Multivariable Calculus from Numerical, Graphical, and Symbolic Points of View, Volume 3

Ostebee and Zorn's approach applies reform principles to a rigorous calculus text. Conceptual understanding is the main goal of the text, and looking at mathematics from many representations (graphical, symbolic, numerical) is the main strategy for achieving this type of understanding. The key strengths of the text include combining symbolic manipulation with graphical and numerical representation, exercises of a varied nature and difficulty, and explanations written to be understandable to student readers.

• A student-friendly and approachable tone, numerous examples, critical-thinking questions, and supportive details and commentary help students successfully read and use the text.
• Representation of mathematical concepts through a variety of viewpoints supports different learning styles. Students see the math worked out through multiple representations--graphically, numerically, and symbolically--to enhance conceptual understanding.
• Proofs presented at point of use contribute significantly to helping students understand rigorous calculus concepts and develop analytic skills.
• Varied exercise sets offer instructors more options for creating homework assignments. Basic Exercises, which are straightforward and focus on a single idea, help students build basic skills.
• Further Exercises are a little more ambitious and may require the synthesis of several ideas, a deeper or more sophisticated understanding of basic concepts, or the use of a computer algebra system such as Maple or Mathematica. These are available for professors to assign when they would like to challenge their students and incorporate technology into their course.
• Answers to Select Exercises can be found in the back of the text, enabling students to get immediate feedback and assess their understanding of the material.
• Interludes are brief project-oriented expositions, with related exercises, that extend the concepts presented in the chapter. Professors have the opportunity to include these topics found at the end of the chapter as independent work, group work, or as a classroom activity. The Interludes include theoretical problems and proofs intended to enhance student understanding of the key calculus concepts.

Note: Each chapter contains a Summary.

11. Infinite Series

11.1 Sequences and Their Limits
11.2 Infinite Series, Convergence, and Divergence
11.3 Testing for Convergence; Estimating Limits
11.4 Absolute Convergence; Alternating Series
11.5 Power Series
11.6 Power Series as Functions
11.7 Taylor Series
Interlude: Fourier series

12. Curves and Vectors

12.1 Three-Dimensional Space
12.2 Curves and Parametric Equations
12.3 Polar Coordinates and Polar Curves
12.4 Vectors
12.5 Vector-Valued Functions, Derivatives, and Integrals
12.6 Modeling Motion
12.7 The Dot Product
12.8 Lines and Planes in Three Dimensions
12.9 The Cross Product
Interlude: Beyond Free Fall

13. Derivatives

13.1 Functions of Several Variables
13.2 Partial Derivatives
13.3 Linear Approximation in Several Variables
13.4 The Gradient and Directional Derivatives
13.5 Higher-Order Derivatives and Quadratic Approximation
13.6 Maxima, Minima, and Quadratic Approximation
13.7 The Chain Rule
13.8 Local Linearity: Some Theory of the Derivative

14. Integrals

14.1 Multiple Integrals and Approximating Sums
14.2 Calculating Integrals by Iteration
14.3 Integrals over Nonrectangular Regions
14.4 Double Integrals in Polar Coordinates
14.5 Triple Integrals
14.6 More Triple Integrals; Cylindrical and Spherical Coordinates
14.7 Multiple Integrals Overviewed: Change of Variables
Interlude: Mass and Center of Mass

15. Other Topics

15.1 Linear, Circular, and Combined Motion
Interlude: Cycloids and Epicycloids
15.2 New Curves from Old
15.3 Curvature
15.4 Lagrange Multipliers and Constrained Optimization
15.5 Improper Multivariable Integrals
Interlude: Constructing Pedal Curves

16. Vector Calculus

16.1 Line Integrals
16.2 More on Line Integrals; A Fundamental Theorem
16.3 Green's Theorem: Relating Line and Area Integrals
16.4 Surfaces and Their Parameterizations
16.5 Surface Integrals
16.6 Derivatives and Integrals of Vector Fields
16.7 Back to Fundamentals: Stokes's Theorem and the Divergence Theorem

Appendixes

A. Matrices and Matrd Integralsix Algebra: A Crash Course
B. Theory of Multivariable Calculus: Brief Glimpses
C. Table of Derivatives