Advanced Calculus using Mathematica

Keith Stroyan, University of Iowa

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Advanced Calculus using Mathematica

Advanced Calculus using Mathematica: NoteBook Edition is a complete text on calculus of several variables written in Mathematica NoteBooks.  The “eBook” has large movable figures and interactive programs to illustrate things like “zooming in” to see “local linearity.”  In addition to lots of traditional style exercises, the eText also has sections on student computing with Mathematica.

You need a copy of Mathematica to use the text.

Audience

We use the text in a basic multivariable calculus course and a more advanced second course on vector calculus for engineering students.

At the Tech Conference, people may be interested in our simple “security” method:  Users are authenticated using a .cdf program.

Contents & Organization

Mathematical Themes

Vectors throughout the course

"Just in time"

Computer work

Distribution

Advanced Calculus using Mathematica: NoteBook Edition is available in the Wolfram Store.

Contents

Chapter 0: An Introduction to Mathematica & Calculus

Chapter 1: Basic Equations and Graphs

Chapter 2: Vector Geometry

Chapter 3: Derivatives and Graphs of Explicit Functions

Chapter 4: Implicit Curves, Surfaces, and Contour Plots

Chapter 5: Inverse and Implicit Functions

Chapter 6: Taylor’s Formula in Several Variables

Chapter 7: Max-min in Several Variables

Chapter 8: Multiple Integrals in Cartesian Coordinates

Chapter 9: Parametric Curves

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

Chapter 11: Vector Fields and Velocity Flows in 2D

Chapter 12: Green’s Theorem, 2-D Divergence and Swirl

Chapter 13: Coordinate Systems in 2 Dimensions

Chapter 14: Path Integrals & Vector Fields in 3D

Chapter 15: Parametric Surfaces

Chapter 16: Curvature of Surfaces

Chapter 17: Coordinate Systems in 3 Dimensions

Chapter 18: Differential Forms

Chapter 19: Flow in 3D, Divergence & Curl

Chapter 20: Partial Differential Equations

Chapter 21: Infinite Series

Mathematical Themes

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.  
We treat the geometry of  equations in the explicit, implicit, and parametric cases by increasing dimension.  Each chapter begins with the linear case. This allows us to treat all cases in a uniform way.

http://homepage.divms.uiowa.edu/~stroyan/AdvCalcWebsite/AdvCalcThemes.html

http://homepage.divms.uiowa.edu/~stroyan/AdvCalcWebsite/AdvCalcLocalLinearity.html

Mathematica helps students “see” local linearity.  Example 4.2.2 Magnification of a Contour

Vectors 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.  This keeps being useful throughout the course, such as in a description of motion on a sphere.

"Just in time"

Another theme in these materials is a “just in time” approach to some topics.  For example, rather than developing all the linear material in the beginning, we study implicit linear equations just before implicit nonlinear equations.  In the more elementary course we teach with these materials, many of the “just in time” topics such as discussion of eigenvalues are skipped.  In the second advanced course, students take a matrix algebra couse in the interim and gain some familiarity with eigenvalues in particular.  This allows us to take advantage of Mathematica in some higher dimensional MAX-min problems where hand computation of eigenvalues is burdensome.

Abstract use of the Product Rules and Chain Rules

Perpendicularity of the Curvature Vector

Conservation of Energy under F = m A

Planets Orbit in a Plane

Computer work

Most of the day-to-day work in our courses is traditional paper and pencil work.  The text mostly looks like an on-screen old-fashioned math book.  For example, Section 3.3  Solutions to many “standard” exercises are in Mathematica in closed cells.

Links back to the text are in the exercise sets.

We start student computing with some very simple programs like solutions to many exercises.  These often befin with a team solved review paper and pencil exercise.

About once a week we have an "eExercise" such as graphing a surface z=f[x,y] and its tangent plane.  In these basic programming assignments, we strive to show the students how the textbook material written as Mathematica commands gives the solution.  At the end of this eExercise students should have a clearer idea of the algorithm that produces the tangent plane.

We also have “higher level” computer problems mainly aimed at a conceptual math idea like the meaning of the gradient.  Chapter 4 Lab 3

The advanced course uses Mathematica to solve some higher dimension probllems (like finding eigenvalues in 5 dimensions for a design optimization.)

Novel Topics

The text also has a number of topics that often are not included in a traditional “calc 3” book or even a stand-alone “Vector Calculus” book.  Mathematica makes a number of these advanced topics more accessible.

Differentiation with Constraints

Local linearity of flows

Functional Dependence

Vector Potentials

More information

http://homepage.math.uiowa.edu/~stroyan/AdvCalcWebsite/AdvCalcHomeWeb.html

Distribution

Advanced Calculus using Mathematica: NoteBook Edition is available in the Wolfram Store.

.cdf installer

WRI wrote a .cdf that installs the style sheets and utilities in hidden directories. from the Wolfram cloud.

Two views of a section.