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曲线积分和曲面积分

2009-12-20 39页 pdf 575KB 270阅读

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曲线积分和曲面积分 � 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 8797:7?7@BA L ff7C7D A E7F7D B G7H7I7J7K7)7*7L t = α E t = β. M...
曲线积分和曲面积分
� 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 ƒ ~ - t ff�„�*�…†i�‡�G�I�=Sˆ�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. Œ7 I7K<; & [α, β] =7ff7j7Ž t, r7s x′2(t) + y′2(t) + z′2(t) 7K77@‘^ l(t) [ t ff�’�“�„�*�@ z X�N�8S’S“�ffS”S„�* 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@_šS› 0 x = x(l), y = y(l), z = z(l), . r = r(l) (0 ≤ l ≤ l0). ‚�œ��- ����������ž�Ÿ� �¡ . ` 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 LS„S* r(l) I�€�c l ff�Q�R�[ '�( L 8�D M(l) © ff7ª7«7Ž 1�2 @¬•�­ 1 €Sc l ff7“7®7+ 1 …  ›S`�:�Ž 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)) g7‹7´ 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� ( x = R cos t, y = R sin t, z = kt 8 0 ≤ t ≤ 2pi ff�j ¹ €�c�… ¿ 10.1.2 I�à (�¹ L : x = a1t + b1, y = a2t + b2, z = a3t + b3 (0 ≤ t ≤ 1), i €Sc�Ä�Å�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�Æ�Ç�… � § 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�a†sSˆSc†ff »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, ŒS oS]SsSáS€ ¹ ffSäS†cSå λ æ 1 KS p @_`S:S¾S*SI†m†çSff†GSY†E w D†r s�è�j�ff�é�ˆ I, ^� I - f(x, y, z) 8 '�( L =�ff Ú j�Û '�( ‡�G�@ê›S- I = ∫ L f(x, y, z)dl = lim λ→0 n∑ i=1 f(xi, yi, zi)∆li, 5 � dl �-�€�c�ë�ì�… Ú j†Û '_( ‡†G†s†E ~ ‡†Gîí†ïîð†‡îGî]†ñîs†ffîò†óît†ôî@ Œ ( t†@êõîö t�@†��L�t�ô�÷�@êø Œ ∫ L dl ÷�K L ff�€�c�…_M Œ ∫ 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 [7Á7 ( 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�€�@_€�=�Ž 1�2� ß�- ∆α. ^ κ¯ = ∣∣∣∆α∆l ∣∣∣ x [�` :S€ ¹ =SªS«S€†c �� ff ²�� L†@ κ¯ ffSSá x�� ú Ù `Sj ¹ €†ff�� ' ,Så†@�� 5 ` ¹ €�ff ²�� '�� … ¾���þ�� p�� å�j���@�� ∆l → 0, x g7‹7h7™ '"( 8�jSD�ff '�� κ. ff L =�fi7D M1 ¾ M2 G7H7T7ª7«7Ž 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�à ( ff '�� … � à ( ff 1 $ 4�+�,�- r = r(t) = r0 + tv, t ∈ R. fl�¦�s r′′(t) = 0, l κ = 0. � ” ff�% @ Œ� '�( L ff '���& -��@_^�‚ ¢ [SjS:�à (�¹ ( '�Ð 10.2.1). ¿ 10.1.5 À�Á� ( 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†@ê‚SgS‹S[�/†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 Â�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�; Â�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 Ø�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�… Œ� 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 @+U�V�W�X�Y�Z�[�ff�\ � … ¿ 10.1.7 $ L [�Â�] (�* 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 ffSŽ 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. e†² S†K�����cîffîà ( u = ui, v = vj d G†Yà;+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. ` p |r′u × r ′ v| = ∣∣∣∣∂(x, y)∂(u, v) ∣∣∣∣ , ³ ‡�ë�ì dσ = ∣∣∣∣∂(x, y)∂(u, v) ∣∣∣∣ dudv. · ¸ h�™�ï�ð�‡�G�ff�FSë�fS4#��ff ³ ‡�ëSì�… ¿ 10.2.1 À #�$ - R ff�l�ff�ffi ³ ‡�… ¿ 10.2.2 À�l ³ x2 + y2 + z2 = R2 œ�m ³ x2 + y2 = Rx ]�n�¨�ff ' ³�³ ‡ ( fi 10.9). Œ7 ' ³ S ff7+7,7[ ~� 8�;`e D =7ff7ï7ë7„7* z = f(x, y), A7„7* f(x, y) 8 D =�s�O�P�ff�j�ý�oSQSR�@ê` p g – x, y ´�T�)�*�@_X ' ³ S ff�)�*�+�, x w µ H�ff�3�4 x = x, y = y, z = f(x, y). K�[�À�h E = 1 + z′2x , G = 1 + z ′2 y , F = z ′ xz ′ y. i { Õ�u dS = √ 1 + z′2x + z ′2 y dxdy í S = ∫ ∫ D √ 1 + z′2x + z ′2 y dxdy. 10 Œ� ' ³ ff�+†,S- x = g(y, z) . y = h(z, x), ` p g�G�H�Ô ' ³�p�q ™ ²S³ Oyz = . ²7³ Ozx =7@ ]7h7ff p�q ;�e�›�T D1 . D2, ^<èr�7h7™7ò�ó�ff�Ä�Å ' ³�³ ‡�ff�f�4 S = ∫ ∫ D1 √ 1 + x′2y + x ′2 z dydz. . S = ∫ ∫ D2 √ 1 + y′2x + y ′2 z dxdz. § 10.2.2 È�É�Ê�� ) Ë�Ì Œ è Ú j7Û '�( ‡�G�j���@ÒÔ ²�³ ;�e�ff�ï�ðS‡�G�Õ�Ö�™ %�& ff ' ³ =�@ x h�™ Ú j�Û ' ³ ‡�G�ffSÜ�ÝS… Þ7ß 10.2.3 $ S [7j�b7s�s7ff »7¼<' ³ @ f(x, y, z) [ ~7 8 S =7ff7„7*7… e m7ç7G7‰7Ô S G 0 n t ' ³ S1, S2, · · · , Sn ( fi 10.11), `7â7á ' ³ t7ff ³ ‡�›�- ∆Si. 8�ã�t�á ' ³ Si =�m w j�D Mi(xi, yi, zi), T�¾�* n∑ i=1 f(xi, yi, zi)∆Si, ŒS oS]SsSá�t ' ³ ff†äS†Ã $ λ æ 1 KS p @_`S:S¾S*†ISm†çSff†G†YS¾ w D r�s�è�j�ff�é�ˆ A, ^� A - f(x, y, z) 8 ' ³ S =�ff Ú j�Û ' ³ ‡�G�@ê› 0 A = ∫ ∫ S f(x, y, z)dS = lim λ→0 n∑ i=1 f(xi, yi, zi)∆Si, 5 � dS �- ' ³ ff ³ ‡�ë�ì�… ø Œ @�u�u ' ³ S =�G���s�9 � ô�@ 5 ³ � åS„S*�- ρ(x, y, z), Ó�Ä�Å S ff ô 2 M , x�v E�-�À ρ(x, y, z) 8 S =�ff Ú j�Û ' ³ ‡�G�@ y s M = ∫ ∫ S ρ(x, y, z)dS. 0 Œ Ú j7Û '6( ‡7G�g�‹�? 0 ~ ‡�G % Ä�Å�j���@ Ú j7Û ' ³ ‡7G ¢ g�‹7? 0 L�w�ff�ï�ð�‡�G % ÄSÅ�@ê•SA�s�_Só�ffSÄ�ÅS‰S^�… $�»�¼�' ³ ff�)�*�+�,S- x = x(u, v), y = y(u, v), z = z(u, v) (u, v) ∈ D, 11 5 � D [ ²7³ O′uv =7ff7s�syx7;�eS… ŒS „�* f(x, y, z) 8 S =7O7P7@v^7‚�8 S =�ff Ú j�Û ' ³ ‡�G�NS8�@êASs∫ ∫ S f(x, y, z)dS = ∫ ∫ D f(x(u, v), y(u, v), z(u, v)) √ EG− F 2dudv. `�:�f�4�ff�z!{ ¢ E Ú j�Û '�( ‡SG�ff�_SJ�fS4�|�}Sò�óS… µ H�g�h�@  »�¼#' ³ S i�Ã�ß�����+�, z = z(x, y) n Ø @_X D [ S 8 ² ³ Oxy =�ff p�q ;�e�@_^�s∫ ∫ S f(x, y, z)dS = ∫ ∫ D f(x, y, z(x, y)) √ 1 + z′2x + z ′2 y dxdy. ¿ 10.2.3 $ S [ Ú j�~�ˆ�ff�l ³ x2 + y2 + z2 = R2 (x ≥ 0, y ≥ 0, z ≥ 0), Ä Å ' ³ ‡�G ∫ ∫ S (x2 + y2)dS. ò7ó7K7ï7ð�‡�GS@ Ú j�Û ' ³ ‡�G ¢ sSj�â�S��t�ôS@ Œ ( t�@ õ�ö�tS@ g ‡7„7*�N7s�sS@‘* 1 ff ' ³ ‡7G�÷�K ' ³�³ ‡�@¬‹�í�‡�GSff���LSt�ô�÷�…¬G�€ » ¼<' ³ =7ff Ú j�Û ' ³ ‡�G�g d�e ‡�G7I�‡�G ' ³ ff�g�®�t % Ä�Å�@ ø Œ ' ³ S [�i »�¼�' ³ S1 ¾ S2  V�X 0 @+‚�ƒ∫ ∫ S f(x, y, z)dS = ∫ ∫ S1 f(x, y, z)dS + ∫ ∫ S2 f(x, y, z)dS. § 10.3 �…„� ������� § 10.3.1 Þ�† ��� $ L [�O�V A,B ff '�( @ ŒS ­ ~ A [ '�( ff�CSDS@ B [�F�D�@ x › 0 L AB , ”�½�› 0 L BA . ` p '�( x s Ù j�:�ƒ ~ ffS+ 1 @_S- ~ 1 '"( … $ O�V A,B ff '�( L ff�)�*�+�,�- L : x = x(t), y = y(t), z = z(t), α ≤ t ≤ β A(x(α), y(α), z(α)), B(x(β), y(β), z(β)). 12 g�‹�Ô�‚�´ 0 [ ~ 1 '�( L AB ff�)�*�+�,�@6‚�ffi���)S* t i α “�®�™ β,  - L ‡�)�*�“�®�ff�+ 1 @�'`ˆS=S�-S)�* '"( ff 0 + 1 …_X L BA › 0 L BA : x = x(t), y = y(t), z = z(t), β ≥ t ≥ α. ‚�ffi���)�*�i β 9�: ™ α, �- '�( ff�/�+ 1 … $ r(t) = (x(t), y(t), z(t)), r ′(t) OüPüA‰V7-ü 1 2 @ ` p L - »ü¼û' ( @ r ′(t) [ ‚7ff7j7:7Ž 162 @ •7A�[�­ 1 )�* t “7®7+ 1 ff7… ]7‹7­ ~ ª�«7Ž 1�2 τ = r′(t) |r′(t)| _7o7K�Ł�‹ L ff 0 + 1 @ ­ ~ ª�«�Ž 1�2 τ 1 = − r′(t) |r(t)| _7o7K�Ł�‹ L ff�/7+ 1 … Œ� L [ Oxy ²�³ =�j�a�Œ!x '"( @'`ˆS=S 5 ” p�H + 1 - 0 + 1 …ê` p L ff�U�Ž�;�e�8 L S��+ 1 ff ; k�… § 10.3.2 È��Ê�����Ë�Ì�� Þ�ß $ü%û& ;�e V =üs‰‘‰’ F (M), L AB [ V � j ¹ü»ü¼ ~ 1 '�( @ ô�D78 F (M) ff�T e ¨�‡ L AB i A “ � ™ B( fi 10.13). À�‘�’�]�T�ff�”�… Ô L AB G 0 á ¹ €7@–•�— F (M) 87€7Q7ë dl =7]7T7ff7Q7ë�” dw. z - F 8 ª�«�Ž 1�2 τ =�ff p�q - F · τ , K�[ x s dw = F · τdl. ]�‹ W = ∫ L F · τdl. Þ7ß 10.4.1 $ v [ %<& ;re V �6ff 162 ’7@ L AB [ V � »7¼ ~ 1 '�( @ τ [7E L ~ 1 _7j7k�ff�ª�«SŽ 1�2 @ ^ ∫ L v · τdl 7- v ‡ ~ 1 '6( L AB ff Ú ï Û '6( ‡7G�…6o L [�Œyx '6( p @ ∫ L v · τdl 7- v ‡y˜r™ L ff�š 2 @ •�w�› 0 ∮ L v · τdl. § 10.3.3 È��Ê�����Ë�Ì���›�œ��ž�Ÿ $ 8 w ~ ff�Ã�ß�����6S¨�@ v = P (x, y, z)i + Q(x, y, z)j + R(x, y, z)k, L : r = r(t) = (x(t), y(t), z(t)), α ≤ t ≤ β. 13 i τ = r′(t) |r′(t)| , dl = |r′(t)|dt g�u ∫ L v · τdl = ∫ β α (P (x(t), y(t), z(t))x′(t) + Q(x(t), y(t), z(t))y′(t) + R(x(t), y(t), z(t))z ′(t)))dt. ]�‹�g�‹�› ∫ L v · τdl = ∫ L Pdx + Qdy + Rdz. Ä�Å�ff�+�‰ ¢ gS‹ài"=†4 5 kSÆVShS™†…f‚ x [SÔ L ff�)�*�+�,S4S—†˜Sœ ‡7ffi��74 Pdx + Qdy + Rdz �6@ ? 0 IS> 2 t ff Riemann ‡7G7@ ‡7G�¨Sˆ�¾S=�ˆ ^�G�H�[�I�J�K�C�D�¾SF�DSff�)S*SL�… Œ7 Ô τdl ´ 0 [7s 1 €Së�ìS@ ^ τdl = (dx, dy, dz), 5 � dx, dy, dz x [ τdl 8 Ox,Oy,Oz + 1 =�ff p�q … ï�Û '�( ‡�G�s�‹�¨�jSâ�tSô 1) ( t * y  v = c1v1+c2v2,v = (P,Q,R),v1 = (P1, Q1, R1),v2 = (P2, Q2, R2), ^�s ∫ L v · τdl = c1 ∫ L v1 · τdl + c2 ∫ L v2 · τdl. ]�‹�@ Œ� Ô P,Q,R ´ 0 (P, 0, 0), (0, Q, 0), (0, 0, R) ff�¾�@ x s∫ L Pdx + Qdy + Rdz = ∫ L Pdx + ∫ L Qdy + ∫ L Rdz. 2) I�‡�G '�( ff�g�®�t *  L AC [�i L AB ¾ L BC O�V�X 0 ff�@_^�s∫ LAC v · τdl = ∫ LAB v · τdl + ∫ LBC v · τdl. 3) ‡�G�ff�+ 1 t * ∫ LAB v · τdl = − ∫ LBA v · τdl. 4) µ H7@ Œ7 '�( L 8� 7Ã7K x c7ff ²7³ U�@ ^ L ff7)7*7+7,7- x = c, y = y(t), z = z(t) (α ≤ t ≤ β), i�Ä�Å�f�4�¡�¢Sg†h ∫ L Pdx = 0. ò�ó Œ L 8�E y c ( . z c )  �Ã�ff ²�³ U�@_^ ∫ L Qdy = 0( . ∫ L Rdz = 0). 14 ¿ 10.3.1 $ ô 2 - m ff7ô7D787ð�‘�’Sff�T e ¨�‡�m�ç�™ $ L Ł D A “ � ™�D B, À�ð�‘�I�£�ô�D�]�T�ff�”�… � w�¤�³ - Oxy ²�³ @ z c�¥�à 1 = ( fi 10.16), ^�ô 2 - m ff�ô�D�8�m j7D M © ]�¦7ff7ð�‘ p = −mgk, `�§ g [7ð�‘7® � å�… l�ô�D�‡�mSç�™ $ L Ł A ™ B p @_`�‘�]�T�ff�”�- W = ∫ LAB p · τdl = ∫ LAB (−mg)dz = −mg ∫ zB zA dz = −mg(zB − zA). � i { g�u�@_ð�‘�’ p ]�T�ff�”�E�™ $ L ¨ ù @+©�ª ~ KSD A í�D B. Œ� Ô�D B « ~ 8 ¤7²�³ =�@r‚�¬�ô�D m Ł D A ‡7m7ç�™ $�­ ™ ¤�³ p @6ð�‘ p ]7T7ff ”�÷�K W (A) = mgzA. `�:�” x �-�ô�D m 8�D A © ff�«�®�… ¿ 10.3.2 ¯!°�I ¤ l�ff × ‘�- F = −k mM r3 r, 5 � m [ ¤ l7ô 2 @ M [�¯y°6ô 2 @ k [ × ‘�w7*7@ r [ Ł ¯y°6­ 1 ¤ l�ff�« 1�2 @+±�À ¤ l Ł i³² D A ™�´µ²§D B “�S # b p @ × ‘�]�TSff�”S… � |7} u�<7@ ¤ l�¶�¯!°�“�S�ff�M��?�@ ŒS o M ˜�™ M0 p @ w ™�ff�ª�«†‰ 1"2 ÑS[ n(M0), x  S [�j�b�Ð�8 ' ³ …+Õ�^S@ x  S [�j�b�ª�8 ' ³ … Ö�× ff ”Mo¨bius Ø ” [7ª�8 ' ³ ff7j7:�Ù7Û�ff�ø�g7@ – j�c�+�3�Ú�a ABCD Û 180◦, · ‡ AB ¾ CD fi�k�Ü�C % @ A ¾ C ð�Ý�@ B ¾ D ð�Ý�@ x h�™ Ù `�Þ�Ø737ff�ß�Û ( fi 10.22). ` p @vc7+73�=�ff�j�a��"« ( @§œ�Ü�Ý 0 Ø�=�ff�j ayx '6( @ o � D�‡�£!x '�( š�S�jSb p @ � D © ff�ª�«�‰ 1"2�Ł j�:�+ 1 O�P >�?�-�_�”�ff�+ 1 @ z"{ `�Þ�Ø�3S[Sª�8 ' ³ … 8�à�á |�} ¾���þ�Ð�8 ' ³ … 2) Ð�8 ' ³ ff ~ 1 1◦ e j�:�D�ff�‰ 1�2 % ƒ ~ Ð�8 ' ³ ffS+ 1 … $ S [7j7:�Ð�8 ' ³ @ S =7m7j7D M0 s�fi7:7ª7«7‰7+ 1 n(M0) ¾ −n(M0), |�}�w ~ j�:�ø Œ n(M0), ^ ' ³ =�]�s�ffSD M x g�‹�ƒ ~ _�I�JSff n(M). ` � ' ³ =�ffS]†sSDS¾�L ff = ³ ffS+S‰†]Sh†™Sff†‰S+ 1 x [ S ff�j�8�…+_�”†ff�8 x [ ' ³ =�]�s�D M r w ‰�+ 1 −n(M). 2◦ »�¼�' ³ ff ~ 1 … $ S [�j�b »�¼ ff�)�* ' ³ x = x(u, v), y = y(u, v), z = z(u, v) (u, v) ∈ D. ^�‚�s�fi�:�‰ 1�2 n = ε ( ∂(y, z) ∂(u, v) , ∂(z, x) ∂(u, v) , ∂(x, y) ∂(u, v) , ) , ε = ±1. ` p g�‹�­ ~ 5 ��j�:S—�ffi 0 8S+ 1 @fâSj�: x —�ffi�/�8�+ 1 … 3◦ Œ!x ' ³ ff ~ 1 Œ!x ' ³ G�-�U�8�¾�ã�8�@'`ˆS=�Ô�ã�8SS
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