?= E=cò?sW￥ ó"D? ?? x1ò?sW￥ ó" 1.?¨ì  T9 ?/ s (1) HL xy2dy ?x2ydx ?L1??x2a2 + y2b2 = 1 |?_ (2) HL (x+y)dx+(x?y)dyL](1) (3) HL (x+y)2dx?(x2 +y2)dyL ^??1A(1;1);B(3;2);C(2;5)￥ ? ??￥H? |?_ (4) HL (x2 +y3)dx?(x3 ?y2)dyL1x2 +y2 = 1 |?_ (5) HL ey sinxdx+e?x sinydyL1 ?a ? x ? b;c ? y ? d￥H? | ?_ (6) HL exy [(ysinxy +cos(x+y))dx+(xsinxy +cos(x+y))dy] ?L ^ ?i? ;á> wL 2. ?¨ì  T9 ?/  wL ?1m?￥ ? (1) ? gLr2 = a2 cos2  (2)2 58=?Lx3 +y3 = 3axy(a > 0) (3) x = a(1+cos2 t)sint;y = asin2 tcost;0 ? t ? 2…: 3. ?¨ú ? T p/ s (1) RR S x2dydz +y2dzdx+z2dxdy ? (a) S1 ?Z80 ? x;y;z ? a￥H? w ??§ (b) S1 ?x2 +y2 = z2(0 ? z ? h)/§ 1 (2) RR S x3dydz +y3dzdx+z3dxdy ?S ^?ê o ?￥?§ (3) !S ^ ? o ?z = pa2 ?x2 ?y2￥ § p (a) RR S xdydz +ydzdx+zdxdy (b) RR S xz2dydz +?x2y ?z2￠dzdx+?2xy +y2z￠dxdy (4)RR S ?x?y2 +z2￠dydz +?y ?z2 +x2￠dzdx+?z ?x2 +y2￠dxdyS ^(x?a)2+ (y ?b)2 +(z ?c)2 = R2￥?§ 4.¨ ?? X ? T9 ?/ s (1) HL x2y3dx+dy +zdz ? (a) L1??x2 +y2 = a2;z = 0Z_ ^ I H? (b) L1y2 + z2 = 1;x = y ??￥??Vxà?_ A ?? I H?Z _ (2) HL (y ?z)dx+(z ?x)dy +(x?y)dzL ^V(a;0;0)ü(0;a;0) à(0;0;a)í?(a;0;0)￥ ??? (3) HL?y2 +z2￠dx+?x2 +z2￠dy +?x2 +y2￠dz ? (a) L1x + y + z = 1D ?USà￥?L Z_D ?? ü ? u× § ?· mE5 (b) L ^ wLx2 + y2 + z2 = 2Rx;x2 + y2 = 2rx(0 < r < R;z > 0) ?￥ Z_D ?? w ?￥ §?· mE5 (4) HL ydx+zdy +xdzL ^x2 + y2 + z2 = a2;x + y + z = 0Vxà? _ A ??? ^ I H?Z_ 2 5. !L1 ü ? > wLl1 ü ?  ?iZ_￡ ü I L cos(n,l)ds = 0 ?n ^L￥?ELZ_ 6. !S ^> w ?l1 ?i%?Z_￡ ü ZZ S cos(n;l)dS = 0 7. pI = HL [xcos(n;x)+ycos(n;y)]dsL1?μ? u×D￥;á> wLn1L￥?E_ 8￡ üú ?sI L cos(r;n) r ds = 0; ?L ^ ü ? B? ?Y u× ￥H?7r ^L B?? ? B?? ￥  ?n ^L￥?ELZ_? ?rV UL B?? = B??￥  ?5??s-′??2…: 99 ?ú ?sZZ S cos(r;n) r2 dS; ?S1e?>;á w ?n1 w ?S ?(?;·;?))￥?E_r = (? ?x)i+(· ?y)j+(? ?z)k;r = jrj k/  ? f?é?) ? (1) w ?S?￥ u×?c(x;y;z)? (2) w ?S?￥ u×c(x;y;z)? 10 p￡ZZZ V dxdydz r = 1 2 ZZ S cos(r;n)dS; ?S ^?V￥s ?;á> w ?n1S￥?ELZ_r = (x;y;z);r = jrj .s/  ? f?ú?) ? 3 (1) V??ce?(0;0;0) (2) V?ce?(0;0;0) H 7 ZZZ V dxdydz r = lim"!0+ ZZZ V?V" dxdydz r ; ?V" ^[e?1?[varepsilon1??￥ o 11 ?¨ú ? TMD[/s (1) RR S xydxdy +xzdzdx+yzdydz; (2) RR S (@u@x cosfi + @u@y cosfl + @u@z cos ) ?cosfi;cosfl;cos ^ w ?￥? ELZ_?? 12 !u(x;y);v(x;y) ^ μ=¨ ?? ê? ?￥f ?i ! ￠u = @ 2u @x2 + @2u @y2 ￡ ü (1) RR S ￠udxdy = Rl @u@nds; (2) RR v￠udxdy = RR (@u@x @v@x + @u@y @v@x)dxdy +Hl v@u@n; (3) RR (u￠v ?v￠u)dxdy = ?Hl(v@u@n ?u@v@n)ds ? 1> wLl ??￥ ü ? u×@u@n; @v@n1l?EL￥Z_? ? 13 !￠u = @2u@x2 + @2u@y2 + @2u@z2, S ^V￥H? w ?￡ ü (1) RRR V ￠udxdydz = RR S @u @n (2) RR S u@u@n = RRR V [(@u@x)2 +(@u@y)2 +(@u@z)2]+RRR V u￠udxdydz T?uV# H? w ?S μ ??￥=¨ ê? ?@u@n1 w ?S￥?EL￥Z_? ? 4 149 ?/  w ?s (1)RR S (x2?y2)dydz+(y2?z2)dzdx+2z(y?x)dxdy ?S ^x2a2 +y2b2 +z2c2 = 1(z ? 0)/§ (2) RR S (x+cosy)dydz +(y +cosz)dzdx+(z +cosx)dxdy;S ^ ?8?￥H ? ?7 ?8??x+y +z = 1? ?US ??? (3) RR S F￠ndS ?F = x3i+y3j+z3k;n ^S￥?E_S1x2+y2+ z2 = a2 (z ? 0) § (4) RR S (x3a2 +yz)dydz+(y3b2 +z3x2)dzdx+(z3c2 +x3y3)dxdy;S ^x2a2 + y2b2 + z2c2 = 1(x ? 0)a§ 15￡ ü? w ?S ??￥8??V = 13 RR S (xcosfi + ycosfl + zcos )dS T?cosfi;cosfl;cos 1 w ?S￥?EL￥Z_?? 16 ?L ^ ü ?xcosfi + ycosfl + zcos ?p = 0 ￥> wL ? ? ? u×￥ ?1S p I L flfl flfl flfl flfl fl dx dy dz cosfi cosfl cos x y z flfl flfl flfl flfl fl ; ?LG?_é? 17 !P;Q;Rμ ?? ê? ? O ?i;á> w ?SμRR S Pdydz + Qdzdx+Rdxdy = 0;￡ ü@P@x + @Q@y + @R@z = 0: 18 !P(x;y);Q(x;y) ? ü ? μ ?? ê? ?7 O[ ?i ?(x0;y0)1??[ ?i? ?r1??￥ ??l : x = x0 + rcos ;y = y0 +rsin (0 ? ? …)?μ Z l P(x;y)dx+Q(x;y)dy = 0; 5 p￡P(x;y) = 0; @Q@x = 0: x2sD ^?í1 1.￡/ sD ^?í1i p ? ì￥′ (1) R(1;1)(0;0) (x?y)(dx?dy) (2) R(1;2)(2;1) ydx?xdyx2·? ü ?￥ ^? (3) R(6;8)(1;0) xdx+ydyx2+y2?YVe?￥ ^? (4) R(a;b)(0;0) f (x+y)(dx+dy) T?f(u) ^ ??f ? (5) R(1;2)(2;1) ’(x)dx+?(y)dy ?’?1 ??f ? (6) R(6;1;1)(1;2;3) yzdx+xzdy +xydz (7) R(2;3;?4)(1;1;1) xdx+y2dy ?z3dz (8) R(x2;y2;z2)(x 1;y1;z1) xdx+ydy+zdzp x2+y2+z2 ?(x1;y1;z1)(x2;y2;z2) o ?x 2 + y2 + z2 = a2  2 p/  ?±s￥ef ? (1) ?x2 +2xy ?y2￠dx+?x2 ?2xy ?y2￠dy (2) ?2xcosy ?y2 sinx￠dx+?2ycosx?x2 siny￠dy (3) azdx+ bzdy + ?by?axz2 dz (4) ?x2 ?2yz￠dx+?y2 ?2xz￠dy +?z2 ?2xy￠dz (5) ?ex siny +2xy2￠dx+?ex cosy +2x2y￠dy (6) h x (x2?y2)2 ? 1 x +2x 2 i dx+ h 1 y ? y (x2?y2)2 +3y 3 i dy +5z3dz 6 3f ?F (x;y)? ?@ I 1Hq? ? P±s TF (x;y)(xdx+ydy) ^ ? ±s 4￡ Pdx+Qdy = 12 xdy ?ydxAx2 +2Bxy +Cy2 a?Hq@P@y = @Q@x ?ABC1è ?AC ?B2 > 0 p ?(0;0)￥ ?ìè ? 5 pI = HL xdx+ydyx2+y2 ?L ^?üVe?￥e?> wL |?_ !L??￥ u×1D (1) D?ce? (2) Dce?  =? 6 p I = I L " y (x?2)2 +y2 + y (x+2)2 +y2 # dx + " 2?x (2?x)2 +y2 + ?(2+x) (2+x)2 +y2 # dy ?L ^?üV(?2;0)?(2;0)?￥e?> wL 7 !u(x;y)? ?Y u×D μ=¨ ?? ê? ?￡ üu(x;y)D = μ @2u @x2 + @2u @y2 = 0 ￥ 1Hq ^D = ?Be?;á> wLL?μ I L @u @nds = 0 ?@u@n1L?EL￥Z_? ? 89 ?s I = Z L (x+y)dx+(x?y)dy x2 +y2 7 ?L ^V?A(?1;0)?B(1;0)￥BH?YVe?￥;á wL ?￥Z? ^ y = f (x) (?1 ? x ? 1) 99 ?s I = Z L xln?x2 +y2 ?1￠dx+yln?x2 +y2 ?1￠dy ?L ^$f ?￥?l× =V?(2;0)à(0;2)￥? ;á wL x3? ?? 1 pu = x2 + 2y2 + 3z2 + 2xy ? 4x + 2y ? 4z?O(0;0;0), A(1;1;1), B(?1;?1;?1)￥0i p01 ,￥? 29 ?/ _ ?F￥ ??è (1) F = (y2 +z2;z2 +x2;x2 +y2) (2) F = (x2yz;xy2z;xyz2) (3) F = ( xyz; yzx; zxy) 3￡ üF = (yz(2x+y+z);xz(x+2y+z);xy(x+y+2z)) ^μ ]?i p ]f ? 4 !P = x2 +5?y +3yz;Q = 5x+3?xz ?2;R = (?+2)xy ?4z (1)9 ?R L Pdx+Qdy +Rdz ?L ^ ?èLx = acost;y = asint;z = ct(0 ? t ? 2…) (2) !F = (P;Q;R) protF (3) I 1Hq/F1μ ]?i p ]f ? 8 5 !’1 V±f ?r = (x;y;z), r = jrj pgrad’(r)div (’(r)r) rot(’(r)r) 6 p_ ?F = (?y;x;z) wLL￥ì @  (1) L1Oxy ü ? ￥??x2 +y2 = 1z = 0 I H?Z_ (2) L1Oxy ü ? ￥??(x?2)2 +y2 = R2z = 0 I H?Z_ (3) L1Oxy ü ?  ?B? ;áe?> wL ???￥ ü ? u×D￥ ?1S￡ üFL￥ì @ 12S (4) !μB ü ?… : ax + by + cz = d(c 6= 0) |…1 §… μB? ;áe?> wLL Z_1?…1?_L??￥ ü ? u×￥ ?1S ùFL￥ì @  ^ I 1$ 7 p_ ?F = grad(arctan yx) wLL￥ì @  (1) L?ì ?zà (2) Lì ?zàB ? (3) Lì ?zàn ? 8 !_ ?F = fP;Q;Rg"e?(0;0;0)?μ ??￥ ê? ? o ?x2 +y2 +z2 = t2 F￥é ?B%?′F￥Z_D O?r = (x;y;z)M ]7 OF￥ ??1 ,￡ üN_ ?1F = kr3r k ^è ? 9 !μB ? ?u = (x;y;z)"(0,0,0)??μ ?? ê? ? ?′ ? ^[e?1??￥ o ?? ? ?￥0?￥ ?1 ,￡ üN ? ? Dc1r (r = px2 +y2 +z2)?μB?è ? ?c11 %?è ? 10 !G ^ bW 7 u×u = (x;y;z)G μ=¨ ?? ê? ?￡ üu = (x;y;z)G =??￥ 1Hq ^G = ?ie?s ?;á w ?S? 9 μZZ s @u @ndS = 0 10