Numerical Methods for Engineers and Scientists 3rd Edition

Chapter 2

Mathematical Background

Core Topics

Concepts from calculus (2.2).

Vectors (2.3).

Matrices and linear algebra (2.4).

Ordinary differential equations (ODEs) (2.5).

Functions of two or more independent variables, partial differentiation (2.6)

Taylor series (2.7)

Inner product and orthogonality (2.8)

2.1 BACKGROUND

The present book focuses on the numerical methods used for calculating approximated solutions to problems that cannot be solved (or are difficult to solve) analytically. It is assumed that the reader has knowledge of calculus, linear algebra, and differential equations. This chapter has two objectives. One is to present, as a reference, some background useful for analysis of numerical methods. The second objective is to review some fundamental concepts and terms from calculus that are useful in the derivation of the numerical methods themselves. The chapter also defines many of the mathematical terms and the notation used in the rest of the book.

It should be emphasized that the topics that are presented in this section are meant as a review, and so are not covered thoroughly. This section is by no means a substitute for a text in calculus where the subjects are covered more rigorously. For each topic, the objective is to have a basic definition with a short explanation. This will serve as a reminder of concepts with which the reader is assumed to be familiar. If needed, the reader may use this information as the basis for seeking for a reference text on linear ...