Interactive Multivariable Calculus
by
Keith D. Stroyan
Publication data:
copyright © 2006 & 2008 by Keith D. Stroyan
Published by Keith D. Stroyan for Shohola Labs
Iowa City, Iowa 52242
ISBN 0-9787763-0-5
Printed in Iowa City, Iowa by Zephyr Copies
Second revised printing January 2008
An early draft of these materials was written with support from NSF grant DUE 99-50164
When the gods want to punish you, they answer your prayers.
Isak Dinesen (Karen Blixen), Out of Africa
Preface
This print book is part of a larger collection of materials also including an electronic text written in Mathematica NoteBooks. The print text is the core material we use in a basic introduction to calculus of several variables. "eChapters", "eSections", "eTopics", and various extra details only appear in the "eText".
The eText has large color figures, often movable (or "live"), and animations to illustrate things like "zooming in" described here in print. The eText has sections on computing the main calculus topics with Mathematica. There are also solutions to the exercises in the closed cells of the eText. You will need a licensed copy of Mathematica to do these computations, but the eText and live figures can be read with the free trial version of Mathematica at http://www.wolfram.com/products/mathematica/trial.cgi
The iMultiCalc website has interactive versions of many of the text examples. These allow you to modify the functions or points in a text example and re-compute the example with webMathematica (running on our server. You don't need Mathematica for this. The output uses MathML and Live3D java graphics, so you need to read the set-up instructions on the site so your browser will display formulas and graphs correctly.)
Mathematical Themes
Explicit, implicit, and parametric representations
Students learning calculus of several variables face a bewildering number of formulas if they try to learn the subject as a special formula for every case. "The gradient is perpendicular to the tangent" is false for explicit surfaces, z=f[x,y], but true for implicit surfaces, F[x,y,z]=c. It seems there are many equations for tangents. However, if we are conscious of the explicit, implicit, parametric classification of equations and graphs, there is only one procedure for finding the equation of the tangent.
Uniform procedures that apply to explicit, implicit, and parametric cases
In the case of an explicit nonlinear equation and graph, the tangent equation procedure produces an explicit linear equation and graph. The tangent procedure applied to an implicit nonlinear equation and graph produces an implicit linear tangent equation and graph. A nonlinear parametric equation and graph produces a linear parametric tangent equation and graph.
Local linearity: small approximations of smooth problems as vector algebra
The association between the various cases of nonlinear equations and their linear tangent equations is the fundamental idea of differential calculus: smooth functions are "locally linear." More graphically speaking, a sufficiently magnified view of a smooth nonlinear graph appears to be the same as the graph of the linear tangent equation. To understand this idea, we first need to understand the geometry of the linear equations in the explicit, implicit, and parametric cases, so the chapters organized by the explicit, implicit, parametric theme in increasing dimension each begin with the linear case.
Building vector geometry translation skills throughout the course
Our development of the interaction between linear and nonlinear equations and graphs is based on vector algebra and geometry. Basic vector geometry and algebra is not only a helpful way to understand the equations of calculus, it is a powerful tool in many parts of math, science and engineering. We urge our reader to develop a working lexicon (or "translation dictionary") that contains the equivalent "words" of the two languages.
“Just in time”
Another theme in these materials is a "just in time" approach to some topics. Rather than developing all the linear material in the beginning, we study implicit linear equations just before implicit nonlinear equations. The general multidimensional product and chain rules of calculus are also postponed. Basic partial derivatives can be computed with the one variable rules, and the real importance of the higher dimensional rules is abstract such as in understanding curvature or conservative vector fields. The abstract rules are introduced when they are needed.
Calculus: The Language of Change
Calculus is one of the great achievements of the human intellect. It has served as the language of change in the development of scientific thought for more than three centuries. "Infinitesimal calculus" grew hand in hand with physical science and technology. Many problems in math, science, engineering, and other subjects involve the change in output as many inputs vary and this book deals with that part of the subject. The geometry and algebra of curves and surfaces in 3D is a concrete, useful, and beautiful way to begin to learn this subject.
Modern computing helps make the subject even more concrete and can help with fast accurate calculations. I beleive that courses that incorporate modern computing can train future scientists to be better users of mathematics. I hope Interactive Multivariable Calculus takes a step in that direction.
Best wishes in learning this wonderful and useful subject.
eChapter 0: An eTour of the Materials
Chapter on CD only
Chapter 1: Basic Equations & Graphs
01 Explicit Line, Parabola, and Circle
02 Implicit Line in 2-D and Implicit Circle
03 Parametric Line in 2-D and Parametric Circle
04 Sliding and Squashing Equations in 2-D
05 Mathematica Plots in 2D
06 Graphing in 3-D by Slicing
07 Mathematica Plots in 3D
08 Mathematica Labs for Chapter 1
01 Position and Displacement Vectors
02 Geometry of Vector Sum
03 Displacement Vectors and Differences
04 Geometry of Scalar Multiplication
05 Angles, Perpendiculars, and Projections
06 Cross Product
07 A Lexicon for Translation and Mathematica
08 Abstract Vector Algebra
09 Just in Time Algebra & Geometry (area, volume, and determinants)
10 Mathematica Labs for Chapter 2
Chapter 3: Explicit Functions of Several Variables
01 Explicit Linear Functions: Graphs, Slopes, and Gradient
02 Practical Functions of Several Variables
03 Derivatives & Linear Approximation, The Explicit Tangent
04 Differentiation Skills and Mathematica
05 Nonlinear Gradients & Directional Derivatives
06 Gradients and Tangents with Mathematica
07 Approximation by Differentials
Remainder of Chapter on CD only
08 Functional Derivatives: Superposition, Product, and Chain Rules
09 Advanced Examples of Continuity, Tangency, and Smoothness
10 Mathematica Labs for Chapter 3
Chapter 4: Implicit Curves, Surfaces, and Contour Plots
01 Implicit Linear Functions
02 Contour Graphs, Gradients, and Tangents
03 Mathematica Contour Plots
04 Implicit Planes
05 Implicit Surfaces
06 Mathematica Implicit Tangent Planes
07 Differentiation Review
08 Mathematica Labs for Chapter 4
eChapter 5: Inverse and Implicit Functions
01 m Linear Equations in n Unknowns
02 Local Solution of Implicit Equations
03 The Inverse Function Theorem
04 Implicit Differential Approximations
05 Differentiation with Constraints
06 Local Flows and Inverse Functions
07 Functional Dependence
Chapter 6: Taylor's Formula in Several Variables
01 Second Order Derivatives, Symmetry
Remainder of Chapter on CD only
02 Principal Axes for 2×2 Symmetric Matrices
03 Local Convexity and Taylor's Second Order Formula in 2D
04 Mathematica and Taylor's Formula
05 Second Order Taylor's Formula in nD
06 Higher Order Taylor's Formula
07 Mathematica Labs for Chapter 6
Chapter 7: Max-min in Several Variables
01 A First Look at Theory
02 Interior Critical Points
Remainder of Chapter on CD only
03 Compact Domains
04 Boundary Extrema & Lagrange Multipliers
05 MAX-min with Mathematica
06 MAX-min Problems
07 Second Derivatives and Local Extrema
08 Extensions of the Theory
09 Mathematica Labs for Chapter 7
Chapter 8: Multiple Integrals in Cartesian Coordinates
01 Riemann Sums in 1-D
02 Riemann Sums in 2-D
03 Iterated Integrals
Remainder of Chapter on CD only
04 Integration in 3-D
05 Integration with Mathematica
06 Improper Integrals
07 Applications of Multidimensional Integrals
01 Parametric Lines
02 Parametric Curves in 2D
03 Mathematica 2D Parametric Curves
04 Parametric Curves in 3D
05 Mathematica 3D Parametric Curves
06 Parametric Tangents
07 Arclength
08 Product Rules, Curvature, Torsion
09 Mathematica Labs for Chapter 9
Chapter 10: Motion along Curves: Gas, Brakes, & Tires
01 Speed, Velocity, Acceleration
02 Gas, Brakes, and Turns
Remainder of Chapter on CD only
04 Kelper's Laws
Chapter 11: Vector Fields and Velocity Flows in 2D
01 Vector Fields
02 Vector Fields with Mathematica
03 Velocity Flows
04 Flow Along and Across a Curve
05 Path Independence and The Chain Rule
06 Gradient Flows and Max-min Revisited
07 Mathematica Labs for Chapter 11
Chapter 12: Green's Theorem, 2-D Divergence and Swirl
01 Introduction to the Three Forms of Green's Theorem
02 Formal Divergence and Swirl
03 Swirl & Geometry of Domains in 2D
04 Green Applications
05 The Microscopic View of Divergence and Swirl
06 Mathematica Labs for Chapter 12
Chapter 13: Coordinate Systems in 2 Dimensions
01 Oriented Area of a Parallelogram & Linear Coordinates
02 Area Integrals in Polar Coordinates
03 Area Integrals for General Coordinates in 2D
04 Differentiation in Coordinates
06 Mathematica Labs for Chapter 13
eChapter 14: Path Integrals & Vector Fields in 3D
01 Vector Fields in 3 D
02 Path Integrals in 3 D and nD
03 The Chain Rule and Gradient (again)
04 Vector Field Identities
05 Mathematica Labs for Chapter 14
eChapter 15: Parametric Surfaces
01 Parametric Planes
02 Parametric Surfaces
03 Surfaces with Mathematica
04 Parametric Surface Tangents
05 Area of Parametric Surfaces
06 Orientation and Flux Integrals
07 Mathematica Labs for Chapter 15
eChapter 16: Curvature of Surfaces
01 Gaussian Curvature
02 Motion Constrained to a Surface
03 Differentiation on a Surface
eChapter 17: Coordinate Systems in 3 Dimensions
01 Oriented Volume of a Parallelepiped & Linear Coordinates
02 Volume Integrals in Cylindrical Coordinates
03 Volume Integrals in Spherical Coordinates
04 Volume Integrals for General Coordinates in 3D
05 Derivatives in Different Coordinate Systems
eChapter 18: Differential Forms
01 Differential Forms in 2D & 3D
02 Exterior Derivatives of Differential Forms
03 Path Integrals of 1-Forms
05 Surface Flux Integrals of 2-Forms
05 Volume Integrals of 3-Forms
06 Differential Forms in n-Dimensions
eChapter 19: Flow in 3D, Divergence & Curl
01 Gradient Fields in 3D
02 Stokes' Theorem and Curl in 3D
03 Simply Connected Domains in 3D & Gradient Fields
04 Gauss's Theorem and Divergence in 3D
05 Handles and Bubbles
06 deRahm Cohomology
07 Microscopic Divergence and Curl
eChapter 20: Partial Differential Equations
01 The Heat Equation and Divergence
02 Laplace's Equation
02 Maxwell's Equations
04 The Wave Equation
05 Euler's and Navier-Stokes' Equations
06 Brownian Motion and the Diffusion Equation
eChapter 21: Theoretical Fine Points
Coninuity in General
The Extreme Value Theorem
Proof of the Principal Axes Theorem
Matrix Norms
Uniform vs. Pointwise Limits and Approximation
Smoothness and Continuity of Partial Derivatives
Mixed Second Partials are Equal
01 Geometric Series
02 Comparison and Weierstrass' M-test
03 The Classical Power series
04Convergence and the Ratio Test
05 Integration and differentiation of Series
06 Series with Mathematica
