Mathematical Physics Vol 1

4.6 Examples

163

If we now apply the mean value theorem to the previous integral, we obtain ( x + ∆ x , y , z ) Z ( x , y , z ) F 1 d x = ∆ xF 1 ( x + θ ∆ x , y , z ) , 0 < θ < 1 . (4.188) By substituting the relation (4.188) into (4.187), and letting ∆ x → 0, we obtain

∂φ ∂ x

= F 1 .

(4.189)

∂φ ∂ y

∂φ ∂ z

Analogously, we obtain

= F 2 and

= F 3 . Finally

∂φ ∂ y

∂φ ∂ z

∂φ ∂ x

k = ∇ φ .

F =

i +

j +

Note. If F is a force field, then in mechanics the integral R c F · d r represents the work of the force. Forces whose work does not depend on the trajectories, which pass through points P 1 and P 2 , are called conservative forces , and the corresponding field is called a conservative field . Exercise 100 Observe the vector field F . a) Let the integral R F · d r be independent of the path. Prove that in that case rot F (= ∇ × F )= 0 . b) Conversely, if ∇ × F = 0 , prove that F is a conservative field. Proof a) If F is a conservative field, then F = ∇ φ , according to (4.56) (see p. 93), and it follows that rot F = 0 (see (4.69), p. 98). b) Conversely, if ∇ × F = 0 , then i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z F 1 F 2 F 3 = 0 , that is ∂ F 3 ∂ y = ∂ F 2 ∂ z , ∂ F 1 ∂ z = ∂ F 3 ∂ x , ∂ F 2 ∂ x = ∂ F 1 ∂ y . (*) It follows from the first of these two equations that

∂ϕ ∂ z

∂ϕ ∂ y

where ϕ = ϕ ( x , y , z ) .

F 3 =

i F 2 =

,