Interactive Multivariable Calculus

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


    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

    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

Chapter 2: Vector Geometry

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

Chapter 9: Parametric Curves

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

Chapter 22: Infinite Series

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

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