Multivariable Calculus and Mathematica®: With Applications to Geometry and Physics

ISBN-13: 9780387983608 / Angielski / Miękka / 1998 / 283 str.

Kevin R. Coombes;Ronald L. Lipsman;Jonathan M. Rosenberg

Opis

One 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.

Spis treści

Benefits of Mathematical Software.- What’s in This Book.- Descriptions.- What’s Not in This Book.- Required Mathematica Background.- How to Use This Book.- A Word About Versions of Mathematica.- Problem Set A: Review of One-Variable Calculus.- Vectors and Graphics.- Vectors.- Applications of Vectors.- Parametric Curves.- Graphing Surfaces.- Parametric Surfaces.- Problem Set B: Vectors and Graphics.- Geometry of Curves.- Parametric Curves.- Geometric Invariants.- Arclength.- The Frenet Frame.- Curvature and Torsion.- Differential Geometry of Curves.- The Osculating Circle.- Plane Curves.- Spherical Curves.- Helical Curves.- Congruence.- Two More Examples.- The Astroid.- The Cycloid.- Problem Set C: Curves.- Kinematics.- Newton’s Laws of Motion.- Kepler’s Laws of Planetary Motion.- Studying Equations of Motion with Mathematica.- Problem Set D: Kinematics.- Directional Derivatives.- Visualizing Functions of Two Variables.- Three-Dimensional Graphs.- Graphing Level Curves.- The Gradient of a Function of Two Variables.- Partial Derivatives and the Gradient.- Directional Derivatives.- Functions of Three or More Variables.- Problem Set E: Directional Derivatives and the Gradient.- Geometry of Surfaces.- The Concept of a Surface.- Basic Examples.- The Implicit Function Theorem.- Geometric Invariants.- Curvature Calculations with Mathematica.- Problem Set F: Surfaces.- Optimization in Several Variables.- The One-Variable Case.- Analytic Methods.- Numerical Methods.- Newton’s Method.- Functions of Two Variables.- Second Derivative Test.- Steepest Descent.- Multivariable Newton’s Method.- Three or More Variables.- Problem Set G: Optimization.- Multiple Integrals.- Automation and Integration.- Regions in the Plane.- Viewing Simple Regions.- Polar Regions.- Viewing Solid Regions.- A More Complicated Example.- Cylindrical Coordinates.- More General Changes of Coordinates.- Problem Set H: Multiple Integrals.- Physical Applications of Vector Calculus.- Motion in a Central Force Field.- Newtonian Gravitation.- Electricity and Magnetism.- Fluid Flow.- Problem Set I: Physical Applications.- Appendix: Energy Minimization and Laplace’s Equation.- Mathematica Tips.- Sample Notebook Solutions.