## Multivariable Calculus and Mathematica®: With Applications to Geometry and PhysicsOne of the authors' stated goals for this publication is to "modernize" the course through the integration of Mathematica. Besides introducing students to the multivariable uses of Mathematica, and instructing them on how to use it as a tool in simplifying calculations, they also present intoductions to geometry, mathematical physics, and kinematics, topics of particular interest to engineering and physical science students. In using Mathematica as a tool, the authors take pains not to use it simply to define things as a whole bunch of new "gadgets" streamlined to the taste of the authors, but rather they exploit the tremendous resources built into the program. They also make it clear that Mathematica is not algorithms. At the same time, they clearly see the ways in which Mathematica can make things cleaner, clearer and simpler. The problem sets give students an opportunity to practice their newly learned skills, covering simple calculations with Mathematica, simple plots, a review of one-variable calculus using Mathematica for symbolic differentiation, integration and numberical integration. They also cover the practice of incorporating text and headings into a Mathematica notebook. A DOS-formatted diskette accompanies the printed work, containing both Mathematica 2.2 and 3.0 version notebooks, as well as sample examination problems for students. This supplementary work can be used with any standard multivariable calculus textbook. It is assumed that in most cases students will also have access to an introductory primer for Mathematica. |

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### Contents

INTRODUCTION | 1 |

Benefits of Mathematical Software | 2 |

Whats in This Book | 3 |

Whats Not in This Book | 4 |

How to Use This Book | 5 |

A Word About Versions of Mathematica | 7 |

REVIEW OF ONEVARIABLE CALCULUS | 9 |

Glossary of Some Useful Mathematics Objects Commands | 13 |

DIRECTIONAL DERIVATIVES AND THE GRADIENT | 95 |

Glossary of Some Useful Mathematics Objects Commands | 101 |

GEOMETRY OF SURFACES | 103 |

The Concept of a Surface | 104 |

Basic Examples | 105 |

The Implicit Function Theorem | 110 |

Geometric Invariants | 112 |

Curvature Calculations with Mathematica | 119 |

VECTORS AND GRAPHICS | 17 |

Applications of Vectors | 19 |

Parametric Curves | 20 |

Graphing Surfaces | 25 |

Parametric Surfaces | 27 |

VECTORS AND GRAPHICS | 29 |

Glossary of Some Useful Mathematics Objects Commands | 33 |

GEOMETRY OF CURVES | 35 |

Geometric Invariants | 38 |

The Frenet Frame | 39 |

Curvature and Torsion | 41 |

Differential Geometry of Curves | 43 |

The Osculating Circle | 44 |

Spherical Curves | 45 |

Helical Curves | 46 |

Congruence | 47 |

Two More Examples | 48 |

The Astroid | 49 |

The Cycloid | 50 |

CURVES | 53 |

Glossary of Some Useful Mathematica Objects Commands | 62 |

KINEMATICS | 65 |

Keplers Laws of Planetary Motion | 68 |

Studying Equations of Motion with Mathematica | 69 |

KINEMATICS | 73 |

Glossary of Some Useful Mathematica Objects Commands | 79 |

DIRECTIONAL DERIVATIVES | 81 |

ThreeDimensional Graphs | 82 |

Graphing Level Curves | 83 |

The Gradient of a Function of Two Variables | 86 |

Directional Derivatives | 89 |

Functions of Three or More Variables | 91 |

SURFACES | 123 |

Glossary of Some Useful Mathematica Objects Commands | 129 |

OPTIMIZATION IN SEVERAL VARIABLES | 133 |

Analytic Methods | 134 |

Newtons Method | 135 |

Functions of Two Variables | 137 |

Second Derivative Test | 138 |

Steepest Descent | 139 |

Multivariable Newtons Method | 141 |

Three or More Variables | 142 |

OPTIMIZATION | 145 |

Glossary of Some Useful Mathematica Objects | 150 |

MULTIPLE INTEGRALS | 153 |

Regions in the Plane | 154 |

Viewing Simple Regions | 156 |

Polar Regions | 158 |

Viewing Solid Regions | 161 |

A More Complicated Example | 165 |

Cylindrical Coordinates | 169 |

MULTIPLE INTEGRALS | 173 |

Glossary of Some Useful Mathematica Objects Commands | 182 |

PHYSICAL APPLICATIONS OF VECTOR CALCULUS | 185 |

Newtonian Gravitation | 188 |

Electricity and Magnetism | 192 |

PHYSICAL APPLICATIONS | 197 |

Glossary of Some Useful Mathematica Objects Commands | 207 |

Energy Minimization and Laplaces Equation | 208 |

MATHEMATICA TIPS | 211 |

Glossary | 225 |

Sample Notebook Solutions | 239 |

275 | |

### Other editions - View all

Multivariable Calculus and Mathematica(r) Kevin R. Coombes,Ronald Lipsman,Jonathan Rosenberg No preview available - 2014 |

### Common terms and phrases

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