Multivariable Calculus and Mathematica: With Applications to Geometry and Physics / Edition 1

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Overview

Aiming to "modernise" the course through the integration of Mathematica, this publication introduces students to its multivariable uses, instructs them on its use as a tool in simplifying calculations, and presents introductions to geometry, mathematical physics, and kinematics. The authors make it clear that Mathematica is not algorithms, but at the same time, they clearly see the ways in which Mathematica can make things cleaner, clearer and simpler. The sets of problems give students an opportunity to practice their newly learned skills, covering simple calculations, simple plots, a review of one-variable calculus using Mathematica for symbolic differentiation, integration and numerical integration, and also cover the practice of incorporating text and headings into a Mathematica notebook. The accompanying diskette contains both Mathematica 2.2 and 3.0 version notebooks, as well as sample examination problems for students, which can be used with any standard multivariable calculus textbook. It is assumed that students will also have access to an introductory primer for Mathematica.

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Product Details

  • ISBN-13: 9780387983608
  • Publisher: Springer New York
  • Publication date: 5/15/1998
  • Edition description: Softcover reprint of the original 1st ed. 1998
  • Edition number: 1
  • Pages: 296
  • Product dimensions: 9.61 (w) x 6.69 (h) x 0.63 (d)

Table of Contents

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.

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