Mathematical Physics Vol 1

Chapter 4. Field theory

94

On basis of these relations we obtain rot v = 0, given that rot v = = ∂ v z ∂ y − ∂ v y ∂ z i + ∂ v x ∂ z − ∂ v z ∂ x j + ∂ v y ∂ x − ∂ v x ∂ y k = = 0 .

(4.60)

As, in general, the following is true

div v =

∂ 2 ϕ ∂ x 2

∂ 2 ϕ ∂ y 2

∂ 2 ϕ ∂ z 2

∂ v y ∂ y

∂ v x ∂ x

∂ v z ∂ z

(4.61)̸

=

+

+

=

+

+

= 0

the theorem is proven.

4.2.5 Examples of potential

Potential of a position vector Observe the vector field v = r = x i + y j + z k . As ∂ v x ∂ x = ∂ v y ∂ y = ∂ v z ∂ z

= 1 it follows that div v = 3̸ = 0.

It further follows that rot v =

∂ z

i +

∂ v z ∂ x

j +

∂ v x ∂ y

∂ v y

∂ v y ∂ x −

∂ v z ∂ y −

∂ v x ∂ z −

k = 0 ,

(4.62)

so we can conclude that the vector field v = r is potential. Potential forces If there exists a scalar function U such that the force S can be represented in the form S = grad U , (4.63) then it is said that the force is conservative and that there exists a potential of the force U . Observe, for example, the gravitational force S = − γ m · m 0 R 2 r , r = R | R | , (4.64) where m and m 0 are masses that are mutually attracted, γ is the gravitational constant, R the position vector of one material point with respect to another, and R the magnitude of the position vector. The potential of this force is given by the expression (see (4.98)) U = γ m 0 R . (4.65) Stationary electrostatic field In electrodynamics, the problem of determining the strength of electric and magnetic fields can be reduced to determining their potential. Let’s start with Maxwell’s 9 equations for electromagnetic field in vacuum: 9 Maxwell James Clark (1831-1879), British physicist. He researched in many areas of physics, and his most significant works are related to electromagnetic phenomena. He formulated four equations laying out the principle