ESCI 342 – Atmospheric Dynamics
Answers to Selected Exercises for Lesson 7
1
D
δ xδ yδ z ) = ∇ • V . Hint: You will need to convince
(
δ x ,δ y ,δ z →0 δ xδ yδ z Dt
D
D
D
yourself that
(δ x ) = δ u;
(δ y ) = δ v;
(δ z ) = δ w .
Dt
Dt
Dt
2. Show that
Answer:
lim
D (δ x )
D (δ y )
D (δ z )
D
+ δ xδ z
+ δ xδ y
(δ xδ yδ z ) = δ yδ z
Dt
Dt
Dt
Dt
but
D
D
(δ x ) = ( x2 − x1 ) = u2 − u1 = δ u
Dt
Dt
and similarly
D
(δ y ) = δ v;
Dt
D
(δ z ) = δ w
Dt
so we have
D
(δ xδ yδ z ) = δ yδ zδ u + δ xδ zδ v + δ xδ yδ w
Dt
and therefore
1
D
δu δv δ w
(δ xδ yδ z ) = + +
δ xδ yδ z Dt
δx δy δz
which in the limit as δx, δy, and δz approach zero becomes ∇ • V .