S : r(u, v) (a ! u ! b, c ! v ! d) 1 (u, v)(x, y, z) = r(u, v) f(x, y, z)f(r(u, v)) f(r(u, v))
S S S : r(u, v) (a ! u ! b, c ! v ! d)
[a, b] [c, d] a = u0 < u1 < · · · < uN = b,
c = v0 < v1 < · · · < vM = d.
4 (9.1) r(ui, vj), r(ui+1, vj), r(ui, vj+1), r(ui+1, vj+1)

|(r(ui+1, vj)" r(ui, vj))# (r(ui, vj+1)" r(ui, vj))|
!ui = ui+1 " ui, !vj = vj+1 " vj r(ui+1, vj)" r(ui, vj) =
!r
!v (ui, vj)!vj
9.1
=
!!!!!ui!vj(9.2)
N $ %, M $ % !ui $ 0 (i = 0, 1, 2, · · · , N " 1) ,(9.3)
!vj $ 0 (j = 0, 1, 2, · · · ,M " 1)(9.4)

dS =
!!!! !r
f(ui, vj)$ %& '
c
# b
!!!! dudv
# #
9.1.
S : r(u, v) = (v cosu, v sinu, v) (0 ! u < 2", 0 ! v ! 1)
f(x, y, z) = 1 + 2z ( (S f(r(u, v))dS f(r(u, v)) = 1 + 2v
!!!! !r
= |(v cosu, v sinu,"v)|
9.2
u, v S : r(u, v) (a ! u ! b, c ! v ! d)
(u, v) = (u0, v0) (9.7)
!r
!v (u0, v0)
S r(u0, v0) 2 u, v (9.8)
!r
!u (u, v)# !r
!v (u, v) &= 0
u ' [a, b] , v ' [c, d] (9.8) S r(u0, v0) S2 (9.7) 1
) # !r
*
r(u0, v0) r(u0, v0) S S r(u0, v0)
± !r
n(u0, v0) :=
"v (u0, v0)!! "r "u (u0, v0)# "r
"v (u0, v0) !!
n(u, v) = "r "u (u, v)#
"r "v (u, v)!! "r
"u (u, v)# "r "v (u, v)
!!
9.3 1
f(x, y, z) S : r(u, v) (a ! u ! b, c ! v ! d) dS
f(r(u, v)) · n(u, v)dS
"r "v (u, v)!! "r
"u (u, v)# "r "v (u, v)
!!
f(r(ui, vj)) · n(ui, vj)
f(r(u, v)) · n(u, v)dS
"r "v (u, v)!! "r
"u (u, v)# "r "v (u, v)
!!
, dudv
# #
=
"u (u, v)# "r "v (u, v)
!!
, dudv
9.2. S : r(u, v) = (cosu, sinu, v) (0 ! u < 2", 0 ! v ! 1)
f(x, y, z) = (ezx, ezy, 1) ( ( S f(r(u, v)) · n(u, v)dS
!r
!u (u, v)# !r
!v (u, v) = (" sinu, cosu, 0)# (0, 0, 1) = (cosu, sinu, 0)
f(r(u, v)) ·
62
# 1
0
# 2!