Calculus
Table of Contents
Prerequisite
- Sets, equivalence relations and functions
- Order relations, upper and lower bounds, supremum and infimum
- Countable sets
- Real numbers
Introduction to topology and metric spaces
- Topological spaces
- Compact and connected sets
- Basis of a topology
- Countable spaces
- Continuity
- Notion of distance and induced topology
Sequences in Metric Spaces
- Cauchy sequences.
- Limit of sequences and sub-sequences.
- Comparison theroem for limit of sequences.
- The number e as a limit of a sequence.
Series
- Series as limit of partial sums.
- Root, ratio and comparison test for series.
Functions and Limits on Metric Spaces
- Notion of limit of function in a metric space.
- Properties of limit (sum, ratio and product of limits)
Limits of Real Functions
- Some important limits.
- Properties of limits of functions (comparison test)
- The number e as limit of a function.
Continuity of Real Functions
- Uniform continuity.
- Weiestrass Theorem.
Derivative of Real Functions
- Properties of derivatives.
- Continuity and derivability.
- Lagrange theorem.
- Rule de L'Hopital and application.
- Derivatives of fundamental functions.
Functions of several real variables
- Continuity in Euclidean spaces
- Derivative and differential
- Concavity and convexity
- Local and global maxima and minima
The Riemann-Steltjes Integral
- Characterization of integrable functions.
- Properties of the integral.
- Fundamental Theorems of Calulus.
- Improper integral and convergence criteria.