# Advanced Mathematics for Engineers and Scientists/todo

The purpose of this page is to keep track of the major advances needed in this wikibooks project. It is not a part of the book itself, but is meant to help assist in its production. In this way the future plans of the project will be in an open access for collaboration among authors. This is not a discussion page, but should reflect the current interests.

## Outline of method

For each chapter we will include an exposition on what will be covered and what are the key applications of this material. The book will put heavy emphasis on worked examples from the literature, with only the most important ideas stated separately. Usually new ideas should be carefully highlighted in the examples. Usually we will go for around five major sub areas in each chapter illustrating solution of major problems in the area and some discussion of the generalization of this material. Each section should be Include reference area in each chapter and some (~10) homework problems at the end.

History is important in each of these disciplines, but will only be mentioned at some times to help with clarification of the material in a section and the applications.

## Outline of material

Recommended Parts: Core material (C), additional Recommended topics (R), Minor topic (M). May put together recommended TOCs for different disciplines, also a map of dependencies of material may be helpful. Linked contents for different disciplines may be helpful to have in the intro.

1. Core material
1. Introduction (R)
1. Goals
2. Approach
3. Various Disciplines
2. Vectors and Essential Linear Algebra (C)
1. Vectors
2. Matrices (from systems of equations perspective)
3. Vector Analysis (C) [DNP notes]
1. Div
3. Curl
4. Green's Theorem
2. Useful Concepts
1. Calculus of Complex Variables (R)
1. Complex variables
2. Complex equations
3. Complex differential equations
2. Fractional Calculus
1. Review of Derivatives and Integrals
2. Fractional Differentigrals
3. Fractional Differential Equations
3. Non-Euclidian Geometry
4. Differential Geometry
5. Variable Transforms [DNP notes]
3. Linear Theory
1. Applied Matrix Theory (R)
1. Gaussian Elimination
2. PLU Decomposition
3. Inner Products
4. Orthogonal Functions
5. Least Squares
6. QR decomposition
7. Singular Value Decomposition
8. Diagonalizable matrices
2. Statistics (C)
3. Ordinary Differential Equations (C) [DNP notes]
1. First Order
2. Second Order
3. Nth order
4. Series solutions
5. Euler-Cauchy
6. Bessel's Equation and Bessel functions
7. Legendre's Equation and Legendre polynomials
4. Fourier Series and Integral Transforms (C) [DNP notes]
5. Partial Differential Equations (C) [DNP notes]
1. Classifications
6. Green's Functions (R)
7. Calculus of Variations (R)
1. Euler Lagrange Equation
2. Side note: the Hamiltonian formulation
4. Nonlinear Theory
1. Asymptotics and Perturbation Theory (R) [DNP notes] [MR notes?]
1. Asymptotic Analysis
2. Perturbation Theory
3. Singular Perturbations
2. Chaos Theory and Bifurcation Theory (M)
1. Fractals
2. Characteristics
3. Bifurcations
4. Stability
3. Stochastic Differential Equations (M) [DNP trans notes]
4. Recent Methods in Nonlinear Differential Equations (R)
1. Adomian Decomposition Method
2. Homotopy Perturbation Method
3. Homotopy Analysis Method
4. Variational Iteration Method
1. Numerical Methods (R)
1. Coding
2. Finite Difference Methods
3. Finite Volume Methods
4. Finite Element Methods (Background on Variational Calculus)
5. Single-step Methods?
6. Multistep Methods?
7. Implicit Methods?
8. Monte-Carlo Methods
9. Comparison of methods applied to heat equation
2. Fundamentals of Topology (M)
1. Metric Space
2. Topological Space
3. Group Theory and Symmetry (M)
1. Finite Groups
2. Symmetry in Chemical compounds
4. Lie Groups (M)
1. Continuous symmetric groups
6. Appendices
1. Solutions to exercises
2. Variable transforms (Moon and Spencer)
3. References

## To Do

• from original PDE's material
1. Italic inline variables
2. TOC Template?
3. TOC box for pages?
4. Modify equations layout