Mathematical Physics Vol 1

4.2 Vector field

99

c)

grad ( div v )=

∂ ∂ x ∂ ∂ z

∂ v z ∂ z ∂ v z ∂ z ∂ 2 v z ∂ x ∂ z z ∂ z 2 ∂ 2 v

∂ ∂ y

∂ v z ∂ z

∂ v y ∂ y

∂ v y ∂ y

∂ v x ∂ x

∂ v x ∂ x

i +

j +

=

+

+

+

+

∂ v y ∂ y

∂ v x ∂ x

k =

+

+

+

= +

i +

∂ 2 v z ∂ y ∂ z

∂ 2 v y ∂ x ∂ y y ∂ z ∂ y ∂ 2 v

∂ 2 v

∂ 2 v

∂ 2 v x ∂ y ∂ x

y

x

j +

+

+

+

+

∂ x 2

∂ y 2

∂ 2 v x ∂ z ∂ x

k .

+

+

d)

div ( rot v )= ?

Let

v = v x i + v y j + v z k , from where, according to definition (4.46) on page 92, for rot v we obtain rot v = ∂ v z ∂ y − ∂ v y ∂ z i + ∂ v x ∂ z − ∂ v z ∂ x j + ∂ v y ∂ x − ∂ v x ∂ y k ⇒ ⇒ rot v = a 1 i + a 2 j + a 3 k ≡ a ,

(4.70)

where we have introduced the following notation ∂ v z ∂ y − ∂ v y ∂ z ≡ a 1 , ∂ v x ∂ z − ∂ v z ∂ x ≡

a 2 ,

∂ v x ∂ y ≡

∂ v y ∂ x −

a 3 .

Further, as

∂ a 1 ∂ x

∂ a 2 ∂ y

∂ a 3 ∂ z

div a =

+

+

,

we finally obtain

div ( rot v )=

(4.71)

= ≡ 0 .

∂ 2 v y ∂ x ∂ z

+

∂ 2 v z ∂ x ∂ y

+

∂ 2 v x ∂ y ∂ z

∂ 2 v y ∂ x ∂ z −

∂ 2 v x ∂ y ∂ z −

∂ 2 v z ∂ x ∂ y −

4.2.7 Spatial derivation The procedure of generalizing the derivative in a direction is called spatial derivation , and the result of this procedure is called spatial derivative . Observe a function ϕ ( r ) , which can be a scalar or vector position function.

Let us note in the field of this function a point A and a region V (part of the space) bordered by a closed oriented surface S , such that A ∈ V . Let mes V = V (4.72) be the measure of the volume of this region, and d S the vector surface element on the closed oriented surface S .

Figure 4.10