�
10 � ��������� ��
����
§ 10.1
����
�������
§ 10.1.1 �������������
( ������ff�fi�fl�ffi��� �!��"ff#fi�fl )
$�%�&�'�(
L ff�)�*�+�,�-
x = x(t), y = y(t), z = z(t),
.�/�0�1�2
ff�3�4
r = r(t),
5
�6)7* t 8797:<;
&
[α, β] =7>7?7@BA L ff7C7D A E7F7D B G7H7I7J7K7)7*7L
t = α E t = β. M
$
r(t) 8<;
&
[α, β] =7N787O7P�ff�QSR r′(t) = (x′(t), y′(t), z′(t)).
8
'�(
L =�T�U�V�W
(
AM1M2 · · ·Mn−1B( fi 10.1). X�G�Y T :
t0 = α < t1 < t2 < · · · < ti−1 < ti < · · · < tn−1 < tn ≤ β
ff�Z�G�D�[�Z�\�D�]�ISJ�ffS)�*SLS@_^S`Sa�W
(
ff�bScS-
l(T ) =
n∑
i=1
|r(ti)− r(ti−1)|
=
n∑
i=1
√
[x(ti)− x(ti−1)]2 + [y(ti)− y(ti−1)]2 + [z(ti)− z(ti−1)]2.
d�e
Q�G���L�f�4�g�h
l(T ) =
n∑
i=1
√
x′2(ξi) + y′
2(ηi) + z′
2(ζi)∆ti,
5
� ti−1 < ξi, ηi, ζi < ti, ∆ti = ti − ti−1. i"K
√
x′2(ξ) + y′2(η) + z′2(ζ) 8 [α, β]3
=SjSkSOSPS@_lSISmSn ε > 0, NS8 δ1 > 0. o ‖T‖ < δ1 p @_ISmSq (ξi, ηi, ζi) ∈
[ti−1, ti]
3(i = 1, 2, · · · , n) r�s∣∣∣√x′2(ξi) + y′2(ηi) + z′2(ξi)−√x′2(ti) + y′2(ti) + z′2(ti)∣∣∣ < ε.
1
l�o ‖T‖ < δ1 p∣∣∣∣∣l(T )−
n∑
i=1
√
x′2(ti) + y′2(ti) + z′2(ti)∆ti
∣∣∣∣∣ < ε
n∑
i=1
∆ti = ε(β − α). (10.1.1)
M 0, o ‖T‖ <
δ2 p ∣∣∣∣∣
n∑
i=1
√
x′2(ti) + y′2(ti) + z′2(ti)∆ti −
∫ β
α
√
x′2(t) + y′2(t) + z′2(t)dt
∣∣∣∣∣ < ε. (10.1.2)
w
δ = min(δ1, δ2). K�[�i (10.1.1),(10.1.2) g�u�@_o ‖T‖ < δ p�x s∣∣∣∣l(T )− ∫ β
α
√
x′2(t) + y′2(t) + z′2(t)dt
∣∣∣∣ < ε(β − α + 1).
y
[
lim
‖T‖→0
l(T ) =
∫ β
α
√
x′2(t) + y′2(t) + z′2(t)dt.
z�{�|�}
x�~�
L ff��c
l0 = lim
‖T‖→0
n∑
i=1
√
x′2(ξi) + y′
2(ηi) + z′
2(ζi)∆ti
=
∫ β
α
√
x′2(t) + y′2(t) + z′2(t)dt.
8 L =
w
j�D M , �I�J�ff�)�*�L�- t, ^� AM ff�c�-
l(t) =
∫ t
α
√
x′2(τ) + y′2(τ) + z′2(τ)dτ.
o M 8 L =�>�?
p
@_�c l
~
- t ff��*�
i��G�I�=S�ffSQSG�S^�h
d l
dt
=
√
x′2(t) + y′2(t) + z′2(t).
Ł
X
d l =
√
x′2(t) + y′2(t) + z′2(t)dt,
.
d l2 = dx2 + dy2 + dz2.
`
x
[
%�&�'�(
ff��cSQ�GSf�4S
2
z
-
|dr| =
√
dx2 + dy2 + dz2
=
√
x′2(t) + y′2(t) + z′2(t)|dt| = |r′(t)| · |dt|,
]��s
dl = ±|dr| = |r′(t)|dt.
7
I7K<;
&
[α, β] =7ff7j7 t, r7s x′2(t) + y′2(t) + z′2(t) 7K77@^ l(t) [
t ff����*�@
z
X�N�8SS�ffSS�* t = t(l), �A��I l ff�Q�R�g��ffi
0
dt
dl
=
1√
x′2(t) + y′2(t) + z′2(t)
.
t = t(l) ��
'�(
L ff�)�*�+�,��
x
h��S�cS-�)S*Sff�+S,S@_S
0
x = x(l), y = y(l), z = z(l),
.
r = r(l) (0 ≤ l ≤ l0).
���- ������������ �¡ . `
p
Ł
�c�ff�Q�G�f�4�s(
dx
dl
)2
+
(
dy
dl
)2
+
(
dz
dl
)2
= 1.
.�/�0 ∣∣∣∣drdl
∣∣∣∣ = 1.
¢
x
[�£�@_8¥¤§¦S+S,S¨�@
1"2
LSS* r(l) I��c l ff�Q�R�[
'�(
L 8�D M(l)
©
ff7ª7«7
1�2
@¬�
1
Sc l ff77®7+
1
S`�:�
1"2
ff�¯S:�+
1"°�±
-
(cos α, cos β, cos γ), ^�s
dx
dl
= cos α,
dy
dl
= cosβ,
dz
dl
= cos γ.
²7³
'6(
r = (x(t), y(t)) g77´
07µ�¶
ff
%�&S'�(
r = r(x(t), y(t), 0), i
{7·
¸
h�
'�(�¹
L : r = (x(t), y(t)), t ∈ [α, β] ff��c�f�4
l0 =
∫ β
α
√
x′2(t) + y′2(t)dt.
3
I�K�º
¹�»�¼�'�(
@
5
Sc�g
~S
-SZ
¹S»S¼�'"(
�cS½�¾S
¿
10.1.1 À�Á�Â
(
x = R cos t, y = R sin t, z = kt
8 0 ≤ t ≤ 2pi ff�j
¹
�c�
¿
10.1.2 I�Ã
(�¹
L : x = a1t + b1, y = a2t + b2, z = a3t + b3 (0 ≤ t ≤ 1), i
Sc���f�4�h
5
c l =
∫ 1
0
√
a21 + a
2
2 + a
2
3dt =
√
a21 + a
2
2 + a
2
3, x [
(�¹
ff�CSD
(b1, b2, b3) E�F�D (a1 + b1, a2 + b2, a3 + b3)
&
ff���
�
§ 10.1.2 È�É�Ê�����Ë�Ì
87Í7Î<Ï6Ð<��@ÒÑ�Ó�À�Ô�8
(�¹
=�ff
~
�G�ÕSÖ�
²�³
.
%�&
ff
'�(
=�
Ò`
x�×�Ø�Ù�Ú
j�Û
'�(
SG�ffSÜ�ÝS
Þ�ß
10.1.1
$
L [
%�&
��j�asSScff
»S¼à'"(
@ f(x, y, z) [
~�
8 L
=�ff��*�
e
G�D N0, N1, · · · , Nn Ô
'�(
L G
0
n :�á�
¹
l1, l2, · · · , ln( fi 10.2),
��`�â�
¹
ff��c�- ∆li. 8�ã�:�á�
¹
li =�m
w
j�D Mi(xi, yi, zi), T�¾�*
n∑
i=1
f(xi, yi, zi)∆li,
S
oS]SsSáS
¹
ffSäScSå λ æ
1
KS
p
@_`S:S¾S*SImçSffGSYE
w
Dr
s�è�j�ff�é� I, ^� I - f(x, y, z) 8
'�(
L =�ff
Ú
j�Û
'�(
�G�@êS-
I =
∫
L
f(x, y, z)dl = lim
λ→0
n∑
i=1
f(xi, yi, zi)∆li,
5
� dl �-��c�ë�ì�
Ú
jÛ
'_(
GsE
~
GîíïîðîGî]ñîsffîòóîtôî@
(
t@êõîö
t�@��L�t�ô�÷�@êø
∫
L
dl ÷�K L ff��c�
_M
∫
L
f(x, y, z)d l = f(ξ, η, ζ)l0,
5
� l0 - L ff��c�@_X (ξ, η, ζ) [ L =�ff�j�D�
ù
K
Ú
j�Û
'�(
�G�ffSÄ�ÅS@êsS¨�úSð�ÓSfS4�
$�'�(
L ff�)�*�+�,�-
x = x(t), y = y(t), z = z(t) (α ≤ t ≤ β),
4
A x(t), y(t), z(t) 8û;
&
[α, β] =üsüOüPüffüjüýüQüR x′ (t), y′(t), z′(t).
ü* f(x, y, z)
8 L =�O�P�@_^�8 L =�ff
Ú
j�Û
'�(
�G�NS8S@_ASs∫
L
f(x, y, z)dl =
∫ β
α
f(x(t), y(t), z(t))
√
x′2(t) + y′2(t) + z′2(t)dt.
T�-�Õ�þ�@
$
²�³
'"(
L ff�Ã�ß�����+�,�- y = y(x) (a ≤ x ≤ b), A y(x) s
O�P�ff�Q�R�@_^�s ∫
L
f(x, y)dl =
∫ b
a
f(x, y(x))
√
1 + y′2(x)dx.
²�³
'�(
L ff�é�����+�,�- r = r(θ) (α ≤ θ ≤ β), A r(θ) s�O�P�ff�Q�R�@
^�s ∫
L
f(x, y)dl =
∫ β
α
f(r(θ) cos θ, r(θ) sin θ)
√
r2(θ) + r′2(θ)dθ.
¿
10.1.3 Ä7Å
'6(
�G
∫
L
(x2 + y2 + z2)dl,
5
� L [7Á7Â
(
x = R cos t, y =
R sin t, z = kt 8 0 ≤ t ≤ 2pi ff�
¹
�
L ff��c�ë�ì�[
dl =
√
(−R sin t)2 + (R cos t)2 + k2dt =
√
R2 + k2dt,
]� ∫
L
(x2 + y2 + z2)dl =
∫ 2pi
0
(R2 + k2t2)
√
R2 + k2dt
= 2pi
(
R2 +
4
3
pi2k2
)√
R2 + k2. �
��
ô7
(��
å ρ(x, y, z) G��787j7a
»7¼�'�(
L =7@ ò7ó 9.4 ff��7þ7g7u
5
ð� �«�
�[
x
G
=
∫
L
xρ(x, y, z)dl∫
L
ρ(x, y, z)dl
, y
G
=
∫
L
yρ(x, y, z)dl∫
L
ρ(x, y, z)dl
, z
G
=
∫
L
zρ(x, y, z)dl∫
L
ρ(x, y, z)dl
.
§ 10.1.3 ���������
'�(
L =�j
¹
c�å�- ∆l ff��@_�=�
1�2�
ß�- ∆α. ^ κ¯ =
∣∣∣∆α∆l ∣∣∣ x [�`
:S
¹
=SªS«Sc
��
ff
²��
L@ κ¯ ffSSá
x��
ú
Ù
`Sj
¹
ff��
'
,Så@��
5
`
¹
�ff
²��
'��
¾���þ��
p��
å�j���@�� ∆l → 0,
x
g77h7
'"(
8�jSD�ff
'��
κ.
ff
L =�fi7D M1 ¾ M2 G7H7T7ª7«7
1�2
r˙1 ¾ r˙2 ( fi 10.3). M̂1M2 ff7�c7-
∆l, ∆r˙ = r˙2 − r˙1. fl�¦�s
|∆α| ≈ |∆r˙|,
]�
κ = lim
∆l→0
∣∣∣∣∆α∆l
∣∣∣∣ = lim∆l→0
∣∣∣∣∆r˙∆l
∣∣∣∣ = |r¨|
= |κ|.
K�[
x
h�
'�(�'��
8Sj�ffiS)�*S3S4�¨SffSffi��S4
κ =
|r′(t)× r′′(t)|
|r′(t)|3
.
κ = 0 ff�D!�
'�(
ff�Ã�?�D�" κ 6= 0
p
@ R = 1κ �
'�(
ff
'���#�$
¿
10.1.4 À�Ã
(
ff
'��
�
Ã
(
ff
1
$
4�+�,�-
r = r(t) = r0 + tv, t ∈ R.
fl�¦�s r′′(t) = 0, l κ = 0. �
ff�%
@
�
'�(
L ff
'���&
-��@_^�
¢
[SjS:�Ã
(�¹
( '�Р10.2.1).
¿
10.1.5 À�Á�Â
(
r(t) = (a cos t, a sin t, bt) (a, b > 0)
ff
'��
(�)
���������
$
L [ Oxy
²�³
=�ff�)�*
'�(�*
x = x(t), y = y(t) (α ≤ t ≤ b).
^�s
κ =
(x′(t), y′(t), 0) × (x′′(t), y′′(t), 0)
(x′2(t) + y′2(t))3/2
=
x′(t)y′′(t)− x′′(t)y′(t)
(x′2(t) + y′2(t))3/2
k.
6
κ =
x′(t)y′′(t)− x′′(t)y′(t)
(x′2(t) + y′2(t))3/2
- L ff
'��
²�³
'"(
ff
'+��,
sS®�-SISL�.Sfl@êSgSS[�/ffS@ κ ff
w�0�w
/�@
¢
s
5�1
q�ff�ç
i�K
x′(t)y′′(t)− x′′(t)y′(t)
x′2(t)
=
(
y′(t)
x′(t)
)′
,
]�
1◦ o κ > 0
p
@
'"(�2���3
S®æ�4S@ê`
p
r′(t), r′′(t),k
0�5
�6S@_]S
r′′(t)
1
'�(!7
ff�j�8 ( fi 10.4).
2◦ o κ < 0
p
@
'"(�2���3�9�:
æ�4S@ê`
p
r′(t), r′′(t),k
0�;
�6S@_]S
r′′(t) Ñ�[�
1
'�(!7
ff�j�8 ( fi 10.5).
µ
H7@
w
)�*�[�Sc
p
@
x
u�<�=S
1�2�>
[�
1
'"(!7
ffSj�8�
jS:�ð�Ó
ff
µ
ø�[
'�(
i�fl��* y = f(x) (a ≤ x ≤ b) n
�p
@
κ =
f ′′(x)
(1 + f ′2(x))3/2
k,
κ =
f ′′(x)
(1 + f ′2(x))3/2
.
κ > 0
p
@ f ′′(x) > 0,
'�(�3!7
3�" κ < 0
p
@ f ′′(x) < 0,
'�(�3!?
3�
8
²�³
'�(
ff�@�3�@ k 6= 0
p
@ R = 1
|k|
�
'�(
ff
'���#�$
¿
10.1.6 À
#�$
- R ff!A�ff
'��
� $
A�ff
1
$
4�+�,�[
r = r(t) = (a, b) + R(cos t, sin t) 0 ≤ t ≤ 2pi.
^�s
κ =
R2 sin2 t + R2 cos2 t
(R2 sin2 t + R2 cos2 t)3/2
=
1
R
. �
]�
#�$
- R ff!A�ff
'��
-
1
R
,
'���#�$
y
[!A�ff
#�$
R.
#�$�B
á�@�A�ff
'��
x
B
�@+�
'
å
¢�B
�@ê`Sâ�rS[�C�Ã�D�ff�ESþ�
�
Ô�ø 1 ��ff!A�F
0�G
p�H
+
1
@
5#1
$
+�,S4�-
r = r(t) = (a, b) + R(cos t,− sin t) 0 ≤ t ≤ 2pi,
7
^�`�:!A�ff
'��
x
[ − 1
R
.
8�I�J
²7³
=�ff�K�L�M�<
p
@ON�P
M�<�ff
'��
O�P�>�?�@Ò`���Q�R�8�M�<
=�S�T
p
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�
¿
10.1.7
$
L [��]
(�*
x = a(t− sin t), y = a(1 − cos t) (a > 0). À L ff
'���#�$
§ 10.1.4
(�)
����������^
$
L [ Oxy
²�³
=�ff
'�(
@
ff
L =�j�D M , A�8 M E L s�_�è�=�
1�2
ff!A�� L(
ff
M) ff
'��
A�@
'��
A�ff!A` !� M
ù
K L ff
'��
�� �
i"K r′(t) = (x′(t), y′(t)) [ L ffS
1"2
@_]S (−y′(t), x′(t))
x
[ L ffS
1
2
1�2
n =
κ√
x′2(t) + y′2(t)
(−y′(t), x′(t))
ff�c�- |κ|, �E�=�
1�2
r¨
²
S�@_]��N�s
n = ±r¨.
a
i�K
r′′(t) · n =
κ
|r′(t)|
(x′(t)y′′(t)− x′′(t)y′(t))
=
κ
|r′(t)|
κ|r′(t)|3 = κ2r′2(t) > 0,
]� n
¢
1
'�(!7
ff�j�8 ( fi 10.6),
z�{
n
x
[�=�
1�2
i�K R = 1
|κ|
= 1
|n|
, ]� Rn
x
[�=��+
1
=�ff�ª�«
1"2
êKS[
x
hS
'��
�� �ff
1
$
-
rκ = r(t) + R
2n = r(t) +
n
κ2
=
(
x(t)−
y′(t)
κ
√
x′2(t) + y′2(t)
, y(t) +
x′(t)
κ
√
x′2(t) + y′2(t)
)
.
8
§ 10.2
����
��
����
§ 10.2.1 �
)
�
)
Ë
$
S [�j�b
»�¼
ff�)�*
'
³
*
r = r(u, v) (u, v) ∈ D,
.�/�0
)�*�+�,
x = x(u, v), y = y(u, v), z = z(u, v) (u, v) ∈ D.
e²
SK�����cîffîÃ
(
u = ui, v = vj d GYà;+e D,
w
5
�_j:áà;fe
Dij : ui ≤ u ≤ ui+1, vj ≤ v ≤ vj+1( fi 10.7). I�J�8 S = x h��jS:�g
'
³
Sij,
�i�fi�a u
'�(
v = vj , v = vj+1 ¾�fi�a v
'�(
u = ui, u = ui+1 h
0
_o ∆ui =
ui+1 − ui ¾ ∆vj = vj+1 − vj r�C�á p @ Sij g��i�ó�´
0
i r(ui+1, vj)− r(ui, vj)
¾ r(ui, vj+1)− r(ui, vj) b
0
ff
²
S!j�k�3 ( fi 10.8).
z
-
r(ui+1, vj)− r(ui, vj) = r
′
u(ui, vj)∆ui + o(∆ui),
r(ui, vj+1)− r(ui, vj) = r
′
v(ui, vj)∆vj + o(∆vj),
]��@ Sij ff
³
∆Sij ≈ |r
′
u(ui, vj)× r
′
v(ui, vj)|∆ui∆vj.
K�[
'
³
S ff
³
S =
∫ ∫
D
|r′u(u, v) × r
′
v(u, v)|dudv.
i�K
(r′u × r
′
v)
2 = r′2u r
′2
v − (r
′
u · r
′
v)
2,
�
E = r′2u = x
′2
u + y
′2
u + z
′2
u ,
G = r′2v = x
′2
v + y
′2
v + z
′2
v ,
F = r′u · r
′
v = x
′
ux
′
v + y
′
uy
′
v + z
′
uz
′
v,
9
x
h�
'
³
S =�ff
³
�ë�ì
dS =
√
EG− F 2dudv
¾
'
³
S ff
³
S =
∫ ∫
D
√
EG− F 2dudv.
o
'
³
S [
²�³
;�e
p
@ S ff�)�*�+�,�-
x = x(u, v), y = y(u, v), z = 0, (u, v) ∈ D.
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