Mathematical Physics Vol 1

Chapter 4. Field theory

136

Given that

i

j

k

∂ ∂ x V x

∂ ∂ y V y

∂ ∂ z V z

∇ × V =

=

= i

∂ z

+ j

∂ V z ∂ x

+ k

∂ y

∂ V y

∂ V y ∂ x −

∂ V z ∂ y −

∂ V x

∂ V x

∂ z −

,

and similarly for the vector field W , the relation (4.159), using (4.160) becomes ∇ · ( V × W )=( ∇ × V ) · W − V · ( ∇ × W ) , (4.161) which was to be proved. b) By replacing W with r in relation (4.161), we obtain ∇ · ( V × r )=( ∇ × V ) · r − V · ( ∇ × r ) . (4.162) Given that, according to the initial assumption ∇ × V = 0, and according to (4.62) ∇ × r = 0 (see p. 94), it follows that ∇ · ( V × r )= r · ( ∇ × V )= 0 . c) Based on the relation (4.161), we can conclude that the divergence of the vector product is equal to zero, because fields V and W are irrotational, i.e. their rotors are equal to zero. d) Let v be an irrotational field i.e. ∇ × v = 0. The task is to examine the vector field w , defined by w = r × v . Let us first compute the gradient of this field ∇ · w = ∇ · ( r × v ) . Using the result under a), the relation (4.62) on p. 94, as well as the initial assumption ( ∇ × v = 0 ) , we obtain ∇ · w = ∇ · ( r × v )=( ∇ × r ) · v − r · ( ∇ × v )= 0 . It can be proved that, in the general case, ∇ × w̸ = 0. For proving this, use a property of the nabla operator (see p. 88). Thus, the field w is a solenoidal (rotational) field. e) Observe the vector field M = r × S , where S is the force, and r the position vector. As, according to the initial assumption, the force is conservative, i.e. ∇ × S = 0, it follows that ∇ · M = ∇ · ( r × S )=( ∇ × r ) · S − r · ( ∇ × S )= 0 .

Exercise 65 Find rot ( r f ( r )) , if f ( r ) is a differentiable function.