Mathematical Methods for Physics and Engineering : A Comprehensive Guide

VECTOR CALCULUS

∇(φ+ψ)=∇φ+∇ψ

∇·(a+b)=∇·a+∇·b

∇×(a+b)=∇×a+∇×b

∇(φψ)=φ∇ψ+ψ∇φ

∇(a·b)=a×(∇×b)+b×(∇×a)+(a·∇)b+(b·∇)a

∇·(φa)=φ∇·a+a·∇φ

∇·(a×b)=b·(∇×a)−a·(∇×b)

∇×(φa)=∇φ×a+φ∇×a

∇×(a×b)=a(∇·b)−b(∇·a)+(b·∇)a−(a·∇)b

Table 10.1 Vector operators acting on sums and products. The operator∇is

defined in (10.25);φandψare scalar fields,aandbare vector fields.

Therefore the curl of the velocity field is a vector equal to twice the angular

velocity vector of the rigid body about its axis of rotation. We give a full

geometrical discussion of the curl of a vector in the next chapter.

10.8 Vector operator formulae

In the same way as for ordinary vectors (chapter 7), for vector operators certain

identities exist. In addition, we must consider various relations involving the

action of vector operators on sums and products of scalar and vector fields. Some

of these relations have been mentioned earlier, but we list all the most important

ones here for convenience. The validity of these relations may be easily verified

by direct calculation (a quick method of deriving them using tensor notation is

given in chapter 26).

Although some of the following vector relations are expressed in Cartesian

coordinates, it may be proved that they are all independent of the choice of

coordinate system. This is to be expected since grad, div and curl all have clear

geometrical definitions, which are discussed more fully in the next chapter and

which do not rely on any particular choice of coordinate system.

10.8.1 Vector operators acting on sums and products

Letφandψbe scalar fields andaandbbe vector fields. Assuming these fields

are differentiable, the action of grad, div and curl on various sums and products

of them is presented in table 10.1.

These relations can be proved by direct calculation.