  # Advanced Calculus

## Theory and Practice, 1st Edition

Chapman and Hall/CRC

572 pages | 99 B/W Illus.

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### Description

Suitable for a one- or two-semester course, Advanced Calculus: Theory and Practice expands on the material covered in elementary calculus and presents this material in a rigorous manner. The text improves students’ problem-solving and proof-writing skills, familiarizes them with the historical development of calculus concepts, and helps them understand the connections among different topics.

The book takes a motivating approach that makes ideas less abstract to students. It explains how various topics in calculus may seem unrelated but in reality have common roots. Emphasizing historical perspectives, the text gives students a glimpse into the development of calculus and its ideas from the age of Newton and Leibniz to the twentieth century. Nearly 300 examples lead to important theorems as well as help students develop the necessary skills to closely examine the theorems. Proofs are also presented in an accessible way to students.

By strengthening skills gained through elementary calculus, this textbook leads students toward mastering calculus techniques. It will help them succeed in their future mathematical or engineering studies.

### Table of Contents

Sequences and Their Limits

Computing the Limits

Definition of the Limit

Properties of Limits

Monotone Sequences

The Number e

Cauchy Sequences

Limit Superior and Limit Inferior

Computing the Limits-Part II

Real Numbers

The Axioms of the Set R

Consequences of the Completeness Axiom

Bolzano-Weierstrass Theorem

Some Thoughts about R

Continuity

Computing Limits of Functions

A Review of Functions

Continuous Functions: A Geometric Viewpoint

Limits of Functions

Other Limits

Properties of Continuous Functions

The Continuity of Elementary Functions

Uniform Continuity

Two Properties of Continuous Functions

The Derivative

Computing the Derivatives

The Derivative

Rules of Differentiation

Monotonicity. Local Extrema

Taylor’s Formula

L’Hôpital’s Rule

The Indefinite Integral

Computing Indefinite Integrals

The Antiderivative

The Definite Integral

Computing Definite Integrals

The Definite Integral

Integrable Functions

Riemann Sums

Properties of Definite Integrals

The Fundamental Theorem of Calculus

Infinite and Improper Integrals

Infinite Series

A Review of Infinite Series

Definition of a Series

Series with Positive Terms

The Root and Ratio Tests

Series with Arbitrary Terms

Sequences and Series of Functions

Convergence of a Sequence of Functions

Uniformly Convergent Sequences of Functions

Function Series

Power Series

Power Series Expansions of Elementary Functions

Fourier Series

Introduction

Pointwise Convergence of Fourier Series

The Uniform Convergence of Fourier Series

Cesàro Summability

Mean Square Convergence of Fourier Series

The Influence of Fourier Series

Functions of Several Variables

Subsets of Rn

Functions and Their Limits

Continuous Functions

Boundedness of Continuous Functions

Open Sets in Rn

The Intermediate Value Theorem

Compact Sets

Derivatives

Computing Derivatives

Derivatives and Differentiability

Properties of the Derivative

Functions from Rn to Rm

Taylor’s Formula

Extreme Values

Implicit Functions and Optimization

Implicit Functions

Derivative as a Linear Map

Open Mapping Theorem

Implicit Function Theorem

Constrained Optimization

The Second Derivative Test

Integrals Depending on a Parameter

Uniform Convergence

The Integral as a Function

Uniform Convergence of Improper Integrals

Integral as a Function

Some Important Integrals

Integration in Rn

Double Integrals over Rectangles

Double Integrals over Jordan Sets

Double Integrals as Iterated Integrals

Transformations of Jordan Sets in R2

Change of Variables in Double Integrals

Improper Integrals

Multiple Integrals

Fundamental Theorems

Curves in Rn

Line Integrals

Green’s Theorem

Surface Integrals

The Divergence Theorem

Stokes’ Theorem

Differential Forms on Rn

Exact Differential Forms on Rn

Solutions and Answers to Selected Problems

Bibliography

Subject Index

Author Index

### Subject Categories

##### BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations
MAT037000
MATHEMATICS / Functional Analysis