µÚ¶þÊ®¶þÕ ÇúÏß»ý·ÖÓëÇúÃæ»ý·Ö
P.361 µÚÒ»ÐÍÇúÏß»ý·ÖÓëµÚÒ»ÐÍÇúÃæ»ý·Ö 1. ¼ÆËãÏÂÁеÚÒ»ÐÍÇúÏß»ý·Ö£º £¨1£©£¨2£©
?(x?y)ds,ÆäÖÐLÊÇÒÔO(0,0),A(1,0),B(0,1)Ϊ¶¥µãµÄÈý½ÇÐΣ»
L?(xL2?y)ds£¬ÆäÖÐLÊÇÒÔÔµãΪÖÐÐÄ£¬RΪ°ë¾¶µÄÓÒ°ëÔ²ÖÜ£»
212x2y2£¨3£©?xyds£¬ÆäÖÐLΪÍÖÔ²2?2?1ÔÚµÚÒ»ÏóÏÞÖеIJ¿·Ö£»
Lab£¨4£©
??Lyds£¬ÆäÖÐLΪµ¥Î»Ô²x2?y2?1£»
222£¨5£©(x?y?z)ds£¬ÆäÖÐLΪÂÝÐýÏßx?acost,y?asint,z?bt(0?t?2?)µÄÒ»
L¶Î£» £¨6£©£¨7£©
??Lxyzds£¬ÆäÖÐLΪÇúÏßx?t,y?212t2,z?t2(0?t?1)µÄÒ»¶Î£» 32L2y2?z2ds,ÆäÖÐLÊÇx2?y2?z2?a2Óëx?yÏཻµÄÔ²ÖÜ£®
2. ÇóÇúÏßx?a,y?at,z?12at(0?t?1,a?0)µÄÖÊÁ¿£®ÉèÆäÏßÃܶÈΪ??22z. a3. Çó°ÚÏß??x?a(t?sint)(0?t??)µÄÖØÐÄ£¬ÉèÆäÖÊÁ¿·Ö²¼ÊǾùÔȵģ®
?y?a(1?cost)4. ¼ÆËãÏÂÁеÚÒ»ÀàÐÍÇúÃæ»ý·Ö£º £¨£±£©
2222x?y?z?a,z?0£» ,ÆäÖÐÊÇÉϰëÔ²Ãæ(x?y?z)dSS??S£¨£²£©
2222£¬ÆäÖÐΪÁ¢Ìå(x?y)dSx?y?z?1µÄ±ß½çÇúÃæ£» S??S£¨£³£©£¨£´£©
dS222£¬x?y?RÆäÖÐΪÖùÃæ±»Æ½Ãæz?0,z?HËù½ØÈ¡µÄ²¿·Ö£» S22??Sx?y??xyzdS£¬ÆäÖÐSÎªÆ½Ãæx?y?z?1ÔÚµÚÒ»ØÔÏÞÖеIJ¿·Ö£»
S5. ÈôÇúÏßÒÔ¼«×ø±ê???(?)(?1????2)±íʾ£¬ÊÔ¸ø³ö¼ÆËã
¹«Ê½¼ÆËãÏÂÁÐÇúÏß»ý·Ö£º £¨£±£©eL?Lf(x,y)dsµÄ¹«Ê½£¬²¢ÓôË
?x2?y2ds£¬ÆäÖÐLΪÇúÏß??a(????4)µÄÒ»¶Î£»
k?£¨£²£©xds£¬ÆäÖÐLΪ¶ÔÊýÂÝÏß??ae(k?0)ÔÚÔ²r?aÄڵIJ¿·Ö£®
?L6. ÉèÓÐÒ»ÖÊÁ¿·Ö²¼²»¾ùÔȵİëÔ²»¡x?rcos?,y?rsin?(0????)£¬ÆäÏßÃܶÈ??a?
word
£¨aΪ³£Êý£©£¬ÇóËü¶ÔÔµã(0,0)´¦ÖÊÁ¿ÎªmµÄÖʵãµÄÒýÁ¦£®
7. Ö¤Ã÷£ºÈôº¯ÊýfÔڹ⻬ÇúÏßL:x?x(t),y?y(t),t?[?,?]ÉÏÁ¬Ðø£¬Ôò´æÔÚµã
(x0,y0)?L£¬Ê¹µÃ?f(x,y)dS?f(x0,y0)?L£¬ÆäÖÐ?LΪLµÄ³¤£®
L8. ¼ÆËã
2z??dS£¬ÆäÖÐSΪԲ׶±íÃæµÄÒ»²¿·Ö£º S?x?rcos?sin??0?r?a,? S:?y?rsin?sin?;D:?
?0???2?,?z?rcos??ÕâÀï?Ϊ³£Êý(0????2).
P.371 µÚ¶þÐÍÇúÏß»ý·Ö
1. ¼ÆËãµÚ¶þÐÍÇúÏß»ý·Ö£º £¨£±£©
?xdy?ydx£¬ÆäÖÐLΪ±¾½ÚÀý£²ÖеÄÈýÖÖÇéÐΣ®
LL£¨£²£©(2a?y)dx?dy£¬ÆäÖÐLΪ°ÚÏßx?a(t?sint),y?a(1?cost)(0?t?2?)ÑØtÔö¼Ó·½ÏòµÄÒ»¶Î£» £¨£³£©
??xdx?ydy222?Lx2?y2£¬ÆäÖÐLΪԲÖÜx?y?a£¬ÒÀÄæÊ±Õë·½Ïò£»
£¨£´£©Ïò£» £¨£µ£©
?Lydx?sinxdy£¬ÆäÖÐLΪy?sinx(0?x??)ÓëxÖáËùΧµÄ±ÕÇúÏߣ¬ÒÀ˳ʱÕë·½
?xdx?ydy?zdz£¬ÆäÖÐL£º´Ó£¨£±£¬£±£¬£±£©µ½£¨£²£¬£³£¬£´£©µÄÖ±Ï߶Σ®
L2. ÉèÖʵãÊÜÁ¦×÷Óã¬Á¦µÄ·´·½ÏòÖ¸ÏòԵ㣬´óСÓëÖʵãÀëÔµãµÄ¾àÀë³ÉÕý±È£®ÈôÖʵãÓÉ
(a,0)ÑØÍÖÔ²ÒÆ¶¯µ½(0,b)£¬ÇóÁ¦Ëù×÷µÄ¹¦¡£
3. ÉèÒ»ÖʵãÊÜÁ¦×÷Óã¬Á¦µÄ·½ÏòÖ¸ÏòԵ㣬´óСÓëÖʵ㵽xyÆ½ÃæµÄ¾àÀë³É·´±È¡£ÈôÖʵã
ÑØÖ±Ïßx?at,y?bt,z?ct(c?0)´ÓM(a,b,c)µ½N(2a,2b,2c)£¬ÇóÁ¦×÷µÄ¹¦¡£ 4. Ö¤Ã÷£ºÇúÏß»ý·ÖµÄ¹À¼ÆÊ½£º
?ABPdx?Qdy?LM,ÆäÖÐLΪABµÄ»¡³¤£¬
ydx?xdy£¬²¢Ö¤Ã÷£º
?y2?R2(x2?xy?y2)2M?max(x,y)?ABP2?Q2£¬ÀûÓÃÉÏÊö²»µÈʽ¹À¼Æ»ý·ÖIR??2xR???limIR?0¡£
£µ.¼ÆËãÑØ¿Õ¼äÇúÏߵĵڶþÐÍÇúÏß»ý·Ö£º
word
£¨£±£©
?Lxyzdz£¬ÆäÖУ̣ºx2?y2?z2?1Óëy?zÏཻµÄÔ²£¬Æä·½Ïò°´ÇúÏßÒÀ´Î¾¹ý£±£¬
£²£¬£·£¬£¸ØÔÏÞ£» £¨£²£©
?(yL2?z2)dx?(z2?x2)dy?(x2?y2)dz£¬ÆäÖÐLΪÇòÃæx2?y2?z2?1ÔÚµÚ
Ò»ØÔÏÞ²¿·ÖµÄ±ß½çÇúÏߣ¬Æä·½Ïò°´ÇúÏßÒÀ´Î¾¹ýxyÆ½Ãæ²¿·Ö£¬yzÆ½Ãæ²¿·ÖºÍzxÆ½Ãæ²¿·Ö£®
P.381 ¸ñÁÖ¹«Ê½ ÇúÏß»ý·ÖÓë·ÏßÎÞ¹ØÐÔ 1. Ó¦ÓøñÁÖ¹«Ê½¼ÆËãÏÂÁÐÇúÏß»ý·Ö£º £¨£±£©£¨£²£©
?Lxy2dx?x2ydy£¬ÆäÖÐLΪԲÖÜx2?y2?a2µÄÕýÏò£»
2?(x?y)Ldx?(x2?y2)dy£¬ÆäÖÐLÊÇÒÔA(1,1),B(3,2),C(2,5)Ϊ¶¥µãµÄÈý½ÇÐΣ¬
·½ÏòÈ¡ÕýÏò£» £¨£³£©
?ABexsiny?my)dx?(excosy?m)dy£¬ÆäÖÐmΪ³£Êý£¬ABΪÓÉ(a,0)µ½(0,0)¾
2¹ýÔ²x?y?axÉϰ벿µÄ·Ïߣ¨ÆäÖÐaΪÕýÊý£©¡£ £². Ó¦ÓøñÁÖ¹«Ê½¼ÆËãÏÂÁÐÇúÏßËùΧµÄÆ½ÃæÃæ»ý£º £¨£±£© ÍÖÔ²£ºx?acos?,y?bsin?; £¨£²£© ˫ŤÏߣºr?acos2?£®
£³. ÑéÖ¤ÏÂÁлý·ÖÓë·ÏßÎ޹أ¬²¢ÇóËüÃǵÄÖµ£º £¨1£©£¨2£©£¨3£©
222??(1,1)(0,0)(1,2)(x?y)(dx?dy);
ydx?xdyÑØÔÚÓÒ°ëÆ½ÃæµÄ·Ïߣ»
x2(2,1)?(6,8)xdx?ydyx?y22(1,0)ÑØ²»Í¨¹ýÔµãµÄ·Ïߣ»
£¨4£©
?(1,2)(1,0)?(x)dx??(y)dy£¬ÆäÖÐ?,?ΪÁ¬Ðøº¯Êý.
4. ÇóÏÂÁÐȫ΢·ÖµÄÔº¯Êý£º
£¨1£©(x?2xy?y)dx?(x?2xy?y)dy; £¨2£©e[e(x?y?2)?y]dx?e[e(x?y)?1]dy; £¨3£©f(x?y)xdx?f(x?y)ydy. 5. ΪÁËʹÏß»ý·ÖÌõ¼þ£¿ 6. ¼ÆËãÇúÏß»ý·Ö
word
2222xyxy2222?F(x,y)(ydx?xdy)Óë»ý·Ö·ÏßÎ޹أ¬Ôò¿É΢º¯ÊýF(x,y)Ó¦Âú×ãÔõÑùµÄ
L?AMB[?(y)ex?my]dx?[??(y)ex?m]dyÆäÖÐ ?(y)ºÍ??(y)ΪÁ¬Ðøº¯Êý;
AMBΪÁ¬½ÓµãA(x1,y1)ºÍµãB(x2,y2)µÄÈκηÏߣ¬µ«ÓëÏß¶ÎABΧ³ÉÒÑÖª´óСΪSµÄÃæ»ý. 7. É躯Êýf(u)
¾ßÓÐÒ»½×Á¬Ðøµ¼Êý£¬Ö¤Ã÷¶ÔÈκι⻬±ÕÇúÏßL£¬ÓÐ8. Çó»ý·ÖÖµI?Íâ·¨Ïß·½Ïò.
9. É躯Êýu(x,y)Ôڹ⻬±ÕÇúÏßLËùΧ³ÉµÄÇøÓòDÉϾßÓжþ½×Á¬ÐøÆ«µ¼Êý¡£Ö¤Ã÷
?Lf(xy)(ydx?xdy)?0.
nΪLµÄÆäÖÐLΪ°üΧÓнçÇøÓòµÄ±ÕÇúÏߣ¬?[xcos(n,x)?ycos(n,y)]ds£¬
L?2u?2u?u ??(2?2)dxdy??ds£¬
L?n?x?yDÆäÖÐ
?uÊÇu(x,y)ÑØLÍâ·¨Ïß·½ÏònµÄµ¼Êý¡£ ?n
P.391 µÚ¶þÐÍÇúÃæ»ý·Ö
1. ¼ÆËãÏÂÁеڶþÐÍÇúÃæ»ý·Ö£º
£¨1£©y(x?z)dydz?xdzdx?(y?xz)dxdy£¬ÆäÖÐSΪx?y?z?0,x?y?z?aÁù
s??22¸öÆ½ÃæËùΧµÄÕý·½Ì岢ȡÍâ²àΪÕýÏò£» £¨2£©
??(x?y)dydz?(y?z)dzdx?(z?x)dxdy£¬ÆäÖÐSÊÇÒÔÔµãΪÖÐÐÄ£¬±ß³¤Îª2µÄÕý
s·½Ìå±íÃæ²¢È¡Íâ²àΪÕýÏò£» £¨3£©
??xydydz?yzdzdx?xzdxdy£¬ÆäÖÐSÊÇÓÉÆ½Ãæx?y?z?0,x?y?z?1ËùΧµÄ
sËÄÃæÌå±íÃæ²¢È¡Íâ²àΪÕýÏò£» £¨4£©£¨5£©
??yzdzdx£¬ÆäÖÐSÊÇÇòÃæxss2?y2?z2?1µÄÉϰ벿·Ö²¢È¡Íâ²àΪÕýÏò£»
2222222(x?a)?(y?b)?(z?c)?R£¬ÆäÖÐÊÇÇòÃæ²¢xdydz?ydzdx?zdxdyS??È¡Íâ²àΪÕýÏò¡£
2. ÉèijÁ÷ÌåµÄÁ÷ËÙΪv?(k,y,0)£¬Çóµ¥Î»Ê±¼äÄÚ´ÓÇòÃæx?y?z?4µÄÄÚ²¿Á÷¹ýÇòÃæ
µÄÁ÷Á¿.
3. ¼ÆËãµÚ¶þÐÍÇúÃæ»ý·ÖI?222??f(x)dydz?g(y)dzdx?h(z)dxdy,ÆäÖÐSÊÇÆ½ÃæÁùÃæÌå
S(0?x?a,0?y?b,0?z?c)µÄ±íÃæ²¢È¡Íâ²à£¬f(x),g(y),h(z)ΪSÉϵÄÁ¬Ðøº¯Êý.
4. Éè´Å³¡Ç¿¶ÈΪE(x,y,z)£¬Çó´ÓÇòÄÚ³ö·¢Í¨¹ýÉϰëÇòÃæx?y?z?a,z?0µÄ´Åͨ
Á¿¡£
2222word