuous functions is continuous.	Pages 230-233, 236-250 of Fitzpatrick
15	Apr. 18	R^n, the vector space (R^n,+,.), the Euclidean scalar (inner) product on R^n, the Euclidean norm and distance, Cauchy-Schwarz and triangular inequalities in R^n; Sequences in R^n and their convergence, the i-th component projection functions.	Pages 269-281 of Fitzpatrick
16	Apr. 20	Open balls, interior of a subset of R^n, open and closed subsets of R^n, Theorem: A subset of R^n is open iff its complement is closed; Union and intersection of open and closed subsets, exterior and boundary of a subset of R^n, the Cartesian product of open and closed subsets of R^n	Pages 282-288 of Fitzpatrick
Quiz 3	Apr. 22
17	Apr. 27	Continuous functions mapping R^n to R^m, epsilon-delta criterion for continuity, Thm: A function F is continuous iff inverse image of every open subset under F is open, Sequentially compact and bounded subsets of R^n, Theorem: Every bounded sequence in R^n has a convergent subsequence	Pages 290-300 of Fitzpatrick
18 & 19	Apr. 29	Theorem: A subset of R^n is Sequentially compact iff it is closed and bounded, Theorem: The image of a sequentially compact subset of R^n under a continuous function is sequentially compact ; Extreme-Value theorem for subsets of R^n and the extreme-value property; Uniformly continuous functions F:R^n->R^m; Convex and pathwise-connected subsets of R^n.	Pages 300-310 of Fitzpatrick
20	May 02	Intermediate Value Property, connected subsets of R^n; Limit points of a subset of R^n, limit of a function of several variables, partial derivatives, continuously differential functions, Theorem: If second order partial derivatives are continuous the order in which they are evaluated is not relevant.	Pages 310-312 & 348-361 of Fitzpatrick
21	May 04	Mean-Value Theorem for a real-valued function of several real variables (all related lemmas and propositions), directional derivative, Theorem: Every continuously differentiable function is continuous.	Pages 364-370 of Fitzpatrick
Quiz4	May 06
22	May 09	k-th order approximation of a function, The First Order Approximation Theorem, Tangent plane to the graph of a continuous function and its existence, affine functions	Pages 372-377 of Fitzpatrick
23	May 11	Extra PS
Midterm Exam 2	May 13
24	May 16	Review of Linear Algebra on R^n: Linear maps and their matrix representations, algebra of matrices, invertible linear maps and their associated matrices, determinant and the inverse of an invertible matrix	Pages 394-405 of Fitzpatrick
25	May 18	First order partial derivatives for function F from an open subset of R^n to R^m, derivative matrix, Mean-value theorem and 1st approximation theorem for F, differential of F	Pages 407-412 of Fitzpatrick
26	May 23	Chain Rule for functions from R^n to R^m	Pages 414-420 of Fitzpatrick
27	May 25	Statment of the Inverse function and the implicit function theorems.	Pages 421-427, 440-442 & 449-450 of Fitzpatrick

Note: The pages from the textbook and the reading material listed above may not include some of the subjects covered in the lectures.