Calculus Illustrated. Volume 5: Differential Equations

Description

his is the fifth volume of the series Calculus Illustrated, a textbook for undergraduate students. Mathematical thinking is often visual. The exposition in this book is driven by its 600 color illustrations. Another unique feature of this book is its study of incremental phenomena well in advance of their continuous counterparts. It is called “Discrete Calculus”. Preface Ordinary differential equations Incremental motion Discrete models: how to set up ODEs Discrete forms Differential forms Solution sets of ODEs Separation of variables in ODEs The method of integrating factors Change of variables in ODEs Euler's method: back to the discrete How large is the difference between the discrete and the continuous? Qualitative analysis of ODEs Linearization of ODEs Motion under forces: the acceleration Discrete models: how to set up ODEs of second order Discrete forms, continued Vector variables Where matrices come from Transformations of the plane Linear operators Examining and building linear operators The determinant of a matrix It's a stretch: eigenvalues and eigenvectors The significance of eigenvectors Bases Classification of linear operators according to their eigenvalues Vector and complex variables Algebra of linear operators and matrices Compositions of linear operators How complex numbers emerge Classification of quadratic polynomials The complex plane C is the Euclidean space R2 Multiplication of complex numbers: C isn't just R2 Complex functions Complex linear operators Linear operators with complex eigenvalues Complex calculus Series and power series Solving ODEs with power series Systems of ODEs Parametric curves The predator-prey model Qualitative analysis of the predator-prey model Solving the Lotka–Volterra equations Vector fields and systems of ODEs Discrete systems of ODEs Qualitative analysis of systems of ODEs The vector notation and linear systems Classification of linear systems Classification of linear systems, continued Applications of ODEs Vector-valued forms The pursuit curves ODEs of second order as systems Vector ODEs of second order: a double spring A pendulum Planetary motion The two- and three-body problems A cannon is fired... Boundary value problems Partial differential equations Heat transfer between adjacent objects Heat transfer depends on permeability Heat transfer is caused by temperature differences Heat transfer depends on the geometry The heat PDE Cells and forms in higher dimensions Heat transfer in dimension 2: a plate The heat PDE for dimension 2 Wave propagation in dimension 1: springs and strings The wave PDE Wave propagation in dimension 2: a membrane Exercises Exercises: Basics Exercises: Analytical methods Exercises: Euler's method Exercises: Generalities Exercises: Models and setting up ODEs Exercises: Qualitative analysis Exercises: Systems Exercises: Second order Exercises: Advanced Exercises: PDEs Exercises: Computing Index