# Alignment: Schaum’s Outline • Advanced Calculus • Third Edition

• Chapter 1 Numbers
• Sets
• Real Numbers
• Decimal Representation of Real Numbers
• Geometric Representation of Real Numbers
• Operations with Real Numbers
• Inequalities
• Absolute Value of Real Numbers
• Exponents and Roots
• Logarithms
• Axiomatic Foundations of the Real Number System
• Point Sets, Intervals
• Countability
• Neighborhoods
• Limit Points
• Bounds
• Bolzano-Weierstrass Theorem
• Algebraic and Transcendental Numbers
• The Complex Number System
• Polar Form of Complex Numbers
• Mathematical Induction
• Chapter 2 Sequences
• Definition of a Sequence
• Limit of a Sequence
• Theorems on Limits of Sequences
• Infinity
• Bounded, Monotonic Sequences
• Least Upper Bound and Greatest Lower Bound of a Sequence
• Limit Superior, Limit Inferior
• Nested Intervals
• Cauchy’s Convergence Criterion
• Infinite Series
• Chapter 3 Functions, Limits, and Continuity
• Functions
• Graph of a Function
• Bounded Functions
• Montonic Functions
• Inverse Functions, Principal Values
• Maxima and Minima
• Types of Functions
• Transcendental Functions
• Limits of Functions
• Right- and Left-Hand Limits
• Theorems on Limits
• Infinity
• Special Limits
• Continuity
• Right- and Left-Hand Continuity
• Continuity in an Interval
• Theorems on Continuity
• Piecewise Continuity
• Uniform Continuity
• Chapter 4 Derivatives
• The Concept and Definition of a Derivative
• Right- and Left-Hand Derivatives
• Differentiability in an Interval
• Piecewise Differentiability
• Differentials
• The Differentiation of Composite Functions
• Implicit Differentiation
• Rules for Differentiation
• Derivatives of Elementary Functions
• Higher-Order Derivatives
• Mean Value Theorems
• L’Hospital’s Rules
• Applications
• Chapter 5 Integrals
• Introduction of the Definite Integral
• Measure Zero
• Properties of Definite Integrals
• Mean Value Theorems for Integrals
• Connecting Integral and Differential Calculus
• The Fundamental Theorem of the Calculus
• Generalization of the Limits of Integration
• Change of Variable of Integration
• Integrals of Elementary Functions
• Special Methods of Integration
• Improper Integrals
• Numerical Methods for Evaluating Definite Integrals
• Applications
• Arc Length
• Area
• Volumes of Revolution
• Chapter 6 Partial Derivatives
• Functions of Two or More Variables
• Neighborhoods
• Regions
• Limits
• Iterated Limits
• Continuity
• Uniform Continuity
• Partial Derivatives
• Higher-Order Partial Derivatives
• Differentials
• Theorems on Differentials
• Differentiation of Composite Functions
• Euler’s Theorem on Homogeneous Functions
• Implicit Functions
• Jacobians
• Partial Derivatives Using Jacobians
• Theorems on Jacobians
• Transformations
• Curvilinear Coordinates
• Mean Value Theorems
• Chapter 7 Vectors
• Vectors
• Geometric Properties of Vectors
• Algebraic Properties of Vectors
• Linear Independence and Linear Dependence of a Set of Vectors
• Unit Vectors
• Rectangular (Orthogonal) Unit Vectors
• Components of a Vector
• Dot, Scalar, or Inner Product
• Cross or Vector Product
• Triple Products
• Axiomatic Approach to Vector Analysis
• Vector Functions
• Limits, Continuity, and Derivatives of Vector Functions
• Geometric Interpretation of a Vector Derivative
• Formulas Involving ∇
• Vector Interpretation of Jacobians and Orthogonal Curvilinear Coordinates.
• Gradient, Divergence, Curl, and Laplacian in Orthogonal Curvilinear Coordinates
• Special Curvilinear Coordinates
• Chapter 8 Applications of Partial Derivatives
• Applications To Geometry
• Directional Derivatives
• Differentiation Under the Integral Sign
• Integration Under the Integral Sign
• Maxima and Minima
• Method of Lagrange Multipliers for Maxima and Minima
• Applications to Errors
• Chapter 9 Multiple Integrals
• Double Integrals
• Iterated Integrals
• Triple Integrals
• Transformations of Multiple Integrals
• The Differential Element of Area in Polar Coordinates, Differential Elements of Area in Cylindrical and Spherical Coordinates
• Chapter 10 Line Integrals, Surface Integrals, and Integral Theorems
• Line Integrals
• Evaluation of Line Integrals for Plane Curves
• Properties of Line Integrals Expressed for Plane Curves
• Simple Closed Curves, Simply and Multiply Connected Regions
• Green’s Theorem in the Plane
• Conditions for a Line Integral to Be Independent of the Path
• Surface Integrals
• The Divergence Theorem
• Stokes’s Theorem
• Chapter 11 Infinite Series
• Definitions of Infinite Series and Their Convergence and Divergence.
• Fundamental Facts Concerning Infinite Series
• Special Series
• Tests for Convergence and Divergence of Series of Constants
• Theorems on Absolutely Convergent Series
• Infinite Sequences and Series of Functions, Uniform Convergence
• Special Tests for Uniform Convergence of Series
• Theorems on Uniformly Convergent Series
• Power Series
• Theorems on Power Series
• Operations with Power Series
• Expansion of Functions in Power Series
• Taylor’s Theorem
• Some Important Power Series
• Special Topics
• Taylor’s Theorem (for Two Variables)
• Chapter 12 Improper Integrals
• Definition of an Improper Integral
• Improper Integrals of the First Kind (Unbounded Intervals)
• Convergence or Divergence of Improper Integrals of the First Kind
• Special Improper Integers of the First Kind
• Convergence Tests for Improper Integrals of the First Kind
• Improper Integrals of the Second Kind
• Cauchy Principal Value
• Special Improper Integrals of the Second Kind
• Convergence Tests for Improper Integrals of the Second Kind
• Improper Integrals of the Third Kind
• Improper Integrals Containing a Parameter, Uniform Convergence
• Special Tests for Uniform Convergence of Integrals
• Theorems on Uniformly Convergent Integrals
• Evaluation of Definite Integrals
• Laplace Transforms
• Linearity
• Convergence
• Application
• Improper Multiple Integrals
• Chapter 13 Fourier Series
• Periodic Functions
• Fourier Series
• Orthogonality Conditions for the Sine and Cosine Functions
• Dirichlet Conditions
• Odd and Even Functions
• Half Range Fourier Sine or Cosine Series
• Parseval’s Identity
• Differentiation and Integration of Fourier Series
• Complex Notation for Fourier Series
• Boundary-Value Problems
• Orthogonal Functions
• Chapter 14 Fourier Integrals
• The Fourier Integral
• Equivalent Forms of Fourier’s Integral Theorem
• Fourier Transforms
• Chapter 15 Gamma and Beta Functions
• The Gamma Function
• Table of Values and Graph of the Gamma Function
• The Beta Function
• Dirichlet Integrals
• Chapter 16 Functions of a Complex Variable
• Functions
• Limits and Continuity
• Derivatives
• Cauchy-Riemann Equations
• Integrals
• Cauchy’s Theorem
• Cauchy’s Integral Formulas
• Taylor’s Series
• Singular Points
• Poles
• Laurent’s Series
• Branches and Branch Points
• Residues
• Residue Theorem
• Evaluation of Definite Integrals

Use Nagwa in conjunction with your preferred textbook. The recommended lessons from Nagwa for each section of this textbook are provided above. This alignment is not affiliated with, sponsored by, or endorsed by the publisher of the referenced textbook. Nagwa is a registered trademark of Nagwa Limited. All other trademarks and registered trademarks are the property of their respective owners.