Page 160 - Vector Analysis

156 CHAPTER 5. Additional Topics

where F is a physical quantity in Eulerian variable, and u is the fluid velocity field. Let η
( )
be the flow map associated to u, and define f = F ˝ η ; that is, f (α, t) = F η(α, t), t , then

B f (α, t) = [] = DF ˝η.
Bt Ft + (u ¨ ∇)F ˝ η Dt

Therefore, the composition of the material derivative of a function and the flow map is the

time rate of change of the composition of that function and the flow map.’

5.3 The particle trajectory and streamlines

Not yet completed!!!

5.4 Exercises

In this set of exercise, the Einstein summation convention is used. (4.9)
Problem 1. Complete the following.

1. Let 䨨’s denote the Kronecker deltas. Prove (4.9); that is, show that
εijkεirs = δjrδks ´ δjsδkr .

2. Let O Ď R3 be an open domain, and u : O Ñ R3 be a smooth vector field. Denote
twice the anti-symmetric part of ∇u as Ω; that is, Ωij = ui,j ´uj,i. Show that

Ωkj = εijkωi , (5.10)

where ω = curlu is the vorticity of u.

3. Use (4.9) to show the following identities:

(a) u ˆ (v ˆ w) = (u ¨ w)v ´ (u ¨ v)w if u, v, w are three 3-vectors.
(b) curlcurlu = ´∆u + ∇divu if u : O Ñ R3 is smooth.
(c) u ˆ curlu = 1 ∇(|u|2) ´ (u ¨ ∇)u if u : O Ñ R3 is smooth.

2

4. Use (5.10) to show that
[]

curl (u ¨ ∇)u = (u ¨ ∇)ω ´ (ω ¨ ∇)u + (divu)ω

if u : O Ñ R3 is smooth.