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”.
Author(s): Peter Saveliev
Publisher: Independently published
Year: 2020
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
