3.1 Introduction

Why use approximation:

Because it is needed. In fact everything you do in real life is an approximation. Real life is proably determined by some sort of quantum mechanics. But we do not know what it is. And we would definitely not know how to solve it even if we knew what it was.
To reduce effort.
To increase accuracy. You might be able to solve approximate equations more accurately than more accurate equations, producing a better result.
To get more insight in the problem. If you have computed say an incompressible flow of interest, the only real thing you can say about it is that indeed, it is free from singularities. You can wax about vortices,” “fingers, intestines, etcetera, that you seem to see in the flow, but that you can also see in the clouds in the sky. If you start computing the flow for various parameters, you may start getting somewhere to an inkling of insight. However, doing that is limited by what the computer can do. And the interpretation will always have ambiguity. However, if you start looking at limiting processes of your parameters, you are suddenly getting somewhere. You can now define meaningful boundary layers, vortex layers”, “shocks. etcetera that are not just arbitrary interpretations but have rigorous mathematical definitions.
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This chapter looks at simple approximations using Taylor series (which of course always includes linearization.)