one can express equation (8.60) in the form
∫∫
V
(
ψ∇^2 φ+∇ψ·∇ φ
)
dV =
∫∫
S
ψ∇φ·dS =
∫∫
S
ψ∂φ
∂n
dS (8 .61)
This result is known as Green’s first identity.
In Green’s first identity interchange ψand φto obtain
∫∫
V
(
φ∇^2 ψ+∇φ·∇ ψ
)
dV =
∫∫
S
φ∇ψ·dS =
∫∫
S
φ∂ψ
∂n
dS (8 .62)
Subtracting equation (8.61) from equation (8.62) produces Green’s second identity
∫∫∫
V
(
φ∇^2 ψ−ψ∇^2 φ
)
dV =
∫∫
S
(φ∇ψ−ψ∇φ)·dS (8 .63)
or ∫∫∫
V
(
φ∇^2 ψ−ψ∇^2 φ
)
dV =
∫∫
S
(
φ∂ψ
∂n
−ψ∂φ
∂n
)
dS
Green’s first and second identities have many uses in studying scalar and vector
fields arising in science and engineering.
Additional Operators
The del operator in Cartesian coordinates
∇= ∂
∂x
ˆe 1 + ∂
∂y
ˆe 2 + ∂
∂z
ˆe 3 (8 .64)
has been used to express the gradient of a scalar field and the divergence and curlof
a vector field. There are other operators involving the operator ∇. In the following
list of operators let A denote a vector function of position which is both continuous
and differentiable.
1. The operator A·∇ is defined as
A·∇ = (A 1 ˆe 1 +A 2 ˆe 2 +A 3 ˆe 3 )·
(
∂
∂x
ˆe 1 + ∂
∂y
ˆe 2 + ∂
∂z
ˆe 3
)
=A 1 ∂
∂x
+A 2 ∂
∂y
+A 3 ∂
∂z
(8 .65)
Note that A·∇ is an operator which can operate on vector or scalar quantities.
2. The operator A×∇ is defined as
A×∇ =
∣∣
∣∣
∣∣
ˆe 1 ˆe 2 ˆe 3
A 1 A 2 A 3
∂
∂x
∂
∂y
∂
∂z
∣∣
∣∣
∣∣
=
(
A 2
∂
∂z −A^3
∂
∂y
)
ˆe 1 +
(
A 3
∂
∂x −A^1
∂
∂z
)
ˆe 2 +
(
A 1
∂
∂y −A^2
∂
∂x
)
ˆe 3
(8 .66)
This operator is a vector operator.
