Mathematical Physics Vol 1

Chapter 4. Field theory

164

We can write the second equation in the form ∂ F 1 ∂ z = ∂ ∂ x ∂ϕ ∂ z = ∂ ∂ z ∂ϕ ∂ x from where it follows that F 1 = ∂ϕ ∂ x . It further follows that

∂φ ∂ x

∂φ ∂ y

∂φ ∂ z

k = ∇ φ .

F = F 1 i + F 2 j + F 3 k =

i +

j +

Thus, the field F is conservative iff rot F = ∇ × F = 0.

Exercise 101 Show that the force F =( 2 xy + z 3 ) i + x 2 j + 3 xz 2 k is conservative and find the scalar potential of this field. Calculate the force necessary to move an object (material point) frompoint P ( 1 , − 2 , 1 ) to point Q ( 3 , 1 , 4 ) .

Solution

Given that

i

j

k

∂ ∂ x ∂ ∂ z 2 xy + z 3 x 2 3 xz 2 ∂ ∂ y

∇ × F =

= 0

it follows (see definition (4.47) on p. 92) that the force F is conservative, that is, that the vector field F is potential, and thus F = ∇ φ or ∇ φ =( 2 xy + z 3 ) i + x 2 j + 3 xz 2 k .

∂φ ∂ x ∂φ ∂ y ∂φ ∂ z

= 2 xy + z 3

(4.190)

= x 2

(4.191)

= 3 xz 2 .

(4.192)

By integrating (4.190) we obtain

φ = x 2 y + xz 3 + f ( y , z ) .

(4.193)

By differentiating the equation (4.193) by y , we obtain ∂φ ∂ y = x 2 + ∂ f ( y , z ) ∂ y . From equations (4.194) and (4.191) it follows that ∂ f ( y , z ) ∂ y = 0 .

(4.194)

(4.195)