12
1 Preliminaries
=
1
ψ 2
ψ
∂ϕ
∂ x
− ϕ
∂ψ
∂ x
i +
1
ψ 2
ψ
∂ϕ
∂ y
− ϕ
∂ψ
∂ y
j +
1
ψ 2
ψ
∂ϕ
∂z
− ϕ
∂ψ
∂z
k
=
1
ψ 2 (ψ∇ϕ − ϕ∇ψ)
From the derivation rule of compound function, there is
∇ f (ϕ) =
∂ f (ϕ)
∂ x
i +
∂ f (ϕ)
∂ y
j +
∂ f (ϕ)
∂z
k =
∂ f (ϕ)
∂ϕ
∂ϕ
∂ x
i +
∂ f (ϕ)
∂ϕ
∂ϕ
∂ y
j +
∂ f (ϕ)
∂ϕ
∂ϕ
∂z
k
= f (ϕ)
∂ϕ
∂ x
i +
∂ϕ
∂ y
j +
∂ϕ
∂z
k
= f (ϕ)∇ϕ
For Eq. (1.3.16), let ϕ = r , then there is
∇ f (r ) = f
(r )∇r
Since r = x i + y j + zk, r = |r| =
x 2 + y 2 + z 2 , therefore there are
∂r
∂ x
=
x
x 2 + y 2 + z 2
=
x
r
,
∂r
∂ y
=
y
r
,
∂r
∂z
=
z
r
∇r =
∂r
∂ x
i +
∂r
∂ y
j +
∂r
∂z
k =
x i + y j + zk
r
=
r
r
= r 0
Substituting the above expression into Eqs. (1.3.16) and (1.3.17) is obtained. Quod
erat demonstrandum.
From Eq. (1.3.17), we obtain
∇ f
(n)
(r ) = f
(n+1)
(r )∇r
(1.3.18)
Substituting Eq. (1.3.9) into Eq. (1.3.6), we obtain
∂ϕ
∂ L
= grad ϕ · L
0
= ∇ϕ · L
0
(1.3.19)
Equation (1.3.19) shows that the differential of the function ϕ along the direction L
equals the scalar product of the gradient of ϕ and the unit vector L
0 in the direction L.
If the function ϕ(x, y, z) = C, then it is called the contour surface equation, it
expresses a family of surfaces, Each value corresponding to the constant C expresses
a surface. At various points on the every surface, although the coordinate values are
different, but the function values are equal, these surface are called the contour
surfaces of the function ϕ. Similarly, if the function ψ(x, y) = C, then it is called
the contour line equation or isoline equation. It expresses a family of curves, each
value corresponding to the constant C expresses a curve, it is called the contour
line, isoline or isoplethic curve of the function ψ. Because the function ϕ remains