Direct Energy, 2018a

13 THOMAS FERMI ANALYSIS 293
13.2 Preliminary Ideas
13.2.1 Derivatives and Integrals of Vectors in Spherical Coordinates
The derivation of the Thomas Fermi equation involves derivatives of vectors
in spherical coordinates. For more details on derivatives and vectors see
[11, ch. 1]. Consider a scalar function described in spherical coordinates,
The gradient
of V (r, θ, φ) is dened
V = V ( −→ r )=V (r, θ, φ). (13.8)
−→ ∂V ∇V =
∂r âr + 1 ∂V
r ∂θ âθ + 1 ∂V
r sin θ ∂φ âφ. (13.9)
Gradient was introduced in Section 1.6.1. It returns a vector which points
in the direction of largest change in the function. The Laplacian is dened
in spherical coordinates as
∇ 2 V = 1 (
∂
r 2 ∂V )
r 2 ∂r ∂r
+ 1
r 2 sin θ
(
∂
sin θ ∂V )
+
∂θ ∂θ
1 ∂ 2 V
r 2 sin 2 θ ∂φ . (13.10)
2
Qualitatively, the Laplacian of a scalar is the second derivative with respect
to spatial position. In the derivations of this chapter, we encounter
only functions which are uniform with respect to θ and φ. For functions
of the form V = V (r), the formulas for gradient and Laplacian simplify
signicantly.
−→ ∂V ∇V =
∂r âr (13.11)
∇ 2 V = 1 (
∂
r 2 ∂V )
(13.12)
r 2 ∂r ∂r
We will also need the vector identity of Eq. 1.10,
∇ 2 V = −→ ∇·−→ ∇V. (13.13)
A dierential volume element in spherical coordinates is given by
dV = r 2 sin θdrdθdφ. (13.14)
A volume integral of the function V (r, θ, φ) over a sphere of radius 1 centered
at the origin is denoted
ˆ 1
r=0
ˆ π
θ=0
ˆ 2π
φ=0
V (r, θ, φ)r 2 sin θdrdθdφ. (13.15)