Announcements

Lecture No	Date	Content	Corresponding Reading material
1	Feb. 04	Real Numbers: Field axioms, positivity or order axioms and some of its consequences, intervals of real numbers, Natural Numbers: Inductive sets, a definition of the set of natural numbers, principle of mathematical induction, an application of proof by induction	Pages 1-7 of Fitzpatrick
2	Feb. 06	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. 11	Archimedean property, some 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. 13	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, Limit of the product, ratio, and linear combinations of two convergent sequences	Pages 17-32 of Fitzpatrick
Quiz 1	Feb. 15
5	Feb. 18	Bounded sequences and boundedness of the convergent sequences, harmonic series, sequential denseness, Monotone Convergence Theorem, closed and open subsets of R.	Pages 35-41 of Fitzpatrick
6	Feb. 20	The Nested Interval Theorem, Subsequence of a sequence, Existence theorem for a monotone subsequence of every real sequence, Sequential compactness, Bolzano-Weierstrass Theorem for closed intervals and its generalization to closed and bounded subsets of R.	Pages 43-46 of Fitzpatrick
7	Feb. 25	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	Feb. 27	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, epsilon-delta characterization of continuity; Uniform continuity, examples.	Pages 62-67 & 70-72 of Fitzpatrick
Quiz 2	Mar. 01
9	Mar. 04	Thm: A continuous function is uniformly continuous on a closed interval, epsilon-delta characterization of uniform continuity. Monotone functions, Continuity of f(x)=x^r for any rational number r.	Pages 67-80 of Fitzpatrick
10	Mar. 06	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, derivative of the inverse function	Pages 81-98 of Fitzpatrick
11	Mar. 11	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, and f is strictly increasing if f '>0.	Pages 99-106 of Fitzpatrick
12	Mar. 13	Thm: a is a local minimum (maximum) of f if f'(a)=0 and f''(a)>0 (f''(a)<0); Cauchy Mean-Value Theorem.	Pages 106-113 of Fitzpatrick
Midterm Exam 1	Mar. 15
13	Mar. 18	Cauchy Sequences: Lemma: Every Cauchy sequence is bounded, Theorem: A real sequence converges iff it is a Cauchy sequence, real series and their convergence, absolute convergence	Pages 228-239 of Fitzpatrick
14	Mar. 20	Sequences of functions and their pointwise and uniform convergence, Theorem: The limit of a uniformly convergent sequence of continuous functions is continuous, 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.	Pages 241-250 & 269-275 of Fitzpatrick
15	Mar. 25	Sequences in R^n and their convergence, the i-th component projection functions, 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.	Pages 277-285 of Fitzpatrick
16	Mar. 27	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; Continuous functions mapping R^n to R^m	Pages 285-293 of Fitzpatrick
Quiz 3	Mar. 29
17	Apr. 01	Componentwise continuity and the Epsilon-delta criteria 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, Thm: Every sequentially compact subset of R^n is closed and bounded.	Pages 293-299 of Fitzpatrick
18	Apr. 03	Thm: Every bounded sequence in R^n has a convergent subsequence; Thm: A subset of R^n is Sequentially compact iff it is closed and bounded, Thm: 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	Pages 299-304 of Fitzpatrick
Spring Break	Apr. 08-12
19	Apr. 15	Convex and pathwise-connected subsets of R^n, Intermediate Value Property, connected subsets of R^n; Limit points of a subset of R^n, limit of a function of several variables	Pages 304-313 & 348-352 of Fitzpatrick
20	Apr. 17	Partial derivatives, continuously differential functions, Theorem: If second order partial derivatives are continuous the order in which they are evaluated is not relevant, Mean-Value Lemma and propositions for a real-valued function of several real variables.	Pages 353-366 of Fitzpatrick
21	Apr. 22	Pre-exam problem and review session.
22	Apr. 24	Mean-Value Theorem for a real-valued function of several real variables, directional derivative, Theorem: Every continuously differentiable function is continuous.	Pages 366-370 of Fitzpatrick
Midterm Exam 2	Apr. 26
23	Apr. 29	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; Review of Linear Algebra on R^n: Linear maps and their matrix representations	Pages 372-377 & 394-399 of Fitzpatrick
24	May 06	Algebra of matrices, continuity of linear maps, invertible linear maps and their associated matrices, determinant and the inverse of an invertible matrix, Thm: An n x n matrix A is invertible iff there is a positive real number c such that for every column vector h we have \|\| A h \|\| ≥ c \|\| h \|\|.	Pages 399-405 of Fitzpatrick
25	May 08	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
Quiz4
26	May 13	Chain Rule for functions from R^n to R^m	Pages 414-420 of Fitzpatrick
27	May 15	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.