Mathematical Physics Vol 1

4.2 Vector field

93

Definition A vector field, in which for all its points the following is true rot v̸ = 0 , div v̸ = 0 ,

(4.50)

is called a complex field .

R Note that the study of a complex field can be reduced to one potential and one solenoidal field.

Let us express a complex field in the form

v = v 1 + v 2 ,

(4.51)

so that

rot v 1 = 0 , div v 1̸ = 0 , rot v 2̸ = 0 , div v 2 = 0 .

(4.52) (4.53)

Then, as

div v = div ( v 1 + v 2 )= div v 1 + div v 2 = div v 1̸ = 0 , rot v = rot ( v 1 + v 2 )= rot v 1 + rot v 2 = rot v 2̸ = 0 .

(4.54) (4.55)

the previous statement is proved.

4.2.4 Potential Assume that the vector function v can be represented as the gradient of a scalar position function ϕ ( r ) , i.e. v = grad ϕ . (4.56) The scalar function ϕ defined in this way is called the potential of the vector field v .

Theorem9 The field of the vector function v =grad ϕ is a potential field.

Proof As, according to the assumption

∂ϕ ∂ x

∂ϕ ∂ y

∂ϕ ∂ z

v = grad ϕ ⇒ v x =

v y =

v z =

(4.57)

,

,

,

it follows that

∂ 2 ϕ ∂ z ∂ y

∂ 2 ϕ ∂ z ∂ y

∂ v y ∂ z

∂ v y ∂ z

∂ v z ∂ y

∂ v z ∂ y −

= 0 .

(4.58)

, ⇒

=

,

=

Similarly, we obtain

∂ v y ∂ x −

∂ v z ∂ x

∂ v x ∂ y

∂ v x ∂ z −

= 0 i

= 0 .

(4.59)