Page 159 - Vector Analysis

§5.2 Eulerian and Lagrangian Coordinates 155

Proof. Let J = det(∇η), and A = (∇η)´1. By (1.2) and (1.3),

n JAij B ηti n JAij B (ui ˝ η) n JAji ( B ui ) B ηk
B αj B αj B xk
Jt = ÿ = ÿ = ÿ ˝η .
B αj
i,j=1 i,j=1 i,j,k=1

n

Since A = (∇η)´1, ř Aijη,kj = δik; thus

j=1

Jt = J(divu) ˝ η. (5.8)

The theorem is then concluded by the fact that J|t=0 = 1 since η is the identity map at

t = 0. ˝

Corollary 5.3. Let u(¨, t) : Ω(t) Ñ Rn be a smooth divergence-free vector field, and η be the
corresponding flow map (which is assumed to exist up to time T as well). If U Ď Ω ” Ω(0)
is a smooth domain and

U (t) = ! P Rn ˇ = η(α, t) for some α P )
x ˇx U;
ˇ

that is, U(t) is the image of U under the map η at time t, then

the volume of U = the volume of U(t) @ t P (0, T ) .

Proof. Let |O| denote the Lebesgue measure of set O. Then

żż ż

|U(t)| = dx = det(∇η)(α)dα = dα = |U|. ˝

U (t) U U

Remark 5.4. If the fluid velocity is divergence-free, then the corollary above says that the

volume of a region carried by the fluid is constant in time. For this reason we sometimes

also called solenoidal vector fields incompressible.

5.2.1 The material derivative

In continuum mechanics, the material derivative describes the time rate of change of some

physical quantity (like heat or momentum) for a material element subjected to a space-and-

time-dependent velocity field. To be more precise, the material derivative, sometimes called
the substantial derivative, denoted by D , is defined by

Dt

DF BF n ui B F = Ft + (u ¨ ∇)F, (5.9)
Dt Bt
ÿ

= + i=1 B xi