Multivariable Mathematics with Maple

1 Multivariable Mathematics with MapleLinear algebra , vector Calculusand Di erential Equationsby James A. Carlson and Jennifer M. Johnsonc 1996 Prentice-Hall Introduction:::::::::::::::::::::::::::: ::::::::::::::::::::::::::11. Introduction to Maple ::::::::::::::::::::::::::::::::::: ::::::::::31. A Quick Tour of the Basics:::::::::::::::::::::::::::::::::: ::42. algebra ::::::::::::::::::::::::::::::::: ::::::::::::::::::::::63. Graphing:::::::::::::::::::::::::::::::: :::::::::::::::::::::94. Solving Equations::::::::::::::::::::::::::::::: :::::::::::::125. Functions::::::::::::::::::::::::::::::: :::::::::::::::::::::156. calculus :::::::::::::::::::::::::::::::: :::::::::::::::::::::187.

2 vector and Matrix Operations:::::::::::::::::::::::::::::: :248. Programming in Maple ::::::::::::::::::::::::::::::::::: :::279. Troubleshooting::::::::::::::::::::::::: ::::::::::::::::::::352. Lines and Planes:::::::::::::::::::::::::::::::::: :::::::::::::::361. Lines in the Plane::::::::::::::::::::::::::::::::::: ::::::::362. Lines in 3-space::::::::::::::::::::::::::::::::: :::::::::::::393. Planes in 3-space::::::::::::::::::::::::::::::::: :::::::::::414. More about Planes:::::::::::::::::::::::::::::::::: :::::::::433. Applications of Linear Systems::::::::::::::::::::::::::::::::: ::491. Networks:::::::::::::::::::::::::::::::: ::::::::::::::::::::492. Temperature at Equilibrium::::::::::::::::::::::::::::: ::::523.

3 Curve-Fitting | Polynomial Interpolation::::::::::::::::::::584. Linear Versus Polynomial Interpolation:::::::::::::::::::::::615. Cubic Splines::::::::::::::::::::::::::::::::: :::::::::::::::644. Bases and Coordinates::::::::::::::::::::::::::::: ::::::::::::::671. Coordinates in the Plane::::::::::::::::::::::::::::::::::: ::672. Higher Dimensions:::::::::::::::::::::::::::::: :::::::::::::713. The vector Space of Piecewise Linear Functions::::::::::::::744. Periodic PL Functions::::::::::::::::::::::::::::::: ::::::::775. Temperature at Equilibrium Revisited::::::::::::::::::::::::825. A ne Transformations in the Plane::::::::::::::::::::::::::::::861. Transforming a Square:::::::::::::::::::::::::::::::::: :::::872.

4 Transforming Parallelograms:::::::::::::::::::::::::: :::::::893. Area:::::::::::::::::::::::::::::::::::: :::::::::::::::::::::914. Iterated Mappings | Making Movies with Maple :::::::::::::935. Stretches, Rotations, and Shears:::::::::::::::::::::::::::::956. Appendix: Maple Code foriterandfilm::::::::::::::::::::99i 6. Eigenvalues and Eigenvectors:::::::::::::::::::::::::::: ::::::::1011. Diagonal matrices:::::::::::::::::::::::::::::::: ::::::::::1012. Nondiagonal Matrices:::::::::::::::::::::::::::::::: :::::::1023. Algebraic Methods::::::::::::::::::::::::::::::::: :::::::::1044. Diagonalization::::::::::::::::::::::::: ::::::::::::::::::::1095. Ellipses and Their Equations::::::::::::::::::::::::::::::: :1136.

5 Numerical Methods::::::::::::::::::::::::::::::::: ::::::::1187. Least Squares | Fitting a Curve to Data::::::::::::::::::::::::1241. A Formula for the Line of Best Fit::::::::::::::::::::::::::1252. Solving Inconsistent Equations::::::::::::::::::::::::::::::1 323. The Stats Package::::::::::::::::::::::::::::::::: :::::::::1348. Fourier Series:::::::::::::::::::::::::::::::::: :::::::::::::::::1371. Periodic Functions::::::::::::::::::::::::::::::: :::::::::::1372. Computing Fourier Coe cients:::::::::::::::::::::::::::::1433. Energy:::::::::::::::::::::::::::::::::: :::::::::::::::::::1474. Filtering::::::::::::::::::::::::::::::: :::::::::::::::::::::1495. Approximations:::::::::::::::::::::::::: :::::::::::::::::::1506.

6 Appendix: Almost Periodic Functions:::::::::::::::::::::::1519. Curves and Surfaces:::::::::::::::::::::::::::::::: :::::::::::::1561. Curves in the Plane | Maps fromRtoR2::::::::::::::::::1562. Curves inR3:::::::::::::::::::::::::::::::::::: :::::::::::1603. Surfaces:::::::::::::::::::::::::::::::: ::::::::::::::::::::1604. Parametrizing Surfaces of Revolution::::::::::::::::::::::::16210. Limits, Continuity, and Di erentiability::::::::::::::::::::::::168 1. Limits | Functions fromRtoR::::::::::::::::::::::::::::1682 . Limits | Functions fromR2toR:::::::::::::::::::::::::::1713 . Continuity:::::::::::::::::::::::::::::: ::::::::::::::::::::1724. Tangent Planes:::::::::::::::::::::::::::::::::: :::::::::::1745.

7 Di erentiability::::::::::::::::::::::::::: ::::::::::::::::::17611. Optimizing Functions of Several Variables::::::::::::::::::::::1811. Review of the One-Variable Case::::::::::::::::::::::::::::1812. Critical Points and the Gradient::::::::::::::::::::::::::::1843 . Finding the Critical Points:::::::::::::::::::::::::::::::::: 1844. Quadratic Functions and their Perturbations::::::::::::::::1865. Taylor's Theorem in Two Variables:::::::::::::::::::::::::190ii 6. Completing the Square:::::::::::::::::::::::::::::::::: ::::1937. Constrained Extrema::::::::::::::::::::::::::::::::: ::::::19512. Transformations and their Jacobians:::::::::::::::::::::::::::2011 . Transforming the Coordinate Grid::::::::::::::::::::::::::2022.

8 Area of Transformed Regions:::::::::::::::::::::::::::::::20 53. Di erentiable Transformations::::::::::::::::::::::::: :::::2074. Polar Coordinates::::::::::::::::::::::::::::: :::::::::::::2105. The Area Integral:::::::::::::::::::::::::::::::: :::::::::::2126. The Change-of-Variables Theorem::::::::::::::::::::::::::2147. Appendix: A ne Approximations:::::::::::::::::::::::::: :2168. Appendix: Gridtransform::::::::::::::::::::::::::: ::::::::21713. Solving Equations Numerically::::::::::::::::::::::::::::: ::::2191. Historical Background:::::::::::::::::::::::::::::: ::::::::2192. The Bisection Method:::::::::::::::::::::::::::::::::: ::::2203. Newton's Method for Functions of One Variable:::::::::::::2224.

9 Newton's Method for Solving Systems:::::::::::::::::::::::2245. A Bisection Method for Systems of Equations:::::::::::::::2286. Winding Numbers::::::::::::::::::::::::::::::::: :::::::::22914. First-order Di erential Equations::::::::::::::::::::::::::::::: 2351. Analytic Solutions::::::::::::::::::::::::::::::: :::::::::::2352. Line Fields:::::::::::::::::::::::::::::::::: :::::::::::::::2393. Drawing Line Fields and Solutions with Maple ::::::::::::::24315. Second-order Equations::::::::::::::::::::::::::::::: :::::::::2461. The Physical Basis::::::::::::::::::::::::::::::::::: :::::::2472. Free Oscillations:::::::::::::::::::::::::::: ::::::::::::::::2473. Damped Oscillations:::::::::::::::::::::::::::: ::::::::::::2514.

10 Overdamping::::::::::::::::::::::::::::: ::::::::::::::::::2535. Critical Damping::::::::::::::::::::::::::::::::: ::::::::::2546. Forced Oscillations:::::::::::::::::::::::::::: ::::::::::::::2557. Resonance::::::::::::::::::::::::::::::: :::::::::::::::::::25816. Numerical Methods for Di erential Equations::::::::::::::::::2611. Estimatingewith Euler's Method:::::::::::::::::::::::::::2612. Euler's Method for General First-order Equations:::::::::::2653. Improvements to Euler's Method::::::::::::::::::::::::::::2684. Systems of Equations::::::::::::::::::::::::::::::: ::::::::270iii 17. Systems of Linear Di erential Equations::::::::::::::::::::::::2761. Normal Coordinates::::::::::::::::::::::::::::: :::::::::::2772.