16
1 Preliminaries
ϕ =
¨
S
a · dS =
¨
S
a · ndS =
¨
S
a n dS
(1.3.31)
then ϕ is called the flux of the vector a along the vector n through the surface S.
The projects of the surface S on the three coordinate surfaces are respectively
⎧
⎪ ⎨
⎪ ⎩
dydz = cos(N , x)dS = cos αdS = ldS
dzdx = cos(N , y)dS = cos βdS = mdS
dxdy = cos(N , z)dS = cos γ dS = ndS
(1.3.32)
The directing surface dS can be expressed as
dS = ndS = idydz + j dzdx + kdxdy
(1.3.33)
The flux of the vector a through the surface S can also be written in the following
form
ϕ =
S
[a x cos(N , x) + a y cos(N , y) + a z cos(N , z)]dS
=
S
(a x dydz + a y dzdx + a z dxdy)
(1.3.34)
When the surface S is a closed surface, it has special important meaning. For
obvious reasons, adopting the method of drawing a circle on the integral sign, namely
using
ϕ =
S
a n dS =
S
a · dS
(1.3.35)
to express the flux through the closed surface.
Suppose that there is a vector field a = a(M), in a neighborhood at a point M
of the field, making an arbitray closed surface S containing the point, the volume
surrounded by the closed surface S is V , calculating the flux of the vector a
through the surface S, when V is contracted to point M in any manner, if there
is a limit of ratio
lim
V →0
ϕ
V
= lim
V →0
S a · dS
V
then the limit is called the divergence of the vector function a at the point M, it is
written as div a. When using the Hamiltonian operator, it can be written as ∇ · a,
namely
div a = ∇ · a = lim
V →0
ϕ
V
= lim
V →0
S a · dS
V
(1.3.36)
1 Preliminaries
ϕ =
¨
S
a · dS =
¨
S
a · ndS =
¨
S
a n dS
(1.3.31)
then ϕ is called the flux of the vector a along the vector n through the surface S.
The projects of the surface S on the three coordinate surfaces are respectively
⎧
⎪ ⎨
⎪ ⎩
dydz = cos(N , x)dS = cos αdS = ldS
dzdx = cos(N , y)dS = cos βdS = mdS
dxdy = cos(N , z)dS = cos γ dS = ndS
(1.3.32)
The directing surface dS can be expressed as
dS = ndS = idydz + j dzdx + kdxdy
(1.3.33)
The flux of the vector a through the surface S can also be written in the following
form
ϕ =
S
[a x cos(N , x) + a y cos(N , y) + a z cos(N , z)]dS
=
S
(a x dydz + a y dzdx + a z dxdy)
(1.3.34)
When the surface S is a closed surface, it has special important meaning. For
obvious reasons, adopting the method of drawing a circle on the integral sign, namely
using
ϕ =
S
a n dS =
S
a · dS
(1.3.35)
to express the flux through the closed surface.
Suppose that there is a vector field a = a(M), in a neighborhood at a point M
of the field, making an arbitray closed surface S containing the point, the volume
surrounded by the closed surface S is V , calculating the flux of the vector a
through the surface S, when V is contracted to point M in any manner, if there
is a limit of ratio
lim
V →0
ϕ
V
= lim
V →0
S a · dS
V
then the limit is called the divergence of the vector function a at the point M, it is
written as div a. When using the Hamiltonian operator, it can be written as ∇ · a,
namely
div a = ∇ · a = lim
V →0
ϕ
V
= lim
V →0
S a · dS
V
(1.3.36)