Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.8 VECTOR OPERATOR FORMULAE

Show that

∇×(φa)=∇φ×a+φ∇×a.

Thex-component of the LHS is

∂

∂y

(φaz)−

∂

∂z

(φay)=φ

∂az

∂y

+

∂φ

∂y

az−φ

∂ay

∂z

−

∂φ

∂z

ay,

=φ

(

∂az

∂y

−

∂ay

∂z

)

+

(

∂φ

∂y

az−

∂φ

∂z

ay

)

,

=φ(∇×a)x+(∇φ×a)x,

where, for example, (∇φ×a)xdenotes thex-component of the vector∇φ×a. Incorporating
they-andz- components, which can be similarly found, we obtain the stated result.

Some useful special cases of the relations in table 10.1 are worth noting. Ifris

the position vector relative to some origin andr=|r|,then

∇φ(r)=

dφ

dr

ˆr,

∇·[φ(r)r]=3φ(r)+r

dφ(r)

dr

,

∇^2 φ(r)=

d^2 φ(r)

dr^2

+

2

r

dφ(r)

dr

,

∇×[φ(r)r]= 0.

These results may be proved straightforwardly using Cartesian coordinates but

far more simply using spherical polar coordinates, which are discussed in subsec-

tion 10.9.2. Particular cases of these results are

∇r=ˆr, ∇·r=3, ∇×r= 0 ,

together with

∇

(

1

r

)

=−

ˆr

r^2

,

∇·

(

ˆr

r^2

)

=−∇^2

(

1

r

)

=4πδ(r),

whereδ(r) is the Dirac delta function, discussed in chapter 13. The last equation is

important in the solution of certain partial differential equations and is discussed

further in chapter 20.

10.8.2 Combinations of grad, div and curl

We now consider the action of two vector operators in succession on a scalar or

vector field. We can immediately discard four of the nine obvious combinations of

grad, div and curl, since they clearly do not make sense. Ifφis a scalar field and