This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Real analysis: course

### From Mathematics Is A Science

## 1 Description

This is a one-semester course in real analysis (aka advanced calculus). Certain facts about derivative, integral, the real number system, continuity and differentiability of functions of a single variable, integration of functions of a single variable, infinite series etc are proven. Goal: good familiarity with the foundations of Calculus, ability to work in the definition-theorem-proof format, some "mathematical maturity".

## 2 Prerequisites

- Calculus 1-3,
- Linear algebra: course
- proofs

## 3 Outline

- Metric spaces
- Functions of several variables
- Gradient
- Extrema of functions of several variables
- Functions of several variables: exercises
- Parametric curves
- Independence of path

## 4 Exercises

- Real analysis: test 1
- Real analysis: test 2
- Real analysis: final 1
- Real analysis: test 3
- Real analysis: test 4
- Real analysis: final 2

## 5 Notes

Text: *A First Course in Real Analysis* by Protter and Morrey (Chapters 2-5,9).

Alternative is a more geometric then analytical approach, see Vector calculus: course.