THE OPERATOR HAT "^"
I. Definition
We define the operator hat "^" as follows:
The magnitude of the vector r is ||r|| or |r| with :
|r|= (r.r)1/2 = (x2 + y2 + z2)1/2 (1)
Let's write the vector u as follows:
u = r - r' with r' ( x', y', z').
The magnitude of the vector u is | u | with:
|u| = |r - r'| = ((x - x')2 + ( y - y')2 + ( z - z') 2)1/2 (2)
We have then:
û = u/|u| (3)
Let's recap:
u = r - r'
|u| = |r-r'| = ((x - x')2 + ( y - y')2 + ( z - z') 2)1/2
û = u/|u|
II. Related operations
II.1. The gradient of (1/|u|); ∇(1/|u|)
We know that:
Grad (1/|r - r'|) = ∇ (1/|r - r'|) = - (r - r')/| r- r' |3; then:
∇ (1/|u|) = - u /|u|3 (4)
From (3), we have u = û |u| ; then :
∇ (1/|u|) = - û |u| /|u|3 = - û /|u|2 = - û /u2
∇ (1/|u|) = - û /|u|2 (5)
II.2. The divergence of û: ∇ . û
∇ . û =
∇ . u/|u| = ∂/∂x ( x - x')/|u| + ∂/∂y ( y - y')/|u| + ∂/∂z ( z - z')/|u| (6)
= (|u| - (x-x')(x - x')/|u|)|u|2 + (|u| - (y - y')(y - y')/|u|)|u|2 +
(|u| - (z - z')(z - z')/|u|)/|u|2 (7)
= (3 |u|2 - (x - x')2 - (y - y')2 - (z - z')2)/|u|3
= (3 |u|2 - |u|2)/|u|3 = 2/|u| (8)
∇ . û = 2/|u| (9)
II.3. The divergence of û/|u| : ∇ . (û/|u|)
∇ . (û/|u|) = û ∇ . (1/|u|) + (1/|u|) ∇ . û (10)
We know from (5) ∇ (1/|u|) = - û /|u|2,
and from (9) ∇ . û = 2/|u| ; then:
∇ . (û/|u|) = û (- û /|u|2) + 2/|u| (1/|u|) =
- û2 /|u|2 + 2/|u|2 =
-1/|u|2 + 2/|u|2 = 1/|u|2
∇ . (û/|u|) = 1/|u|2 (11)
II.4. The gradient of |u| : ∇(|u|)
∇(|u|) = ∇| r - r'|=
(∂/∂x (| r - r'|), ∂/∂y (| r - r'|), ∂ (((x - x')2 +
( y - y')2 + ( z - z') 2)1/2)/∂z) (12)
(∂/∂x, ∂/∂y, ∂/∂z)((x - x')2 +
( y - y')2 + ( z - z') 2)1/2) (13)
= ((x - x')/|u|,(y - y')/|u| ,(z - z')/|u| ) = (r - r')/|u| = u/|u| = û
∇|u|= û (14)
II.5. The divergence of û/|u|2 :
∇ . (û/|u|2)
We have seen from Green's function that
Δ G0(r,r') = δ3(r - r') (15)
with G0(r, r') = (-1/4π) /r - r'
Then:
Δ(1 /r - r') = - 4π δ3(r - r') (16)
Δ(1 /u) = - 4π δ3(u) (17)
From (5) we have: ∇ (1/|u|) = - û /|u|2, then:
∇ . (û /|u|2) = - ∇ (∇ (1/|u|) = - Δ (1/|u|) = 4π δ3(u) (18)
∇ . (û /|u|2) = 4π δ3(u) (19)
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