30
1 Preliminaries
where, ξ , η and ζ are all in S. Therefore
rot n a = (∇ × a) n = lim
S→0
Γ
S
= lim
S→0
L a · dL
S
=
∂a z
∂η
−
∂a y
∂ζ
cos(n, i) +
∂a x
∂ζ
−
∂a z
∂ξ
cos(n, j ) +
∂a y
∂ξ
−
∂a x
∂η
cos(n, k) (1.3.83)
When S → 0 with contracting to point M, ξ → x, η → y, ζ → z, Eq. (1.3.83)
becomes
rot n a = (∇ × a) n = lim
S→0
Γ
S
= lim
S→0
L a · dL
S
=
∂a z
∂ y
−
∂a y
∂z
cos(n, i) +
∂a x
∂z
−
∂a z
∂ x
cos(n, j ) +
∂a y
∂ x
−
∂a x
∂ y
cos(n, k) (1.3.84)
The Formula (1.3.84) of calculating the circulation surface density is rewritten as
lim
S→0
L a · dL
S
=
∂a z
∂ y
−
∂a y
∂z
i +
∂a x
∂z
−
∂a z
∂ x
j
+
∂a y
∂ x
−
∂a x
∂ y
k
· (cos αi + cos β j + cos γ k) (1.3.85)
where, cos α = cos(n, i), cos β = cos(n, j ), cos γ = cos(n, k).
The second group vector on the right side of Eq. (1.3.85) is the unit normal vector
n of S at point M. The first group vector is called the rotation, curl or rotor of
the vector field a at point M, it is written as
rot a =
∂a z
∂ y
−
∂a y
∂z
i +
∂a x
∂z
−
∂a z
∂ x
j +
∂a y
∂ x
−
∂a x
∂ y
k
(1.3.86)
Thus, Eq. (1.3.80) can be expressed as
lim
S→0
L a · dL
S
= rot a · n = rot n a
(1.3.87)
Since n is arbitrary, so the above expression gives the definition of projection of
the vector rot a in arbitrary direction, and it has nothing to do with the choice of the
coordinate system. Thus it can be seen clearly that when n is the same as rot a in
direction, the circulation surface density is the largest, the direction of rotation is the
direction of the maximum circulation surface density.
The projects of the vector rot a in the three axes are