From Calculus to Chaos: An Introduction to DynamicsOxford University Press, 1997 - 269 pages What is calculus really for? This book is a highly readable introduction to applications of calculus, from Newton's time to the present day. These often involve questions of dynamics, i.e., of how--and why--things change with time. Problems of this kind lie at the heart of much of applied mathematics, physics, and engineering. From Calculus to Chaos takes a fresh approach to the subject as a whole, by moving from first steps to the frontiers, and by focusing on the many important and interesting ideas which can get lost amid a snowstorm of detail in conventional texts. The book is aimed at a wide readership, and assumes only some knowledge of elementary calculus. There are exercises (with full solutions) and simple but powerful computer programs which are suitable even for readers with no previous computing experience. David Acheson's book will inspire new students by providing a foretaste of more advanced mathematics and some of its liveliest applications. |
Contents
Introduction | 1 |
Elementary oscillations | 5 |
A brief review of calculus | 10 |
Ordinary differential equations 22 | 22 |
Computer solution methods 377 | 37 |
Exercises | 105 |
Exercises | 113 |
Fluid flow | 120 |
Instability and catastrophe | 133 |
Common terms and phrases
amplitude approximation autonomous boundary layer c₁ c₂ calculus chain rule chaos chaotic Chapter constant damping DEFDBL denotes dimensionless direction field double pendulum dx dt dx/dt dynamical ellipse equations of motion equilibrium point Euler method example exp(x flow fluid fnf(x frequency function GOSUB gradually improved Euler method initial conditions initial values INPUT integration inverse-square law inverted limit cycle linear equation linear stability theory LOOP m₁ m₂ mass mathematical nonlinear obtain orbit ordinary differential equations oscillation parameter partial differential equations particle particular phase path phase plane phase space pivot positive PRINT problem PSET QBasic result Reynolds number right-hand side Runge-Kutta method SCREEN 9 Section Setting up graphics shown in Fig shows solution curves solve spring stable step-by-step method Subroutines Taylor series unstable variable vertical vibration viscous x=xo x₁ ду