22
1 Preliminaries
Fig. 1.2 The flux, tangential
vector and normal vector of a
vector
O
x
y
a
a
n
τ
M
Γ
curve Γ
ϕ =
Γ
a · ndΓ =
Γ
a n dΓ
(1.3.48)
then ϕ is called the flux of the vector field a along the normal vector n direction
through the curve Γ , as shown in Fig. 1.2.
It follows from Fig. 1.2 that in rectangular coordinates, the normal unit vector n and
tangential unit vector τ of a curve Γ and the curve have the following relationships
n = n x + n y = n x i + n y j = cos(n, x)i + cos(n, y) j
= cos(τ, y)i + cos(τ, −x) j = cos αi + sin α j =
dy
dΓ
i −
dx
dΓ
j
(1.3.49)
τ = τ x + τ y = τ x i + τ y j = cos(τ, x)i + cos(τ, y) j
= cos(n, −y)i + cos(n, x) j = − sin αi + cos α j =
dx
dΓ
i +
dy
dΓ
j (1.3.50)
where, α is the angle between the normal direction of the curve Γ and the x axis.
If the unit vector i in the x direction and unit vector j in the y direction are
expressed by the normal unit vector n and tangential unit vector τ of a curve Γ , then
there are
i = cos αn − sin ατ =
dy
dΓ
n +
dx
dΓ
τ
(1.3.51)
j = sin αn + cos ατ = −
dx
dΓ
n +
dy
dΓ
τ
(1.3.52)