Mathematical Physics Vol 1

Chapter 4. Field theory

140

Thus the equations (4.164) and (4.165) can be represented in the form

∂ 2 u ∂ t 2

∇ 2 u = a 2

,

which was to be proved 19 .

Exercise 69 Let the vector field F be described by the relation F = −

y i + x j

. Compute

x 2 + y 2

a) ∇ × F . b)

I F · d r along an arbitrary closed path. Explain the results, if F represents the force.

Solution a) Let us start with the definition of rotor

i

j

k

∂ ∂ x

∂ ∂ y

∂ ∂ z

∇ × F =

,

− y x 2 + y 2

x x 2 + y 2

0

from where we obtain rot F = 0 in any area except in point (0,0). Thus, the vector field F is laminar. If F represents the force, then the field of the force is potential, and the force is conservative. b) Observe the integral along a closed line I F · d r = I − y d x + x d y x 2 + y 2 . Due to the shape of the line, it is convenient to switch to the polar coordinate system, which is connected to the Cartesian coordinate system by relations

x = ρ cos ϕ , y = ρ sin ϕ , where ( ρ , ϕ ) are polar coordinates. By differentiating we obtain d x = − ρ sin ϕ d ϕ + d ρ cos ϕ d y = ρ cos ϕ d ϕ + d ρ sin ϕ . The relevant value, expressed in the two coordinate systems, is now − y d x + x d y x 2 + y 2 = d ϕ = d arctan y x . 19 Note that these equations are known as Maxwell’s equations for the electromagnetic field.