# 12 Vector calculus

In Chapter 8 we introduced the idea of a vector as a quantity with both magnitude and direction and we discussed vector algebra, particularly as applied to analytical geometry, and the differentiation and integration of vectors with respect to a scalar parameter. In this chapter we extend our discussion to include directional derivatives and integration over variables that are themselves vectors. This topic is called *vector calculus* or *vector analysis*. It plays a central role in many areas of physics, including fluid mechanics, electromagnetism and potential theory.

## 12.1 Scalar and vector fields

If scalars and vectors can be defined as continuous functions of position throughout a region of space, they are referred to as *fields* and the region of space in which they are defined is called a *domain.* An example of a scalar field would be the distribution of temperature *T* within a fluid. At each point the temperature is represented by a scalar field *T*(**r)** whose value depends on the position **r** at which it is measured. A useful concept when discussing scalar fields is that of an *equipotential surface*, that is, a surface joining points of equal value. This is somewhat analogous to the contour lines on a two-dimensional map, which join points of equal height. An example of a vector field is the distribution of velocity **v**(**r**) in a fluid. At every point **r**, the velocity is represented by a vector of definite magnitude and direction, both of which can change continuously ...

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