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Appendix: All Solutions to the Problems
or
∇v · ∇(gu) = ∇ · (gu∇v) − guv
(4)
Substituting Eqs. (4) into (2), to get
(gu)v = ∇ · [v∇(gu)] − ∇ · (gu∇v) + guv
= ∇ · [v∇(gu) − (gu∇v)] + guv
(5)
Quod erat demonstrandum.
1.8 Prove
Γ (u cos α + v cos β)dΓ =
˜
D
∂u
∂ x
+
∂v
∂ y
dxdy.
Proof In the plane rectangular coordinate system, let a = ui + v j , n = cos αi +
cos β j , according to the Green formula (1.3.55), there is
Γ
a · ndΓ =
¨
D
∇ · adD
(1)
Substituting a and n into the left side of Eq. (1), to obtain
Γ
a · ndΓ =
Γ
(ui + v j ) · (cos αi + cos β j )dΓ =
Γ
(u cos αi + v cos β)dΓ
(2)
Substituting a into the right side of Eq. (1), to get
¨
D
∇ · adD =
¨
D
∂
∂ x
i +
∂
∂ y
j
· (ui + v j )dxdy =
¨
D
∂u
∂ x
+
∂v
∂ x
dxdy
(3)
Thus there is
Γ
(u cos α + v cos β)dΓ =
¨
D
∂u
∂ x
+
∂v
∂ y
dxdy
(4)
Quod erat demonstrandum.
1.9 Prove the following identity
˚
V
(u
2 v − uv)dV =
S
u
∂∂v
∂n
− v
∂∂u
∂n
dS
Proof In Eq. (1.3.74), let ψ = u, ϕ = v, then