Mathematical Physics Vol 1

Chapter 4. Field theory

98

4.2.6 A brief overview of introduced concepts

vector operations of the I type vector operations of the II type

div ( grad u ) rot ( grad u )

u − scalar function →

grad u − vector →

grad ( div a )

a − vector function →    Higher order operations Let u = u ( x , y , z ) be a scalar field, then

div a − scalar →

div ( rot a ) rot ( rot a )

rot a − vector →

∂ u ∂ x

∂ u ∂ y

∂ u ∂ z

k = u ′ x i + u ′ y j + u ′ z k .

i +

j +

grad u =

Let us now determine vector values of II order a)

divgrad u =

∂ ∂ x ∂ 2 u ∂ x 2

∂ u ∂ x

∂ ∂ y ∂ 2 u ∂ z 2

∂ u ∂ y

∂ ∂ z

∂ u ∂ z

(***)

=

+

+

=

∂ 2 u ∂ y 2

= ∇ u − is a scalar.

=

+

+

b)

i

j k

∂ ∂ x ∂ u ∂ x

∂ ∂ y ∂ u ∂ y

∂ ∂ z ∂ u ∂ z

rot ( grad u )=

=

=

∂ 2 u ∂ z ∂ y

i −

∂ 2 u ∂ z ∂ x

j +

∂ 2 u ∂ x ∂ y

∂ 2 u ∂ y ∂ z −

∂ 2 u ∂ x ∂ z −

∂ 2 u ∂ y ∂ x −

k .

As, for continuous functions

∂ 2 u ∂ y ∂ z ∂ 2 u ∂ x ∂ z ∂ 2 u ∂ y ∂ x

∂ 2 u ∂ z ∂ y ∂ 2 u ∂ z ∂ x ∂ 2 u ∂ x ∂ y

=

,

=

,

=

,

we finally obtain

rot ( grad u ) ≡ 0 .

(4.69)