Image Processing Reference
In-Depth Information
Equation 4.11 can now be substituted back into Equation 4.9, and the entire
equation can then be integrated over a small volume ∇ V about the origin as
shown:
Ce
r
ikr
(4.12)
d
Vk
(
∇+
2
2
)
=− =
δ
()
r
d
Vxyz
(()( )( ))
δ δδ
v
v
where the right-hand side can be simplified due to the identity of the delta
function as follows:
Ce
r
ikr
(4.13)
d Vk
(
∇+
)
=−
1
2
2
v
Now the left-hand side of Equation 4.13 can be simplified by applying the dis-
tributive property as follows:
Ce
r
ikr
Ce
r
ikr
(4.14)
2
d
Vk
+
2
d
V
=−
1
v
v
As V becomes smaller and smaller, the second term in Equation 4.14 gradu-
ally goes to zero due to the fact that d V in this term is defined as
d
Vr
= 2 sinθθφ
d dd
r
(4.15)
and as V gets smaller so does the r 2 term in Equation 4.15 which causes the
second term to go to zero as stated earlier. This further simplifies Equation
4.14 as follows:
Ce
r
ikr
2
d
V
= −
1
(4.16)
v
Equation 4.16 can then be rearranged as follows:
Ce
r
ikr

Ce
r
ikr
-
=−
2
d
V
=
d
V
∇⋅ ∇
1
(4.17)
v
v
This now allows the divergence theorem to be applied to Equation 4.17 as
follows:

Ce
r
ikr
-
=∇ =−
Ce
r
ikr
∫∫
d
V
∇⋅ ∇
d
S
1
(4.18)
v
S
which can be expanded as

Ce
r
ikr
-
∫∫
dr
S
2
sinθθφ
dd
=−
1
(4.19)
r
S

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