Question

Irrotational solenoidal and harmonic function for stess strain

Answer 1

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Suppose a vector field A⃗  is irrotational,

∇×A⃗ =0,(1)

and also solenoidal,

∇⋅A⃗ =0;(2)

(1) implies the existence of a function ϕ such that

A⃗ =∇ϕ;(3)

then by (2),

∇2ϕ=∇⋅∇ϕ=∇⋅A⃗ =0;(4)

ϕ is thus a harmonic function; we thus see that a field satisfying (1) and (2) is the gradient of some harmonic ϕ; conversely, if (3) binds for some harmonic ϕ, then

∇⋅A⃗ =∇⋅∇ϕ=∇2ϕ=0,(5)

and

∇×A⃗ =∇×∇ϕ=0,(6)

identically.

We thus see that the class of irrotational, solenoidal vector fields conicides, locally at least, with the class of gradients of harmonic functions.

Such fields are prevalent in electrostatics, in which the Maxwell equation

∇×E⃗ =−∂B⃗ ∂t(7)

becomes

∇×E⃗ =0(8)

in the event that the magnetic field B⃗  is constant in time; the in the absence of charges, e.g., in free space, the equation

∇⋅E⃗ =ρϵ0(9)

yields

∇⋅E⃗ =0;(10)

it thus follows that there exists a (voltage) potential ϕ with

E⃗ =∇ϕ(11)