Multivariable calculus and Mathematica : with applications to geometry and physics
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.
xiii, 283 pages : illustrations ; 24 cm + 1 computer disc (3 1/2 in.)
9780387983608, 0387983600
37903577
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.
Includes index