Mathematical
Analysis, An Introduction
Real Number system
Inductive set and the set
of natural numbers N. Infinite set, countable set. Well ordering
property of N. The integers and the rational numbers.
The real numbers. The Field axioms,
the positivity axioms, the archimedean property and
the (order) completeness axiom. Supremum and infimum
Sequences.
Definition. Definition of
convergence. Properties: sums, product and reciprocal. Comparison test. Squeeze Theorem. Convergence implies boundedness. Monotone convergence
theorem. Cauchy sequence. Cauchy
principle of convergence. Subsequences. Bolzano Weierstrass
Theorem. Every real bounded sequence has a monotone subsequence.
Sequence tending to + or - infinity. Subsets of the real
numbers, the intervals. limit point or cluster
point or accumulation point of a subset of R. Open and closed subsets. Closure. Sequentially compact subset of R.
Heine Borel Theorem. Countable
compactness and sequentially compactness.
Continuous
Functions.
e - d definition of continuity. Sequence
definition of continuity. Properties: Sum, product and quotient. Composition. Consequence of continuity.
Continuous image of compact subset is compact. Extreme Value
Theorem. Maximizer and minimizer. Continuous image
of an interval is an interval. Intermediate Value Theorem.
Monotone functions and continuity Inverse function and continuity. Uniform continuity. Any continuous function on a closed and
bounded interval is uniformly continuous. Limits of a
function. Properties: sum, product and quotient. Composition
and limit. One sided limits. Squeeze Theorem for limits of functions.
Differentiable functions.
Definition
of derivative.
Properties: sum, product and quotient. Chain rule. Relative maximum and relative minimum. Local
minimizer and local maximizer.
Relative Extremum. Rolle's Theorem.
Mean Value Theorem. Consequences of Mean Value Theorem.
Monotone functions. First derivative test and second derivative test for
relative extremum. Concavity.
Concavity and derived function. Derivative
of inverse function. Cauchy mean value Theorem.
L' Hôpital's Rule and an analytic consequence. Taylor Theorem with remainder. Intermediate
Value property of the derived function, Darboux
theorem.
Integration.
Anti-derivative. Properties: additivity.
Change of variable for anti-derivative. Riemann sums and Riemann integral. Lower and Upper Darboux sums. Lower and upper integrals. Refinement
Lemma. Convergence of Lower and upper Darboux
sums. Equivalent definitions of Riemann integrability.
Any continuous function on a closed and bounded interval is integrable.
Additivity.
Properties of integrable functions.
Convergence of Riemann sums. Mean Value Theorem for integrals. Darboux
fundamental theorem of calculus. First fundamental
theorem of Calculus, Second fundamental theorem of calculus. Products and modulus of integral functions. Integration by parts. Change of variable formula for Remain
integrals. Second Mean Value Theorem for integrals. Improper integrals,
convergence, absolute convergence, conditional convergence, tests for
convergence, Lebesgue integral and improper integral,
differentiation under the integral sign, the probability integral.
There are 14 chapters with
exercises. These have been used in a course in mathematical
analysis. Below we give a reasonable lesson plan for self studies and
included typical choice of tutorial questions. We have also included
references to Fitzpatrick's book.
The
chapters are in pdf format.
Here is
the content page Contents.
|
Chapters and references |
Tutorial |
|
|
Chapter 1 Ref: Chapter 2 Dedekind's cuts |
Tutorial 1 |
Tutorial 2 |
|
Chapter 2. |
Tutorial
3 |
Tutorial 4 |
|
Chapter 3
|
Tutorial 5 |
Tutorial 6 |
|
Chapter 4
|
Tutorial 7 |
Tutorial 8 |
|
Chapter 5
|
Tutorial 9 |
Tutorial
10 |
|
Chapter 6 Series Chapter 7 Series of Functions and Power Series Chapter 8 Uniform Convergence and Differentiation |
Chapter 9 Uniform
Convergence Integration and Power Series Chapter 10 Weierstrass Approximation Theorem Chapter 11 The Elementary Functions |
Chapter 12 Arithmetic of Power Series Chapter 13 Special Tests
for Convergence Chapter 14 Improper and Lebesgue Integral |