Interactive Multivariable Calculus

Keith Stroyan
University of Iowa
http://www.math.uiowa.edu/~stroyan/

Abstract

We are developing a text in multivariable calculus that has both a print and "live" version with animations and movable figures.  The majority of the text does not rely on (overt) Mathematica computations.

There are Mathematica sections and short "labs" beyond the main text.

Drafts of the text have been used with 3 different levels of students, where different sections were printed for the different levels.  The electronic version is a more complete reference.

This talk will demonstrate the "live" text and discuss topics that  Mathematica helps students master.

Contents

Chapter 0: An eTour of the Materials

01 An Electronic Overview

02 A Mathematica Primer

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 2-D

06 Graphing in 3-D by Slicing

07 Mathematica Plots in 3-D

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)

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 with Mathematica

05 Directional Derivatives & Nonlinear Gradients

06 Tangents with Mathematica

07 Approximation by Differentials

08 Functional Derivatives: Superposition, Product, and Chain Rules

Chapter 4: Taylor's Formula in Several Variables

01 Second Order Derivatives, Symmetry

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

Chapter 5: 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 Differentiation Review

Chapter 6: 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 7: Max-min in Several Variables

01 A First Look at Theory

02 Interior Critical Points

03 Compact Domains

04 Boundary Extrema& Lagrange Multipliers

05 Extrema with Theory

06 MAX-min withMathematica

07 Practical MAX-min Problems

08 Second Derivatives and Local Extrema

09 Extensions of the Theory

Chapter 8: Multiple Integrals in Cartesian Coordinates

01 Riemann Sums in 1-D

02 Riemann Sums in 2-D

03 Iterated  Integrals

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, Curvature, and Torsion

08 Shapes and Mathematica

Chapter 10: Motion along Curves: Gas, Brakes, & Tires

01 Speed, Velocity, Acceleration

02 Gas, Brakes, and Turns

03 Motion Along Curves with Mathematica

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

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 Proof of Green's Theorem

Chapter 13: Coordinate Systems in 2 Dimensions

01 Oriented Area of a Parallelogram

02 Area Integrals in Polar Coordinates

03 Area Integrals for General Coordinates in 2D

04 Change of Variables for Double Integrals

05 Differentiation in Coordinates

Chapter 14: Path Integrals & Conservative Fields in 3D

01 Path Integrals in 3 D and nD

02 The Chain Rule and Gradient (again)

03 Conservation of "Energy"

Chapter 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

Chapter 16: Curvature of Surfaces

01 Gaussian Curvature

02 Motion Constrained to a Surface

03 Differentiation on a Surface

Chapter 17: Coordinate Systems in 3 Dimensions

01 Oriented Volume of a Parallelepiped

02 Volume Integrals in Cylindrical Coordinates

03 Volume Integrals in Spherical Coordinates

04 Volume Integrals for General Coordinates in 3D

05 Change of Variables for Triple Integrals

06 Derivatives in Different Coordinate Systems

Chapter 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

Chapter 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

Chapter 20: Partial Differential Equations

01 The Heat Equation and Divergence

02 Maxwell's Equations

03 Euler's and Navier-Stokes' Equations

Chapter 21: Theoretical Fine Points

Coninuity in General

The Extreme Value Theorem

Proof of the Principal Axes Theorem

Matrix Norms

The Lipschitz Estimate\(\(|\)\(F[X + dX] - 
            F[X]\)\(|\)\(\(≤\)\(\(||\)\(DF[\(X\_*\)]\)\(||\)\(·\)\)\)\(|\)\(dX\)\(|\)\)

Uniform vs.Pointwise Limits and Approximation

Smoothness and Continuity of Partial Derivatives

Bounded Pointwise Partials are Smooth

Mixed Second Partials are Equal