Mathematical Physics Vol 1

4.2 Vector field

97

On the other hand

1 ε o

ρ ,

div E =

div − grad Φ −

∂ t

∂ A

1 ε o

ρ ,

=

∂ ∂ t

1 ε o

∆Φ +

ρ .

( div A )= −

∂ Φ ∂ t

Given that div A = − ε o µ o

it follows that

∂ 2 Φ ∂ t 2

1 ε o

∆Φ − ε o µ o

ρ .

= −

Thus, instead of four Maxwell partial differential equations that are coupled, in which E and B are unknown, we obtain four uncoupled equations, which are easier to solve, and in which the variables A and Φ are unknown.

Properties of divergence

a) div( c a )= c · div a , c=const. b) div( a + b )=div a +div b , c) div( u a )= u · div a + a · grad u , where u is a scalar function. Proof Here we will prove only property c), while leaving the proof of properties a) and b), as easier ones, to the reader for practice. Given that u a = ua x i + ua y j + ua z k =( ua x ) i +( ua y ) j +( ua z ) k ,

ir follows that

∂ ∂ x

∂ ∂ y

∂ ∂ z

div ( u a )=

( ua x )+

( ua y )+

( ua z )=

∂ a y ∂ y

∂ a x ∂ x

∂ u ∂ x ·

∂ u ∂ y ·

∂ a z ∂ z

∂ u ∂ z ·

= u · = u

a x + u ·

a y + u ·

a z =

+

+

+

∂ z

∂ a y ∂ y

∂ a x ∂ x

∂ a z

∂ u ∂ x ·

∂ u ∂ y ·

∂ u ∂ z ·

a x +

a y +

a z =

+

+

+

= u · div a + a · grad u ,

by which the property c) is proven.

Some properties of rotor a) rot c =0,if c = −−−→ const. b) rot ( c a ) = c rot a , c = const., c) rot( a + b )=rot a +rot b , d) rot( u a )= u rot a + a × grad u , where u is a scalar function.