1.3 Fundamentals of the Theory of Field
37
rot a · n =
L k
a · dL
S k
+ ε k
namely
S k rot a · n =
L k
a · dL + ε k S k
where, the rotation is the value of the element of surface at a point M, the left-handed
side expresses the flux on the element of surface S k ; ε k is a small enough amount,
and when S k → 0, ε k → 0. Summing the above expression with respect to k from
1 to n, we obtain
n
k=1
S k rot a · n =
n
k=1
L k
a · dL +
n
k=1
ε k S k
Within the closed surface L k , a on the public side of the two adjacent closed
curves of the various elements of area is the same, but the direction of the two curves
is opposite, the two integrals cancel each other out, accordingly, on the right side
of the above expression, only keeping the integral of the first term with respect to
the closed surface L. Meanwhile, let n → ∞, such that S k → 0, then the lefthanded side of the above expression takes the surface integral
˜
S rota · ndS as a
limit. Moreover for the vector a, it can be thought that when n is sufficiently large,
there must exist a infinitesimal ε, such that
|ε 1 | < ε, |ε 2 | < ε, . . . , |ε n | < ε
Moreover when n → ∞, there is
lim
n→∞
ε = 0
Therefore
n
k=1
ε k S k
≤
n
k=1
εεS k = ε
n
k=1
S k = εS
Because the product of an infinitesimal and a bounded function is still an
infinitesimal, the limit of the above expression is zero, thus
¨
S
rota · ndS =
L
a · dL
Quod erat demonstrandum.
The Stokes theorem has another vector form, it can be expressed as