Mathematical Physics Vol 1

Chapter 4. Field theory

184

Substituting the equations (4.210) and (4.211) in (4.209), we obtain y V C · ( ∇ × B ) d V = x S C · ( n × B ) d S . As C is aa arbitrary constant vector, it can be placed before the integral, and we finally obtain C · y V ∇ × B d V = C · x S n × B d S ⇒ y V ∇ × B d V = x S n × B d S .

Exercise 127 Let P be a point within a body of volume ∆ V , whose outer boundary is ∆ S . Prove that at point P the following is true a)

s ∆ S

φ n d S ∆ V

∇ φ = lim ∆ V → 0

,

b)

s ∆ S

n × A d S ∆ V

∇ × A = lim ∆ V → 0

,

Solution a) Given that (see Example 125, p. 183) y V by a scalar product with i we obtain y V

∇ φ d V = x ∆ S

φ n d S ,

∇ φ · i d V = x ∆ S

φ n · i d S .

Applying the mean value theorem yields

s ∆ S

φ n · i d S ∆ V

∇ φ · i =

,

where ∇ φ · i is the mean value of ∇ φ · i in the entire ∆ V . Taking the limit value, when ∆ V −→ 0, so that P remains within ∆ V , we obtain ∇ φ · i = lim ∆ V → 0 s ∆ S φ n · i d S ∆ V . (4.212)