# New in Version 4.0

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**Chapter 14: Vector Analysis**. Covers line integrals, Green's, Stokes' and Divergence Theorems, etc. See below for section headings. This content is included in APEX Calculus 3; as many schools do not teach this content, it is omitted in APEX Calculus 3*, the abridged edition. (See the Download and Purchase pages for more information.)**Cylindrical and Spherical Coordinates.**A new section on triple integration with these coordinate systems is included in Chapter 13.**Better numbering of of Definitions, Theorems, etc.**Now "Definition 12.4.3" refers to the 3rd Definition of Chapter 12, Section 4.**Better exercise sets.**The main adjustment here is that all question sets (i.e., "In Exercises 17-25, do ..." ) there are an even number of problems and the first problem number is odd. New problems have been added.*A wholesale changing of the exercises to justify a new edition did not occur.***"Small" corrections.**Numerous misspellings, grammatical errors, dropped minus signs, etc., have been fixed. (Many thanks to all who submitted corrections.)**"Large" corrections.**There were several places in Version 3.0 where a definition or theorem just was not right. A partial list of topics affected: one-sided limits, definition of increasing/decreasing functions, limits of multivariable functions, and multivariable second derivative test.

**Content of APEX Calculus, Version 4.0**

APEX Calculus comprises 14 chapters; each chapter and section is listed below. This text was written to have the same basic organization as most traditional calculus textbooks.

The whole text is 972 pages long. For multiple reasons (lower cost per semester, lower weight per semester, increased likelihood of students bringing their text to class, etc.), APEX Calculus has been broken up into 3 overlapping volumes, roughly along traditional "Calc 1, 2 & 3" lines, as follows:

- Calculus 1: Chapters 1 through 6.1
- Limits through Integration by Substitution

- Calculus 2: Chapters 5 through 8
- Integration through Sequences and Series

- Calculus 3: Chapters 9 through 14
- Curves in the Plane through Vector Analysis

- Calculus 3*: Abridged, Chapters 9 through 13 (for those not needing Vector Analysis)
- Curves in the Plane through Multiple Integration

If this division does not match up well with your department's division of Calculus material:

- Tell me so I know what people are most interested in,
- Consider editing the source files at GitHub to create your own version, and/or
- Consider "just making it work as is"; while not ideal, the book is still a great text at a great price.

- Limits
- An introduction to Limits
- Epsilon-Delta Definition of a Limit
- Finding Limits Analytically
- One-Sided Limits
- Continuity
- Limits Involving Infinity

- Derivatives
- Instantaneous Rates of Change: The Derivative
- Interpretations of the Derivative
- Basic Differentiation Rules
- The Product and Quotient Rules
- The Chain Rule
- Implicit Differentiation
- Derivatives of Inverse Functions

- The Graphical Behavior of Functions
- Extreme Values
- The Mean Value Theorem
- Increasing and Decreasing Functions
- Concavity and the Second Derivative
- Curve Sketching

- Applications of the Derivative
- Newton's Method
- Related Rates
- Optimization
- Differentials

- Integration
- Antiderivatives and Indefinite Integration
- The Definite Integral
- Riemann Sums
- The Fundamental Theorem of Calculus
- Numerical Integration

- Techniques of Integration
- Substitution
- Integration by Parts
- Trigonometric Integrals
- Trigonometric Substitution
- Partial Fraction Decomposition
- Hyperbolic Functions
- L'Hopital's Rule
- Improper Integration

- Applications of Integration
- Area Between Curves
- Volume by Cross-Sectional Area: Disk and Washer Methods
- The Shell Method
- Arc Length and Surface Area
- Work
- Fluid Forces

- Sequences and Series
- Sequences
- Infinite Series
- Integral and Comparison Tests
- Ratio and Root Tests
- Alternating Series and Absolute Convergence
- Power Series
- Taylor Polynomials
- Taylor Series

- Curves in the Plane
- Conic Sections
- Parametric Equation
- Calculus and Parametric Equations
- Introduction to Polar Coordinates
- Calculus and Polar Functions

- Vectors
- Introduction to Cartesian Coordinates in Space
- An Introduction to Vectors
- The Dot Product
- The Cross Product
- Lines
- Planes

- Vector-Valued Functions
- Vector-Valued Functions
- Calculus and Vector-Valued Functions
- The Calculus of Motion
- Unit Tangent and Normal Vectors
- The Arc Length Parameter and Curvature

- Functions of Several Variables
- An Introduction to Multivariable Functions
- Limits and Continuity of Multivariable Functions
- Partial Derivatives
- Differentiability and the Total Differential
- The Multivariable Chain Rule
- Directional Derivatives
- Tangent Lines, Normal Lines, and Tangent Planes
- Extreme Values

- Multiple Integration
- Iterated Integration and Area
- Double Integration and Volume
- Double Integration with Polar Coordinates
- Center of Mass
- Surface Area
- Volume Between Surfaces and Triple Integration
- Triple Integration with Cylindrical and Spherical Coordinates

- Vector Calculus
- Introduction to Line Integrals
- Vector Fields
- Line Integrals over Vector Fields
- Flow, Flux, Green's Theorem and the Divergence Theorem
- Parametrized Surfaces and Surface Area
- Surface Integrals
- The Divergence Theorem and Stokes' Theorem

- Appendix:
- Solutions to Odd Numbered Exercises
- Index
- Useful formulas