Page 284 - Intro to Tensor Calculus

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I 31. Show that in an orthogonal coordinate system the components of ∇(∇· ~u) can be expressed in terms
of physical components by the relation

1 ∂ 1 ∂(h 2 h 3 u(1)) ∂(h 1 h 3 u(2)) ∂(h 1 h 2 u(3))
[∇ (∇· ~u)] = + +
i i 1 2 3
h i ∂x h 1 h 2 h 3 ∂x ∂x ∂x
2
I 32. Show that in orthogonal coordinates the components of ∇ ~u can be written
2 jk
∇ ~u = g u i,jk = A i
i
and in terms of physical components one can write

3 2 3 3
X 1 ∂ (h i u(i)) X m ∂(h m u(m)) X m ∂(h i u(i))
h i A(i)= 2  − 2 −
j
h ∂x ∂x j ij ∂x j jj ∂x m
j=1 j m=1 m=1

3 3 3 !
X ∂ m X m p X m p
− h m u(m) − − 
∂x j ij ip jj jp ij
m=1 p=1 p=1
2
I 33. Use the results in problem 32 to show in Cartesian coordinates the physical components of [∇ ~u] i = A i
can be represented
2
2
2
∂ u ∂ u ∂ u
2
∇ ~u · ˆ e 1 = A(1) = + +
∂x 2 ∂y 2 ∂z 2
2
2
2
∂ v ∂ v ∂ v
2
∇ ~u · ˆ e 2 = A(2) = + +
∂x 2 ∂y 2 ∂z 2
2
2
2
∂ w ∂ w ∂ w
2
∇ ~u · ˆ e 3 = A(3) = 2 + 2 + 2
∂x ∂y ∂z
where (u, v, w) are the components of the displacement vector ~u.
2
I 34. Use the results in problem 32 to show in cylindrical coordinates the physical components of [∇ ~u] i = A i
can be represented
1
2 2
2 ∂u θ
∇ ~u · ˆ e r = A(1) = ∇ u r − 2 u r − 2
r r ∂θ
1
2 2 2 ∂u r
∇ ~u · ˆ e θ = A(2) = ∇ u θ + − u θ
2
r ∂θ r 2
2 2

∇ ~u · ˆ e z = A(3) = ∇ u z
2
2
2
∂ α 1 ∂α 1 ∂ α ∂ α
2
where u r ,u θ ,u z are the physical components of ~u and ∇ α = + + +
2
∂r 2 r ∂r r ∂θ 2 ∂z 2
2
I 35. Use the results in problem 32 to show in spherical coordinates the physical components of [∇ ~u] i = A i
can be represented
2 2cot θ 2
2 2
2 ∂u θ ∂u φ
∇ ~u · ˆ e ρ = A(1) = ∇ u ρ − u ρ − − u θ −
2
2
ρ 2 ρ ∂θ ρ 2 ρ sin θ ∂φ
1
2 2 2 ∂u ρ 2cos θ ∂u θ
∇ ~u · ˆ e θ = A(2) = ∇ u θ + − u θ − 2
2
2
2
ρ ∂θ ρ sin θ ρ sin θ ∂φ
1 2
2 2
∂u ρ 2cosθ ∂u θ
∇ ~u · ˆ e φ = A(3) = ∇ u φ − 2 u φ + 2 + 2
2
2
ρ sin θ ρ sin θ ∂φ ρ sin θ ∂φ
where u ρ ,u θ ,u φ are the physical components of ~u and where
2
2
2
∂ α 2 ∂α 1 ∂ α cot θ ∂α 1 ∂ α
2
∇ α = + + + + 2
2
2
∂ρ 2 ρ ∂ρ ρ ∂θ 2 ρ 2 ∂θ ρ sin θ ∂φ 2