Page 179 - Intro to Tensor Calculus

174

Laplacian

2
The Laplacian ∇ U has the contravariant form

∂U
2
ij
ij
ij
∇ U = g U ,ij =(g U ,i ) ,j = g . (2.1.9)
∂x i
,j
Expanding this expression produces the equations:
∂ ∂U ∂U j
2 ij im
∇ U = g + g
∂x j ∂x i ∂x i mj
√

∂ ij ∂U 1 ∂ g ij ∂U
2
∇ U = g + √ g
∂x j ∂x i g ∂x j ∂x i
(2.1.10)

√

1 √ ∂ ∂U ∂U ∂ g
2 ij ij
∇ U = √ g j g i + g i j
g ∂x ∂x ∂x ∂x
1 ∂ √ ∂U
2 ij
∇ U = √ gg .
g ∂x j ∂x i
In orthogonal coordinates we have g ij =0 for i 6= j and
2
2
g 11 = h , g 22 = h , g 33 = h 2 3
1
2
and so (2.1.10) when expanded reduces to the form
1 ∂ h 2 h 3 ∂U ∂ h 1 h 3 ∂U ∂ h 1 h 2 ∂U
2
∇ U = + + . (2.1.11)
h 1 h 2 h 3 ∂x 1 h 1 ∂x 1 ∂x 2 h 2 ∂x 2 ∂x 3 h 3 ∂x 3
This representation is only valid in an orthogonal system of coordinates.
EXAMPLE 2.1-3. (Laplacian) Find the Laplacian in spherical coordinates.
Solution: Utilizing the results given in the previous example we find the Laplacian in spherical coordinates
has the form
1 ∂ 2 ∂U ∂ ∂U ∂ 1 ∂U
2
∇ U = ρ sin θ + sin θ + . (2.1.12)
2
ρ sin θ ∂ρ ∂ρ ∂θ ∂θ ∂φ sin θ ∂φ
This simplifies to
2
2
2
∂ U 2 ∂U 1 ∂ U cot θ ∂U 1 ∂ U
2
∇ U = + + + + 2 . (2.1.13)
2
2
∂ρ 2 ρ ∂ρ ρ ∂θ 2 ρ 2 ∂θ ρ sin θ ∂φ 2
The table 1 gives the vector and tensor representation for various quantities of interest.