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.
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