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Advanced Calculus

Theory and Practice

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## Book 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 R

^{n}Functions and Their Limits

Continuous Functions

Boundedness of Continuous Functions

Open Sets in R

^{n}The Intermediate Value Theorem

Compact Sets

**Derivatives**Computing Derivatives

Derivatives and Differentiability

Properties of the Derivative

Functions from R

*to R*

^{n}

^{m}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 **R^{n} Double Integrals over Rectangles

Double Integrals over Jordan Sets

Double Integrals as Iterated Integrals

Transformations of Jordan Sets in R

^{2}

Change of Variables in Double Integrals

Improper Integrals

Multiple Integrals

**Fundamental Theorems **Curves in R

^{n}Line Integrals

Green’s Theorem

Surface Integrals

The Divergence Theorem

Stokes’ Theorem

Differential Forms on R

^{n}Exact Differential Forms on R

^{n}

Solutions and Answers to Selected Problems

Bibliography

Subject Index

Author Index