Announcements

Lecture No	Date	Content	Corresponding Reading material
1	Feb. 15	Real Numbers: Field axioms, positivity or order axioms and some of its consequences, intervals; Natural Numbers: Inductive sets, a definition of the set of natural numbers, principle of mathematical induction, an application of proof by induction	Pages 1-6 of Fitzpatrick
2	Feb. 19	Integers, rational numbers, and some of their algebraic properties; Theorem: x^2=2 has no rational solution; upper and lower bounds, bounded subsets of R, supremum and infimum, Completeness Axiom.	Pages 7-10 of Fitzpatrick
3	Feb. 22	The Archimedean Property, theorems on the distribution of integers in R, Dense subsets of R, Theorem: Sets of rational and irrational numbers are dense in R; Absolute value and its basic properties, the triangular inequality.	Pages 12-17 of Fitzpatrick
4	Feb. 24	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, product, ratio, and linear combination of two convergent sequences.	Pages 18-32 of Fitzpatrick
Quiz 1	Feb. 26
5	Mar. 01	Bounded sequences, sequential denseness, closed subsets of R, Monotone Convergence Theorem and its application in proving that the harmonic series is divergent.	Pages 35-40 of Fitzpatrick
6	Mar. 03	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), Continuous functions, sum, scalar product, product, ratio, and composition of of functions and their continuity.	Pages 41-48 and 53-57 of Fitzpatrick
Quiz 2	Mar. 05
7	Mar. 08	The Extreme-Value and Intermediate-Value Theorems. Proof of the existence of a real and positive solution of x^2=c for each positive real number c.	Pages 58-65 of Fitzpatrick
8	Mar. 10	Uniform continuity, 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 66-80 of Fitzpatrick
Quiz 3	Mar. 12
9	Mar. 15	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
10	Mar. 17	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.	Pages 96-104 of Fitzpatrick
Quiz 4	Mar. 19
11	Mar. 22	Applications of Mean-Value Theorem: f is constant iff f '=0, 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 and one of its applications (related to Taylor polynomial).	Pages 104-113 of Fitzpatrick
12	Mar. 24	Cauchy Sequences: Lemma: Every Cauchy sequence is bounded, Theorem: A real sequence converges iff it is a Cauchy sequence; Real series and various tests of their convergence and absolute convergence.	Pages 228-239 of Fitzpatrick
13	Mar. 26	Sequences of functions and their pointwise and uniform convergence, Theorem: The limit of a uniformly convergent sequence of continuous functions is continuous.	Pages 241-246 & 249-250 of Fitzpatrick
Extra PS	Mar. 29
Midterm Exam 1	Mar. 31
Spring Break	Apr. 5-9
14	Apr. 12	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
15	Apr. 14	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
16	Apr. 19	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.	Pages 290-296 of Fitzpatrick
Quiz 5	Apr. 20
17	Apr. 21	Sequentially compact and bounded subsets of R^n, Theorem: Every bounded sequence in R^n has a convergent subsequence, 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.	Pages 298-302 of Fitzpatrick
18	Apr. 26	Uniformly continuous functions F:R^n->R^m; Convex and pathwise-connected subsets of R^n, Intermediate Value Property, connected subsets of R^n.	Pages 303-312 of Fitzpatrick
19	Apr. 28	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 348-361 of Fitzpatrick
Quiz 6	Apr. 30
20	May 03	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
21	May 05	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
Quiz 7	May 07
22	May 10	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
23	May 12	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
Midterm Exam 2	May 17
24	May 24	Chain Rule for functions from R^n to R^m	Pages 414-420 of Fitzpatrick
25	May 26	Statment of the Inverse function and the implicit function theorems.	Pages 421-427, 440-442 & 449-450 of Fitzpatrick
Quiz 8	May 28

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