Advanced Calculus: Theory and Practice

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.

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