It is Chapter 2 that points out the triple product A T CA in the equations of equilibrium, and Chapter 3 finds that framework repeated by differential equations. Applications and examples are given throughout. Where the theory strengthens the understanding, it is provided-but this is not a book about proofs. After the equations are formulated they need to be solved. Chapter 4 uses Fourier methods and complex variables, Chapter 5 uses numerical methods.
Orthogonality is fundamental, as it is for analytical methods.
The book is a text for applied mathematics and advanced calculus and engineering mathematics. It aims to explain what is essential, as far as possible within one book and one year. It solves the piecewise linear equations of mathematical programming through a series of linear equations, whose coefficient matrices are again A T CA . Finite elements and recursive filters and Laplace's equation fit this same pattern. In my experience the framework is seen and understood by the reader, and appreciated.