积分

A4&,~CFl ℄�W 1 ��3�(zMbs�b[�3 2 1.1 �vN' Ω �?;R�T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 ��T�C[:sD�sW�T?Æ� . . . . . . . . . . . . . . . . . . . . . 3 2 (0Mbs�b[�3 7 2.1 CN:sD�T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 CN:sD�TFb . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.2 CN:sD�TÆ� . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 CN:sW�T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.1 sW?$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 CN:sW�TFb . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.3 CN:sW�TÆ� . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 6Æ�3�I%Tq 18 3.1 Green g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Gauss g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Stokes g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4 Q. wEk6M�3 21 4.1 kl��W��� �pD?K*P . . . . . . . . . . . . . . . . . . . . . . . 21 4.1.1 ��T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.1.2 sD�T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.1.3 sW�T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 klM�K*P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5 K� 23 1 §1 �B5�){N t� \B5 §1.1 DW?�Tu n∑ i=1 f(Mi)∆Ωi. {o{bu ; d→ 0 ��C4un0�C�\7l Ω ?~Tu Mi ∈ Ωi ?Ut�w* 0�C6 f(M) u�vN' Ω �?;R�T��6 ∫ Ω f(M)dΩ, ∫ Ω f(M)dΩ = lim d→0 n∑ i=1 f(Mi)∆Ωi, i� f(M) *6��r��Ω *6�Tro�dΩ *6{(3T�W�3T�'�3T. �H1.1. (1) J f(M) ksf�a Ω L�/HbX�&L0eQ�qy�TUsf�a �4"�5�TL0pQE��p���. (2) J f(M) ≡ 1, �pQE�r�- ∫ Ω dΩ �Tsf�a Ω �i.o0poap. �W�T�qD℄'V;p?F�T���T�C[:sD�sW�T: (1) ~�vN' Ω 6� x �?r�J [a, b], \S [a, b] �?�T*6+�3, �6 ∫ [a,b] f(x)dx = ∫ b a f(x)dx = lim d→0 n∑ i=1 f(ξi)∆xi. �v`b6^ [a, b] 6A�sD y = f(x) 6s�?s�$N?W�. (2) ~�vN' Ω 6gW.N D, \S D �?�T*60��3, ����<>�6 ∫∫ D f(x, y)dxdy = lim d→0 n∑ i=1 f(ξi, ηi)∆xi∆yi. �v`b�^ D 6A�sW z = f(x, y) 6E?sEÆ'?'�. (3) ~�vN' Ω 62�?', \S Ω �?�T*6e��3, ����<>�6 ∫∫∫ Ω f(x, y, z)dxdydz = lim d→0 n∑ i=1 f(ξi, ηi, ζi)∆xi∆yi∆zi. 2 (4) ~�vN' Ω 62�sDJ L, \S L �?�T*6(zMbs�3, �6 ∫ L f(x, y, z)ds = lim d→0 n∑ i=1 f(ξi, ηi, ζi)∆si. (1.1) (5) ~�vN' Ω 62�sW Σ, \S Σ �?�T*6(zMb[�3, �6 ∫∫ Σ f(x, y, z)dS = lim d→0 n∑ i=1 f(ξi, ηi, ζi)∆Si. (1.2) §1.2 ��3�(zMbs�b[�3%Ek ��T�C[:sD�sW�T?Æ�k����Tro�Æ-6. (1) N��T�6N1�TÆ�, 2=1+1. u����<>�~��Tro D T��Æ6 D = {(x, y)|a ≤ x ≤ b, ϕ(x) ≤ y ≤ ψ(x)}, D = {(x, y)|c ≤ y ≤ d, ϕ(y) ≤ x ≤ ψ(y)}, wN��TEdB�6 ∫∫ D f(x, y)dxdy = ∫ b a dx ∫ ψ(x) ϕ(x) f(x, y)dy, ∫∫ D f(x, y)dxdy = ∫ d c dy ∫ ψ(y) ϕ(y) f(x, y)dx. ~�Tro D nqoik�w/Kf���< x = ρ cos θ, y = ρ sin θ�0����� <>?N��T�6���<>?N��T6∫∫ D f(x, y)dxdy = ∫∫ D f(ρ cos θ, ρ sin θ)ρdρdθ. ~��Tro D T��Æ6 D = {(ρ, θ)|a ≤ θ ≤ b, ϕ(θ) ≤ ρ ≤ ψ(θ)}, D = {(ρ, θ)|c ≤ ρ ≤ d, ϕ(ρ) ≤ θ ≤ ψ(ρ)}, w���<>?N��TEdB�6 ∫∫ D f(ρ cos θ, ρ sin θ)ρdρdθ = ∫ b a dθ ∫ ψ(θ) ϕ(θ) f(ρ cos θ, ρ sin θ)ρdρ, ∫∫ D f(ρ cos θ, ρ sin θ)ρdρdθ = ∫ d c ρdρ ∫ ψ(ρ) ϕ(ρ) f(ρ cos θ, ρ sin θ)dθ. ~�Tro D n0qoik�w/Kfmb���< x = aρ cos θ, y = bρ sin θ, 3 0� ∫∫ D f(x, y)dxdy = ∫∫ D f(aρ cos θ, bρ sin θ)abρdρdθ, im:�. �{ D 6 x 2 a2 + y 2 b2 ≤ r2�wH x = aρ cos θ, y = bρ sin θ, 0 ≤ θ ≤ 2pi, 0 ≤ ρ ≤ r, Ed?i ∫∫ D f(x, y)dxdy = ∫∫ D f(aρ cos θ, bρ sin θ)abρdρdθ = ab ∫ 2pi 0 dθ ∫ r 0 f(aρ cos θ, bρ sin θ)ρdρ. S1.1. Bh" x 2 a2 + y 2 b2 = r2 ^�jf� D �0p. (2) ��T�61�TÆ�. RO[: ? 3=2+1, wzi�? 2=1+1, �z> 3=1+1+1. �2�?' Ω GY��gW-e��{G xoy gW-e�>-ero Dxy�� Ω � W?��>ÆsWT��Æ6 z = z1(x, y), z = z2(x, y)�w2�?' Ω 0^�Æ6 Ω = {(x, y, z)|(x, y) ∈ Dxy, z1(x, y) ≤ z ≤ z2(x, y)}, Edl���Tro Ω ?�Æ�w��T0^�6 ∫∫∫ Ω f(x, y, z)dxdydz = ∫∫ Dxy dxdy ∫ z2(x,y) z1(x,y) f(x, y, z)dz. RON: ? 3=1+2, wzi�? 2=1+1, �z> 3=1+1+1. �2�?' ΩGY�� -e��{G z -e�>-er� [c, d]�Kx` z ∈ [c, d], gW z = z ! Ω >!3 Dz, 06 Ω ugW z = z T� (x, y) ?�GQ4�w2�?' Ω 0^�Æ6 Ω = {(x, y, z)|z ∈ [c, d], (x, y) ∈ Dz}, Edl�W�Tro Ω ?�Æ�w��T0^�6 ∫∫∫ Ω f(x, y, z)dxdydz = ∫ d c dz ∫∫ Dz f(x, y, z)dxdy. 0RO�fl!3Dz ��z_�Æ��{�qo x 2+y2 ≤ h(z)2�0qo x2 a2 + y 2 b2 ≤ h(z)2 �. �H1.2. % 3=2+1 o 3=1+2 h�;5� 2 �TK7pQ�u\���%/�E"ro (`�) qE"rtW?>&. S1.2. u\ ∫∫∫ Ω zdxdydz, ;5 Ω �D0 z = x2 + y2 e:0 z = 4 ^j0��z�a. 4 (3) C[:sD�T�6F�TÆ�. �2�sD L ?#�R,6 x = x(t), y = y(t), z = z(t), α ≤ t ≤ β, (1.3) wC[:sD�T�6F�T ∫ L f(x, y, z)ds = ∫ β α f [x(t), y(t), z(t)] √ x′(t)2 + y′(t)2 + z′(t)2dt, i�{(3T ds u�WsD L ?#�R,>6 ds = √ x′(t)2 + y′(t)2 + z′(t)2dt. (1.4) #�?�;��r� f(x, y, z) ≡ 1 �>sD L u��#�R,�Æ>?{(Æ�g 6 ∫ L ds = ∫ β α √ x′(t)2 + y′(t)2 + z′(t)2dt, 06+ CI ^��% P282?~ D. � ��YqIsDJ?(I ∆s,~iu x, y, z �-e(IT�6 ∆x,∆y,∆z, w ∆s = √ ∆x2 +∆y2 +∆z2. ~sD?#�R,h (1.3) d-�wh3Tk<� dx = x′(t)dt, dy = y′(t)dt, dz = z′(t)dt, w ∆x ≈ x′(t)∆t, ∆y ≈ y′(t)∆t, ∆z ≈ z′(t)∆t, i ∆s ≈ √ x′(t)2 + y′(t)2 + z′(t)2 ∆t, ~ (1.4) ?h6�0^���a. �H1.3. �;�BÆ�DzpQ�Dz�O}o^�< z = z(t), >-ero Dxy�wsW Σ �Æ6 z = z(x, y), (x, y) ∈ Dxy, l�C[:sW�T�6N��T∫∫ Σ f(x, y, z)dS = ∫∫ Dxy f [x, y, z(x, y)] √ 1 + z2x + z 2 ydxdy. ~�sW Σ G yoz gW-e>-ero Dyz�wsW Σ �Æ6 x = x(y, z), (y, z) ∈ Dyz, l�C[:sW�T�6N��T∫∫ Σ f(x, y, z)dS = ∫∫ Dyz f [x(y, z), y, z] √ 1 + x2y + x 2 zdydz. ~�sW Σ G zox gW-e>-ero Dzx�wsW Σ �Æ6 y = y(z, x), (z, x) ∈ Dzx, l�C[:sW�T�6N��T∫∫ Σ f(x, y, z)dS = ∫∫ Dzx f [x, y(z, x), z] √ 1 + y2z + y 2 xdzdx. #�?�~��r� f(x, y, z) ≡ 1, w~�sW Σ G xoy ��W-e�0>sW?W �Æ�g 6 (0u+ CI ^�>% P165-167iF=$�) SΣ = ∫∫ Σ dS = ∫∫ Dxy √ 1 + z2x + z 2 ydxdy, i� dS = √ 1 + z2x + z 2 ydxdy 6sW Σ : z = z(x, y) ?W�3T. � ��� � dxdy dS = 1√ 1 + z2x + z 2 y , �W� j�*�sW Σ : z = z(x, y) �D (x, y, z) /G�OGCEK z ?RGm� cos γ(� §2.2.1), i dxdy dS = cos γ � ∆x∆y ∆S ≈ cos γ, dS = dxdy cos γ � ∆S ≈ ∆x∆y cos γ . 6 h0*~=DqsWW�g ?h6�0^��<"��a. >W��B:sW�T �k<� �Z�{0 (��� 2.3(2)). :�?0>sWGi!��gW-e�?sW W�Æ�g (L). S1.5. u\ ∫∫ Σ (x+ y + z)dS, ;5 Σ �:0 x = 0, y = 0, z = 0, x+ y + z = 1 ^j�� a#0. S1.6. u\ ∫∫ Σ dS x2+y2 , ;5 Σ k":0 x 2 + y2 = R2, 0 ≤ z ≤ H. (8�k 2piH/R. Ju \ ∫∫ Σ dS x2+y2+z2 , &u\�)M�) §2 )1N t� \B5 §2.1 (0Mbs�3 §2.1.1 (0Mbs�3+} � L 62�� A D< B D?iGl|sD�r� f(x, y, z) u L �i#. u L �x `&|D M0(x0, y0, z0) = A,M1(x1, y1, z1), . . . ,Mn−1(xn−1, yn−1, zn−1),Mn(xn, yn, zn) = B, 2M� L ~T6 n bIiG{J M̂i−1Mi, i = 1, 2, . . . , n, nKd?�D{J6 −−−−−→ Mi−1Mi = (xi − xi−1, yi − yi−1, zi − zi−1), 0UJIiGsD{J M̂i−1Mi u x, y, z �?-eT�6 ∆xi = xi − xi−1, ∆yi = yi − yi−1, ∆zi = zi − zi−1, (2.1) (Y�`?�∆xi , ∆yi, ∆zi0}0Z�t,l L?RG). x`Ut (ξi, ηi, ζi) ∈ M̂i−1Mi, i = 1, 2, . . . , n, ��Tu n∑ i=1 f(ξi, ηi, ζi)∆xi. ~; λ = max 1≤i≤n {|−−−−−→Mi−1Mi|} → 0 �����Tu�C�4un�\7l L ?~TuD (ξi, ηi, ζi) ∈ M̂i−1Mi ?Ut�w*0�C6r� f(x, y, z) uiGsD L �.�� x %b s�3��6 ∫ L f(x, y, z)dx, ∫ L f(x, y, z)dx = lim λ→0 n∑ i=1 f(ξi, ηi, ζi)∆xi. (2.2) ~/K�Tu n∑ i=1 f(ξi, ηi, ζi)∆yi. 7 ~; λ = max 1≤i≤n {|−−−−−→Mi−1Mi|} → 0 �����Tu�C�4un�\7l L ?~TuD (ξi, ηi, ζi) ∈ M̂i−1Mi ?Ut�w*0�C6r� f(x, y, z) uiGsD L �K�� y %b s�3��6 ∫ L f(x, y, z)dy, ∫ L f(x, y, z)dy = lim λ→0 n∑ i=1 f(ξi, ηi, ζi)∆yi. (2.3) :�?�/K�Tu n∑ i=1 f(ξi, ηi, ζi)∆zi, wEdir� f(x, y, z) uiGsD L �.�� z %bs�3��6 ∫ L f(x, y, z)dz, ∫ L f(x, y, z)dz = lim λ→0 n∑ i=1 f(ξi, ηi, ζi)∆zi. (2.4) ^�b�T,*6.��%bs�3�(0Mbs�3. ''-B?�r� P (x, y, z), Q(x, y, z), R(x, y, z) FbuiGsD L �i#�~/K �Tu n∑ i=1 [P (ξi, ηi, ζi)∆xi +Q(ξi, ηi, ζi)∆yi +R(ξi, ηi, ζi)∆zi], wEdi>W?CN:sD�T ∫ L P (x, y, z)dx+Q(x, y, z)dy +R(x, y, z)dz, ∫ L P (x, y, z)dx+Q(x, y, z)dy +R(x, y, z)dz = lim λ→0 n∑ i=1 [P (ξi, ηi, ζi)∆xi +Q(ξi, ηi, ζi)∆yi +R(ξi, ηi, ζi)∆zi]. {XlWT�K x, y, z ?sD�TH��W?#�oN. �H2.1. (1) J L− #R��}Dz L O}{N��}Dz, & ∫ L f(x, y, z)dx = − ∫ L− f(x, y, z)dx, ∫ L f(x, y, z)dy = − ∫ L− f(x, y, z)dy, ∫ L f(x, y, z)dz = − ∫ L− f(x, y, z)dz. 51�;�> (2.1) 5!)� ∆xi,∆yi,∆zi SdV {Nr�-. �5BK�DzpQ �DzO}�^. (2) %L0�BK�DzpQE�5�yM-X�}iI M̂i−1Mi {�� x, y, z 8�e� ∆xi,∆yi,∆zi GT,0, n.���-!�DzpQ.z�^r. M-X� }iI M̂i−1Mi �i.k ∆si (;�5k,0), (ξi, ηi, ζi) ∈ M̂i−1Mi 4�Dz�=}"−→ T (ξi, ηi, ζi) {J� x, y, z 8�O}�vk cosαi, cosβi, cos γi, &H�?> ∆xi = ∆xi ∆si ∆si ≈ cosαi∆si, 8 ∆yi = ∆yi ∆si ∆si ≈ cosβi∆si, ∆zi = ∆zi ∆si ∆si ≈ cos γi∆si, &!�DzpQ�pQe�^r n∑ i=1 f(ξi, ηi, ζi)∆xi ≈ n∑ i=1 f(ξi, ηi, ζi) cosαi∆si, n∑ i=1 f(ξi, ηi, ζi)∆yi ≈ n∑ i=1 f(ξi, ηi, ζi) cosβi∆si, n∑ i=1 f(ξi, ηi, ζi)∆zi ≈ n∑ i=1 f(ξi, ηi, ζi) cos γi∆si. (2.5) ��\Q (1.1) � (2.2),(2.3),(2.4), &?>BK�DzpQ�BÆ�DzpQ.z�^ r: ∫ L f(x, y, z)dx = ∫ L f(x, y, z) cosαds, ∫ L f(x, y, z)dy = ∫ L f(x, y, z) cosβds, ∫ L f(x, y, z)dz = ∫ L f(x, y, z) cosγds, (2.6) ?g<�r ∫ L Pdx+Qdy +Rdz = ∫ L [P cosα+Q cosβds+R cos γ]ds, ;5 cosα, cos β, cos γ Q$kDz L LC (x, y, z) 4=}" −→ T (x, y, z) {J� x, y, z 8� O}�v. v�OQ��� dx ds = cosα, dy ds = cosβ, dz ds = cos γ, ℄ dx = cosαds, dy = cosβds, dz = cos γds. J�Æ�?��r-X�}iI M̂i−1Mi {�� x, y, z 8�e� ∆xi,∆yi,∆zi % ÆE,0, ��:+: �ubs L fZ*"%auVnd2uy�bs L %2uz . *�L0^�!�Dz.z^r�g=e\QGT0��. J�}Dz L L-C4�= }"^EO}�Dz L �O}%Æ3�r{N�&!�DzpQ�LW\Q β, α < β, &�Dz L �O}Æ3�DzL t J�C4:l=}"Ek −→ T (t) = ( x′(t), y′(t), z′(t) ) √ x′(t)2 + y′(t)2 + z′(t)2 , �T!�DzpQ���\Q (2.6) β, α > β, &�Dz L �O}Æ3�DzL t J�C4:l=}"Ek −→ T (t) = − ( x′(t), y′(t), z′(t) ) √ x′(t)2 + y′(t)2 + z′(t)2 , �T!�DzpQ���\Q (2.6) sQqt %5�3)�5pQe�qyTBÆ�DzpQ. }�5���n.o�$ � BK�DzpQ�1T3�xP������n.lT�AQ (2.5) B �pQe2 ~3���BK�DzpQ. *CJt0�BK�D0pQ TI5. S2.1. M L kDz x = t, y = t2, z = t3 L6 t = 0 !> t = 1 ��}DziI��JE "�DzpQ ∫ L Pdx+Qdy +Rdz kkJi.�DzpQ. "&T:: hl t 2 0 �< 1�itEdl℄dsD#�R,? t D/?97mGC 6 −→ T (t) = ( x′(t), y′(t), z′(t) ) √ x′(t)2 + y′(t)2 + z′(t)2 = ( 1, 2t, 3t2 ) √ 1 + 4t2 + 9t4 , �`< t = x, t2 = y, t3 = z, w��mGC�6sD�D (x, y, z) /97mGC6 −→ T (x, y, z) = ( 1, 2x, 3y ) √ 1 + 4x2 + 9y2 . (2.7) l�t cosα = 1√ 1 + 4x2 + 9y2 , cosβ = 2x√ 1 + 4x2 + 9y2 , cos γ = 3y√ 1 + 4x2 + 9y2 , i>% P98 iF= $�. �sWh Σ : F (x, y, z) = 0 d- (~hi!N d-HY�6��R,), wu Σ �D (x, y, z) /?[bOGC6 −→n (x, y, z) = ( Fx, Fy , Fz ) . ! Kd?97GC6 ( Fx, Fy, Fz ) √ F 2x + F 2 y + F 2 z . i0OGCn x, y, z }G��?RGm�6 cosα = Fx√ F 2x + F 2 y + F 2 z , cosβ = Fy√ F 2x + F 2 y + F 2 z , cos γ = Fz√ F 2x + F 2 y + F 2 z . 11 8V"N?��$sW� [bsW+iB$�fsW�?UD/?OGC −→n 6r T. KU�sW Σ�T61$n_$�1$��OGC)1?[$�_$��OGC)_ ?[$, [�t1$6}G. K�U�sW Σ, T>WoN� (1) Σ : z = z(x, y), (x, y) ∈ Dxy, T6�$n>$�� (>) $��OGC)� (>) ? [$. [�t�$6}G�0$?OGC −→n = (−zx,−zy, 1) n z }G?�� γ 6}�� i −→n Edl z ?RGm� cos γ = 1√ 1 + z2x + z 2 y > 0. (2) Σ : x = x(y, z), (y, z) ∈ Dyz, T6l$nz$�l (z) $��OGC)l (z) ? [$. [�tl$6}G�0$?OGC −→n = (1,−xy,−xz) n x }G?�� α 6} ��i −→n Edl x ?RGm� cosα = 1√ 1 + x2y + x 2 z > 0. (3) Σ : y = y(z, x), (z, x) ∈ Dzx, T6�$nj$�� (j) $��OGC)� (j) ? [$. [�tj$6}G�0$?OGC −→n = (−yz, 1,−yx) n y }G?�� β 6}�� i −→n Edl y ?RGm� cosβ = 1√ 1 + y2z + y 2 x > 0. [b�$sW~tFD$�w*6�ub[. S2.3. M Σ k�<7p0 z = 12 (x 2 + y2) �� z = 0 e z = 2 .z�'Q�Et+. U BD0 Σ LC (x, y, z) 4L}"{J� x, y, z 8�O}�v. "&T:: h�W?oN (1) n�`$�i_>6 6 cosα = x√ x2 + y2 + 1 , cosβ = y√ x2 + y2 + 1 , cos γ = −1√ x2 + y2 + 1 . §2.2.2 (0Mb[�3+} >Wd-CN:sW�T?Fb. 2Fb�8V0^.-0:�TnsW?RGi k�Mn2Fb�P�0^.-nC[:sW�T?k<. 8V?/KiGsW Σ `� [bA R, (8V#E5wQO1TXbX#7Q) �Æ-6?�T?Fb. � Σ : z = z(x, y) 6iGsW�r� f(x, y, z) Fbu Σ �i#. Bx`� Σ ~T6 n bI4 Si, i = 1, 2, . . . , n, UI4W��6 ∆Si, MU4 Si u xoy W�?-ero6 (σi)xy, iW��6 ∆(σi)xy. x`t (ξi, ηi, ζi) ∈ Si, /K�Tu n∑ i=1 f(ξi, ηi, ζi)∆(Si)xy, 12 i� ∆(Si)xy =   ∆(σi)xy, Σ t�$; −∆(σi)xy, Σ t>$. ~; λ = max1≤i≤n{Si ?�(} → 0 ���u ?�C4un�\7l Σ ?~TuD (ξi, ηi, ζi) ∈ Si ?Ut�w*0�C6r� f(x, y, z) uiGsW Σ �.�� x, y %b[ �3, �� ∫∫ Σ f(x, y, z)dxdy, ∫∫ Σ f(x, y, z)dxdy = lim λ→0 n∑ i=1 f(ξi, ηi, ζi)∆(Si)xy, �+'B�; Σ t�$�� ∫∫ Σ f(x, y, z)dxdy = lim λ→0 n∑ i=1 f(ξi, ηi, ζi)∆(σi)xy, ; Σ t>$�� ∫∫ Σ f(x, y, z)dxdy = lim λ→0 n∑ i=1 f(ξi, ηi, ζi)(−∆(σi)xy). � Σ : y = y(z, x) 6iGsW�r� f(x, y, z) Fbu Σ �i#. Bx`� Σ ~T6 n bI4 Si, i = 1, 2, . . . , n, UI4W��6 ∆Si, MU4 Si u zox W�?-ero6 (σi)zx, iW��6 ∆(σi)zx. x`t (ξi, ηi, ζi) ∈ Si, /K�Tu n∑ i=1 f(ξi, ηi, ζi)∆(Si)zx, i� ∆(Si)zx =   ∆(σi)zx, Σ tj$; −∆(σi)zx, Σ t�$. ~; λ = max1≤i≤n{Si ?�(} → 0 ���u ?�C4un�\7l Σ ?~TuD (ξi, ηi, ζi) ∈ Si ?Ut�w*0�C6r� f(x, y, z) uiGsW Σ �.�� z, x %b[ �3, �� ∫∫ Σ f(x, y, z)dzdx, ∫∫ Σ f(x, y, z)dzdx = lim λ→0 n∑ i=1 f(ξi, ηi, ζi)∆(Si)zx, �+'B�; Σ tj$�� ∫∫ Σ f(x, y, z)dzdx = lim λ→0 n∑ i=1 f(ξi, ηi, ζi)∆(σi)zx, ; Σ t�$�� ∫∫ Σ f(x, y, z)dzdx = lim λ→0 n∑ i=1 f(ξi, ηi, ζi)(−∆(σi)zx). 13 � Σ : x = x(y, z) 6iGsW�r� f(x, y, z) Fbu Σ �i#. Bx`� Σ ~T6 n bI4 Si, i = 1, 2, . . . , n, UI4W��6 ∆Si, MU4 Si u yoz W�?-ero6 (σi)yz, iW��6 ∆(σi)yz. x`t (ξi, ηi, ζi) ∈ Si, /K�Tu n∑ i=1 f(ξi, ηi, ζi)∆(Si)yz, i� ∆(Si)yz =   ∆(σi)yz , Σ tl$; −∆(σi)yz, Σ tz$. ~; λ = max1≤i≤n{Si ?�(} → 0 ���u ?�C4un�\7l Σ ?~TuD (ξi, ηi, ζi) ∈ Si ?Ut�w*0�C6r� f(x, y, z) uiGsW Σ �.�� y, z %b[ �3, �� ∫∫ Σ f(x, y, z)dydz, ∫∫ Σ f(x, y, z)dydz = lim λ→0 n∑ i=1 f(ξi, ηi, ζi)∆(Si)yz , �+'B�; Σ tl$�� ∫∫ Σ f(x, y, z)dydz = lim λ→0 n∑ i=1 f(ξi, ηi, ζi)∆(σi)yz, ; Σ tz$�� ∫∫ Σ f(x, y, z)dydz = lim λ→0 n∑ i=1 f(ξi, ηi, ζi)(−∆(σi)yz). ^���T,*6.��%b[�3�(0Mb[�3. QY�`?���WFb?CN:sW�T�U�oNhlsW Σ �hA R, z = z(x, y) � y = y(z, x) � x = x(y, z) �Æ� 0�Tuib#D�*�u �? ∆(Si)xy � ∆(Si)zx � ∆(Si)yz H�+t. MKl[�?iGsW Σ���[F`hY[ bA R,�Æ�0��Y�sW Σ T`6iCbsW�Ub`hA R,�Æ�Mn Ub?RGhpsW Σ ,F. y0RO�8V*0^�CN:sW�T?Fb/m< [�?iGsW��|=_$z��ub[f%(0Mb[�3-h� #��. Æ� �Z[X��sWT-6�Æ*0^D. ''-B?�r� P (x, y, z), Q(x, y, z), R(x, y, z)FbuiGsW Σ �?CN:sW �T ∫∫ Σ Pdydz +Qdzdx+Rdxdy. �H2.3. (1) H��pQE��3�J Σ− #R��}D0 Σ E{N+��}D0�& ∫∫ Σ− Pdydz +Qdzdx+Rdxdy = − ∫∫ Σ Pdydz +Qdzdx+Rdxdy. (2) !�D0pQT�^r�, t0�JE" x, y �D0pQk�x�Y1. �n. �3_7��>JE" x, y �D0pQE�4� (t03x�vdG%4�). uyM� 14 }D0 Σ : z = z(x, y) EL+�&pQek n∑ i=1 f(ξi, ηi, ζi)∆(Si)xy = n∑ i=1 f(ξi, ηi, ζi)∆(σi)xy. M%C (ξi, ηi, ζi) ∈ Σi 4�L}" −→n (ξi, ηi, ζi){�� z 8,}w��O}�vk cos γi(5 kPT0�kO+?), &� ∆(σi)xy = ∆(σi)xy ∆Si ∆Si ≈ cos γi∆Si, �T!�pQ�pQe�^r n∑ i=1 f(ξi, ηi, ζi)∆(Si)xy ≈ n∑ i=1 f(ξi, ηi, ζi) cos γi∆Si. (2.9) < λ→ 0 NJ(!�pQ�E��r�∫∫ Σ f(x, y, z)dxdy = ∫∫ Σ f(x, y, z) cosγdS, (2.10) ;5 cos γ kD0 Σ LC (x, y, z) 4L}" −→n (x, y, z) {�� z 8�O}�v. J�}D 0 Σ : z = z(x, y) Et+�&^r (2.10) G0�. SPL�$ cos γ kD0 Σ LC (x, y, z) 4L}" −→n (x, y, z) {�� z 8�O}�v (;�5kP,0), & Σ− EL+>%D0 Σ− LC (x, y, z) 4L}" −→n (x, y, z)− {�� z 8�O}�vk − cosγ��TJ Σ− � \Q (2.10) �? ∫∫ Σ− f(x, y, z)dxdy = ∫∫ Σ− f(x, y, z)(− cosγ)dS, �5 ∫∫ Σ f(x, y, z)dxdy = − ∫∫ Σ− f(x, y, z)dxdy = − ∫∫ Σ− f(x, y, z)(− cosγ)dS = ∫∫ Σ− f(x, y, z) cosγdS = ∫∫ Σ f(x, y, z) cosγdS. ;�L0AhÆ&T�kBÆ�D0pQ�D0O}o^. #i!XBK�D0pQ�BÆ�D0pQ�^rg=�Z. ℄n.� ∫∫ Σ P (x, y, z)dydz = ∫∫ Σ P (x, y, z) cosαdS, ∫∫ Σ Q(x, y, z)dzdx = ∫∫ Σ Q(x, y, z) cosβdS, ∫∫ Σ R(x, y, z)dxdy = ∫∫ Σ R(x, y, z) cosγdS, (2.11) ;5 cosα, cosβ, cos γ kD0 Σ LC (x, y, z) 4L}" −→n (x, y, z) Q${�� x, y, z 8� O}�v. ?g<�k∫∫ Σ Pdydz +Qdzdx+Rdxdy = ∫∫ Σ [ P cosα+Q cosβ +R cos γ ] dS. 15 v�OQ: �� dydz dS = cosα, dzdx dS = cosβ, dxdy dS = cos γ, ℄ dydz = cosαdS, dzdx = cosβdS, dxdy = cos γdS. (2.12) (3) BK�D0pQ�p���TmE%a%2[Ht:lNz a2�zD0� %" (|~dtB&�WXt,�( P221-222)�u\5%"N?>AQ (2.9) � �p Qe�5pQe�qyTBÆ�D0pQ. }�5���n.o�$� BK�D0p Q�1T3�xP������n.lT�AQ (2.9) B �pQe2~3���BK� D0pQ. S2.4. ��!�D0pQ�^ru\ ∫∫ Σ(x − y)dydz + ydzdx − xdxdy, ;5 Σ k:0 x+ y + z = 1 %BÆFy�'Q�EL+. &z��KlsW6gWoN�hli�?OGC�'C�i0^/K�6C[: sW�TÆ�. S2.5. �D0pQ ∫∫ Σ f(x, y, z)dydz 2.3 ℄q>. i dydz = cosα cosγ dxdy = (−x)dxdy, l�><�&?6 6∫∫ Σ f(x, y, z)dydz = ∫∫ Σ f(x, y, z)(−x)dxdy. (2.13) §2.2.3 (0Mb[�3Ek ~iGsW Σ : z = z(x, y) t�$, G xoy W-e>-ero6 Dxy, wCN:sW �T6N��T ∫∫ Σ f(x, y, z)dxdy = ∫∫ Dxy f [x, y, z(x, y)]dxdy. ~sW Σ t>$, wh�� 2.3(1) 0~��� j��Zt. ~iGsW Σ : y = y(z, x) tj$, G zox W-e>-ero6 Dzx, wCN:sW �T6N��T ∫∫ Σ f(x, y, z)dzdx = ∫∫ Dzx f [x, y(z, x), z)]dzdx. 16 ~sW Σ t�$, wh�� 2.3(1) 0~��� j��Zt. ~iGsW Σ : x = x(y, z) tl$, G yoz W-e>-ero6 Dyz, wCN:sW �T6N��T ∫∫ Σ f(x, y, z)dydz = ∫∫ Dyz f [x(y, z), y, z)]dydz. ~sW Σ tz$, wh�� 2.3(1) 0~��� j��Zt. #�?�~sW Σ GY��gW-e�>E+?-ero Dyz, l�∫∫ Σ f(x, y, z)dydz = ∫∫ Σ1 f(x, y, z)dydz + ∫∫ Σ2 f(x, y, z)dydz = ∫∫ Dyz f( √ 2z − y2, y, z)dydz +(−1) ∫∫ Dyz f(− √ 2z − y2, y, z)dydz. ;wi�~+iK*P�wÆ��=fK*P�Æ� (F� §4 $�). : (2.13)_�8VG1[��J�0^*p�q (2.13)� j�?sW�T! Æ� 0�T?s/�)�_�8V?sW Σ �5 z = 12 (x 2 + y2) �QYt��� � Σ G xoy W-e>-ero Dxy = {(x, y)| x2 + y2 ≤ 4}, l� (2.13) � j�?sW�T�6N��T (�`$) 6∫∫ Σ f(x, y, z)(−x)dxdy = − ∫∫ Dxy f [x, y, 1 2 (x2 + y2)](−x)dxdy, l�+X0^�- ∫∫ Σ f(x, y, z)dydz ?�. F=>�0#�^� (+ CI ) >% P228 > 3. &z�: �&d8V%fD�F�?Æ�CN:sW�T?RO. ;wUf�SRO "Y.+'&[�pw�Uf?RO0^�&[`�z_�- 0 —– {QYLA!� �9k��?L>. S2.7. u\ ∫∫ Σ(x−y)dxdy+x(y−z)dydz, ;5 Σ�":0 x2+y2 = 1e:0 z = 0, z = 3 ^j�a Ω �#0�Ei+. S2.8. u\ ∫∫ Σ xydydz + yzdzdx+ xzdxdy, ;5 Σ k:0 x + y + z = 1, x, y, z ≥ 0, E L+. (/�Bo��� 2.4 �OL. 8�k 1/8) 17 §3 7�B5�J'Ur 8V℄'~=B:sD�Tik<�B:sW�TZik<� 0VD{SL?� T�� �9S*�VD��T���T�D�TuW�T. M>W?b Xg w k�?kD{��T�?k<. l�2Y�`b���8V*VD[��T�*�; R�T! �V