¸ßÊýÏÂʮһÕÂÖصã×ܽá+ÀýÌâ

ʱ¼ä£º2024.3.15

µÚʮһÕÂÇúÏß»ý·ÖÓëÇúÃæ»ý·Ö

¡¾½ÌѧĿ±êÓëÒªÇó¡¿

1.Àí½âÁ½ÀàÇúÏß»ý·ÖµÄ¸ÅÄÁ˽âÁ½ÀàÇúÏß»ý·ÖµÄÐÔÖʼ°Á½ÀàÇúÏß»ý·ÖµÄ¹Øϵ¡£

2.ÕÆÎÕ¼ÆËãÁ½ÀàÇúÏß»ý·ÖµÄ·½·¨¡£

3.ÊìÁ·ÕÆÎÕ¸ñÁÖ¹«Ê½²¢»áÔËÓÃƽÃæÇúÏß»ý·ÖÓë·¾¶Î޹صÄÌõ¼þ£¬»áÇóȫ΢·ÖµÄÔ­º¯Êý¡£

4.Á˽âÁ½ÀàÇúÃæ»ý·ÖµÄ¸ÅÄî¡¢ÐÔÖʼ°Á½ÀàÇúÃæ»ý·ÖµÄ¹Øϵ£¬ÕÆÎÕ¼ÆËãÁ½ÀàÇúÃæ»ý·ÖµÄ·½·¨£¬Á˽â¸ß˹¹«Ê½¡¢Ë¹ÍпË˹¹«Ê½£¬»áÓøß˹¹«Ê½¼ÆËãÇúÃæ»ý·Ö¡£

5.ÖªµÀÉ¢¶ÈÓëÐý¶ÈµÄ¸ÅÄ²¢»á¼ÆËã¡£

6.»áÓÃÇúÏß»ý·Ö¼°ÇúÃæ»ý·ÖÇóһЩ¼¸ºÎÁ¿ÓëÎïÀíÁ¿¡£

¡¾½ÌѧÖص㡿

1.Á½ÀàÇúÏß»ý·ÖµÄ¼ÆËã·½·¨£»

2.¸ñÁÖ¹«Ê½¼°ÆäÓ¦Óã»

3.Á½ÀàÇúÃæ»ý·ÖµÄ¼ÆËã·½·¨£»

4.¸ß˹¹«Ê½¡¢Ë¹ÍпË˹¹«Ê½£»

5.Á½ÀàÇúÏß»ý·ÖÓëÁ½ÀàÇúÃæ»ý·ÖµÄÓ¦Óá£

¡¾½ÌѧÄѵ㡿

1.Á½ÀàÇúÏß»ý·ÖµÄ¹Øϵ¼°Á½ÀàÇúÃæ»ý·ÖµÄ¹Øϵ£»

2.¶Ô×ø±êµÄÇúÏß»ý·ÖÓë¶Ô×ø±êµÄÇúÃæ»ý·ÖµÄ¼ÆË㣻

3.Ó¦ÓøñÁÖ¹«Ê½¼ÆËã¶Ô×ø±êµÄÇúÏß»ý·Ö£»

4.Ó¦Óøß˹¹«Ê½¼ÆËã¶Ô×ø±êµÄÇúÃæ»ý·Ö£»

5.Ó¦ÓÃ˹ÍпË˹¹«Ê½¼ÆËã¶Ô×ø±êµÄÇúÏß»ý·Ö¡£

6.Á½ÀàÇúÏß»ý·ÖµÄ¼ÆËã·½·¨£¬Á½ÀàÇúÏß»ý·ÖµÄ¹Øϵ£»

7.¸ñÁÖ¹«Ê½¼°ÆäÓ¦ÓøñÁÖ¹«Ê½¼ÆËã¶Ô×ø±êµÄÇúÏß»ý·Ö£»

8.Á½ÀàÇúÃæ»ý·ÖµÄ¼ÆËã·½·¨¼°Á½ÀàÇúÃæ»ý·ÖµÄ¹Øϵ£»

9.¸ß˹¹«Ê½¡¢Ë¹ÍпË˹¹«Ê½£¬Ó¦Óøß˹¹«Ê½¼ÆËã¶Ô×ø±êµÄÇúÃæ»ý·Ö£»

10.Á½ÀàÇúÏß»ý·ÖÓëÁ½ÀàÇúÃæ»ý·ÖµÄÓ¦Óã»

11.Ó¦ÓÃ˹ÍпË˹¹«Ê½¼ÆËã¶Ô×ø±êµÄÇúÏß»ý·Ö¡£

¡¾½Ìѧ¿Îʱ·ÖÅä¡¿ (14ѧʱ)

µÚ1 ´Î¿Î¡¡ ¡ì£± µÚ2 ´Î¿Î¡¡ ¡ì2 µÚ3 ´Î¿Î¡¡ ¡ì3

µÚ4 ´Î¿Î¡¡ ¡ì4 µÚ5´Î¿Î ¡ì5 µÚ6´Î¿Î ¡ì6

µÚ7´Î¿Î Ï°Ìâ¿Î

¡¾²Î¿¼Êé¡¿

[1]ͬ¼Ã´óѧÊýѧϵ.¡¶¸ßµÈÊýѧ£¨Ï£©¡·£¬µÚÎå°æ.¸ßµÈ½ÌÓý³ö°æÉç.

[2] ͬ¼Ã´óѧÊýѧϵ.¡¶¸ßµÈÊýѧѧϰ¸¨µ¼ÓëÏ°ÌâÑ¡½â¡·£¬µÚÁù°æ.¸ßµÈ½ÌÓý³ö°æÉç.?

[3] ͬ¼Ã´óѧÊýѧϵ.¡¶¸ßµÈÊýѧϰÌâÈ«½âÖ¸ÄÏ£¨Ï£©¡·£¬µÚÁù°æ.¸ßµÈ½ÌÓý³ö°æÉç

¡ì11.1¶Ô»¡³¤µÄÇúÏß»ý·Ö

Ò»¡¢¶Ô»¡³¤µÄÇúÏß»ý·ÖµÄ¸ÅÄîÓëÐÔÖÊ

ÇúÏßÐι¹¼þµÄÖÊÁ¿:

ÉèÒ»ÇúÏßÐι¹¼þËùÕ¼µÄλÖÃÔÚxOyÃæÄÚµÄÒ»¶ÎÇúÏß»¡LÉÏ, ÒÑÖªÇúÏßÐι¹¼þÔÚµã(x,y)´¦µÄÏßÃܶÈΪm(x,y). ÇóÇúÏßÐι¹¼þµÄÖÊÁ¿.

°ÑÇúÏß·Ö³ÉnС¶Î, Ds1, Ds2, ¡Á ¡Á ¡Á, Dsn(DsiÒ²±íʾ»¡³¤);

ÈÎÈ¡(xi,hi)ÎDsi, µÃµÚiС¶ÎÖÊÁ¿µÄ½üËÆÖµm(xi,hi)Dsi;

Õû¸öÎïÖÊÇúÏßµÄÖÊÁ¿½üËÆΪ;

Áîl=max{Ds1, Ds2, ¡Á ¡Á ¡Á, Dsn}®0, ÔòÕû¸öÎïÖÊÇúÏßµÄÖÊÁ¿Îª

.

ÕâÖֺ͵ļ«ÏÞÔÚÑо¿ÆäËüÎÊÌâʱҲ»áÓöµ½.

¶¨ÒåÉ躯Êýf(x,y)¶¨ÒåÔÚ¿ÉÇ󳤶ȵÄÇúÏßLÉÏ, ²¢ÇÒÓнç.£¬½«LÈÎÒâ·Ö³Én¸ö»¡¶Î: Ds1, Ds2, ¡Á ¡Á ¡Á, Dsn, ²¢ÓÃDsi±íʾµÚi¶ÎµÄ»¡³¤; ÔÚÿһ»¡¶ÎDsiÉÏÈÎÈ¡Ò»µã(xi,hi), ×÷ºÍ; Áîl=max{Ds1, Ds2, ¡Á ¡Á ¡Á, Dsn}, Èç¹ûµ±l®0ʱ, ÕâºÍµÄ¼«ÏÞ×Ü´æÔÚ, Ôò³Æ´Ë¼«ÏÞΪº¯Êýf(x,y)ÔÚÇúÏß»¡LÉ϶Ի¡³¤µÄ

ÇúÏß»ý·Ö»òµÚÒ»ÀàÇúÏß»ý·Ö, ¼Ç×÷, ¼´

.

ÆäÖÐf(x,y)½Ð×ö±»»ýº¯Êý,L½Ð×ö»ý·Ö»¡¶Î.

ÇúÏß»ý·ÖµÄ´æÔÚÐÔ:µ±f(x,y)Ôڹ⻬ÇúÏß»¡LÉÏÁ¬Ðøʱ, ¶Ô»¡³¤µÄÇúÏß»ý·ÖÊÇ´æÔÚµÄ. ÒÔºóÎÒÃÇ×ܼٶ¨f(x,y)ÔÚLÉÏÊÇÁ¬ÐøµÄ.

¸ù¾Ý¶Ô»¡³¤µÄÇúÏß»ý·ÖµÄ¶¨Òå,ÇúÏßÐι¹¼þµÄÖÊÁ¿¾ÍÊÇÇúÏß»ý·ÖµÄÖµ, ÆäÖÐm(x,y)ΪÏßÃܶÈ.

¶Ô»¡³¤µÄÇúÏß»ý·ÖµÄÍƹã:.

Èç¹ûL(»òG)ÊǷֶι⻬µÄ, Ôò¹æ¶¨º¯ÊýÔÚL(»òG)ÉϵÄÇúÏß»ý·ÖµÈÓÚº¯ÊýÔڹ⻬µÄ¸÷¶ÎÉϵÄÇúÏß»ý·ÖµÄºÍ. ÀýÈçÉèL¿É·Ö³ÉÁ½¶Î¹â»¬ÇúÏß»¡L1¼°L2, Ôò¹æ¶¨

.

±ÕÇúÏß»ý·Ö:Èç¹ûLÊDZÕÇúÏß, ÄÇôº¯Êýf(x,y)ÔÚ±ÕÇúÏßLÉ϶Ի¡³¤µÄÇúÏß»ý·Ö¼Ç×÷.

¶Ô»¡³¤µÄÇúÏß»ý·ÖµÄÐÔÖÊ:

ÐÔÖÊ1 Éèc1¡¢c2Ϊ³£Êý, Ôò

;

ÐÔÖÊ2 Èô»ý·Ö»¡¶ÎL¿É·Ö³ÉÁ½¶Î¹â»¬ÇúÏß»¡L1ºÍL2, Ôò

;

ÐÔÖÊ3ÉèÔÚLÉÏf(x,yg(x,y), Ôò

.

ÌرðµØ, ÓÐ

¶þ¡¢¶Ô»¡³¤µÄÇúÏß»ý·ÖµÄ¼ÆËã·¨

¸ù¾Ý¶Ô»¡³¤µÄÇúÏß»ý·ÖµÄ¶¨Òå, Èç¹ûÇúÏßÐι¹¼þLµÄÏßÃܶÈΪf(x,y), ÔòÇúÏßÐι¹¼þLµÄÖÊÁ¿Îª.

ÁíÒ»·½Ãæ, ÈôÇúÏßLµÄ²ÎÊý·½³ÌΪ

x=j(t),y=y(t) (a£t£b),

ÔòÖÊÁ¿ÔªËØΪ

,

ÇúÏßµÄÖÊÁ¿Îª

.

¼´.

¶¨ÀíÉèf(x,y)ÔÚÇúÏß»¡LÉÏÓж¨ÒåÇÒÁ¬Ðø,LµÄ²ÎÊý·½³ÌΪx=j(t),y=y(t) (a£t£b),

ÆäÖÐj(t)¡¢y(t)ÔÚ[a,b]ÉϾßÓÐÒ»½×Á¬Ðøµ¼Êý, ÇÒj¢2(t)+y¢2(t)¹0, ÔòÇúÏß»ý·Ö´æÔÚ, ÇÒ

(a<b).

ӦעÒâµÄÎÊÌâ:¶¨»ý·ÖµÄÏÂÏÞaÒ»¶¨ÒªÐ¡ÓÚÉÏÏÞb.

ÌÖÂÛ:

(1)ÈôÇúÏßLµÄ·½³ÌΪy=y(x)(a£x£b), Ôò=?

Ìáʾ:LµÄ²ÎÊý·½³ÌΪx=x,y=y(x)(a£x£b),

.

(2)ÈôÇúÏßLµÄ·½³ÌΪx=j(y)(c£y£d), Ôò=?

Ìáʾ:LµÄ²ÎÊý·½³ÌΪx=j(y),y=y(c£y£d),

.

(3)ÈôÇúGµÄ·½³ÌΪx=j(t),y=y(t),z=w(t)(a£t£b),

Ôò=?

Ìáʾ:.

Àý1¼ÆËã, ÆäÖÐLÊÇÅ×ÎïÏßy=x2ÉϵãO(0, 0)ÓëµãB(1, 1)Ö®¼äµÄÒ»¶Î»¡.

½â ÇúÏߵķ½³ÌΪy=x2(0£x£1), Òò´Ë

.

Àý2¼ÆËã°ë¾¶ÎªR¡¢ÖÐÐĽÇΪ2aµÄÔ²»¡L¶ÔÓÚËüµÄ¶Ô³ÆÖáµÄת¶¯¹ßÁ¿I(ÉèÏßÃܶÈΪm=1).

½â È¡×ø±êϵÈçͼËùʾ, Ôò.ÇúÏßLµÄ²ÎÊý·½³ÌΪ

x=Rcosq,y=Rsinq(-a£q<a).

ÓÚÊÇ

=R3(a-sinacosa).

Àý3¼ÆËãÇúÏß»ý·Ö, ÆäÖÐGΪÂÝÐýÏßx=acost¡¢y=asint¡¢z=ktÉÏÏàÓ¦ÓÚt´Ó0µ½´ï2pµÄÒ»¶Î»¡.

½â ÔÚÇúÏßGÉÏÓÐx2+y2+z2=(acost)2+(asint)2+(kt)2=a2+k2t2, ²¢ÇÒ

,

ÓÚÊÇ

.

С½á

ÓÃÇúÏß»ý·Ö½â¾öÎÊÌâµÄ²½Öè:

(1)½¨Á¢ÇúÏß»ý·Ö;

(2)д³öÇúÏߵIJÎÊý·½³Ì ( »òÖ±½Ç×ø±ê·½³Ì) , È·¶¨²ÎÊýµÄ±ä»¯·¶Î§;

(3)½«ÇúÏß»ý·Ö»¯Îª¶¨»ý·Ö;

(4)¼ÆË㶨»ý·Ö.

½Ìѧ·½Ê½¼°½Ìѧ¹ý³ÌÖÐӦעÒâµÄÎÊÌâ

ÔÚ½Ìѧ¹ý³ÌÖÐҪעÒâÇúÏß»ý·Ö½â¾öÎÊÌâµÄ²½Ö裬Ҫ½áºÏʵÀý£¬·´¸´½²½â¡£

ʦÉú»î¶¯Éè¼Æ

1.ÒÑÖªÍÖÔ²Öܳ¤Îªa£¬Çó¡£

2.ÉèCÊÇÓɼ«×ø±êϵÏÂÇúÏß¼°ËùΧ³ÉÇøÓòµÄ±ß½ç£¬Çó

½²¿ÎÌá¸Ù¡¢°åÊéÉè¼Æ

×÷ÒµP190: 3£¨1£©£¨3£©£¨5£©£¨7£©

¡ì11.2¶Ô×ø±êµÄÇúÏß»ý·Ö

Ò»¡¢¶Ô×ø±êµÄÇúÏß»ý·ÖµÄ¸ÅÄîÓëÐÔÖÊ

±äÁ¦ÑØÇúÏßËù×÷µÄ¹¦:

ÉèÒ»¸öÖʵãÔÚxOyÃæÄÚÔÚ±äÁ¦F(x,y)=P(x,y)i+Q(x,y)jµÄ×÷ÓÃÏ´ӵãAÑع⻬ÇúÏß»¡LÒƶ¯µ½µãB, ÊÔÇó±äÁ¦F(x,y)Ëù×÷µÄ¹¦.

ÓÃÇúÏßLÉϵĵãA=A0,A1,A2, ¡Á ¡Á ¡Á,An-1,An=B°ÑL·Ö³Én¸öС»¡¶Î,

ÉèAk=(xk,yk), ÓÐÏòÏ߶εij¤¶ÈΪDsk, ËüÓëxÖáµÄ¼Ð½ÇΪtk, Ôò

(k=0, 1, 2, ¡Á ¡Á ¡Á,n-1).

ÏÔÈ», ±äÁ¦F(x,y)ÑØÓÐÏòС»¡¶ÎËù×÷µÄ¹¦¿ÉÒÔ½üËÆΪ

;

ÓÚÊÇ, ±äÁ¦F(x,y)Ëù×÷µÄ¹¦

,

´Ó¶ø

.

ÕâÀït=t(x,y), {cost, sint}ÊÇÇúÏßLÔÚµã(x,y)´¦µÄÓëÇúÏß·½ÏòÒ»Öµĵ¥Î»ÇÐÏòÁ¿.

°ÑL·Ö³Én¸öС»¡¶Î:L1,L2, ¡Á ¡Á ¡Á,Ln;±äÁ¦ÔÚLiÉÏËù×÷µÄ¹¦½üËÆΪ:

F(xi,hi)¡ÁDsi=P(xi,hi)Dxi+Q(xi,hi)Dyi;

±äÁ¦ÔÚLÉÏËù×÷µÄ¹¦½üËÆΪ:

;

±äÁ¦ÔÚLÉÏËù×÷µÄ¹¦µÄ¾«È·Öµ:

,

ÆäÖÐlÊǸ÷С»¡¶Î³¤¶ÈµÄ×î´óÖµ.

Ìáʾ:

ÓÃDsi={Dxi,Dyi}±íʾ´ÓLiµÄÆðµãµ½ÆäÖÕµãµÄµÄÏòÁ¿. ÓÃDsi±íʾDsiµÄÄ£.

¶Ô×ø±êµÄÇúÏß»ý·ÖµÄ¶¨Òå:

¶¨ÒåÉ躯Êýf(x,y)ÔÚÓÐÏò¹â»¬ÇúÏßLÉÏÓнç. °ÑL·Ö³Én¸öÓÐÏòС»¡¶ÎL1,L2, ¡Á ¡Á ¡Á,Ln; С»¡¶ÎLiµÄÆðµãΪ(xi-1,yi-1), ÖÕµãΪ(xi,yi), Dxi=xi-xi-1, Dyi=yi-yi-1; (xi,h)ΪLiÉÏÈÎÒâÒ»µã,lΪ¸÷С»¡¶Î³¤¶ÈµÄ×î´óÖµ.

Èç¹û¼«ÏÞ×Ü´æÔÚ, Ôò³Æ´Ë¼«ÏÞΪº¯Êýf(x,y)ÔÚÓÐÏòÇúÏßLÉ϶Ô×ø±êxµÄÇúÏß»ý·Ö, ¼Ç×÷, ¼´,

ÉèLΪxOyÃæÉÏÒ»Ìõ¹â»¬ÓÐÏòÇúÏß, {cost, sint}ÊÇÓëÇúÏß·½ÏòÒ»Öµĵ¥Î»ÇÐÏòÁ¿, º¯ÊýP(x,y)¡¢Q(x,y)ÔÚLÉÏÓж¨Òå. Èç¹ûÏÂÁжþʽÓҶ˵Ļý·Ö´æÔÚ, ÎÒÃǾͶ¨Òå

,

,

Ç°Õß³ÆΪº¯ÊýP(x,y)ÔÚÓÐÏòÇúÏßLÉ϶Ô×ø±êxµÄÇúÏß»ý·Ö, ºóÕß³ÆΪº¯ÊýQ(x,y)ÔÚÓÐÏòÇúÏßLÉ϶Ô×ø±êyµÄÇúÏß»ý·Ö, ¶Ô×ø±êµÄÇúÏß»ý·ÖÒ²½ÐµÚ¶þÀàÇúÏß»ý·Ö.

¶¨ÒåµÄÍƹã:

ÉèGΪ¿Õ¼äÄÚÒ»Ìõ¹â»¬ÓÐÏòÇúÏß, {cosa, cosb, cosg}ÊÇÇúÏßÔÚµã(x,y,z)´¦µÄÓëÇúÏß·½ÏòÒ»Öµĵ¥Î»ÇÐÏòÁ¿, º¯ÊýP(x,y,z)¡¢Q(x,y,z)¡¢R(x,y,z)ÔÚGÉÏÓж¨Òå. ÎÒÃǶ¨Òå(¼ÙÈç¸÷ʽÓҶ˵Ļý·Ö´æÔÚ)

,

,

.

,,

.

¶Ô×ø±êµÄÇúÏß»ý·ÖµÄ¼òдÐÎʽ:

;

.

¶Ô×ø±êµÄÇúÏß»ý·ÖµÄÐÔÖÊ:

(1) Èç¹û°ÑL·Ö³ÉL1ºÍL2, Ôò

.

(2) ÉèLÊÇÓÐÏòÇúÏß»¡, -LÊÇÓëL·½ÏòÏà·´µÄÓÐÏòÇúÏß»¡, Ôò

.

Á½ÀàÇúÏß»ý·ÖÖ®¼äµÄ¹Øϵ:

Éè{costi, sinti}ΪÓëDsiͬÏòµÄµ¥Î»ÏòÁ¿, ÎÒÃÇ×¢Òâµ½{Dxi, Dyi}=Dsi, ËùÒÔ

Dxi=costi¡ÁDsi, Dyi=sinti¡ÁDsi,

,

.

¼´,

»ò.

ÆäÖÐA={P,Q},t={cost, sint}ΪÓÐÏòÇúÏß»¡LÉϵã(x,y)´¦µ¥Î»ÇÐÏòÁ¿,dr=tds={dx,dy}.

ÀàËƵØÓÐ

,

»ò.

ÆäÖÐA={P,Q,R},T={cosa, cosb, cosg}ΪÓÐÏòÇúÏß»¡GÉϵã(x,y,z)´¦µ¥ÃÇÇÐÏòÁ¿,dr=Tds={dx,dy,dz},AtΪÏòÁ¿AÔÚÏòÁ¿tÉϵÄͶӰ.

¶þ¡¢¶Ô×ø±êµÄÇúÏß»ý·ÖµÄ¼ÆËã:

¶¨Àí: ÉèP(x,y)¡¢Q(x,y)ÊǶ¨ÒåÔڹ⻬ÓÐÏòÇúÏßL:x=j(t),y=y(t), ÉϵÄÁ¬Ðøº¯Êý, µ±²ÎÊýtµ¥µ÷µØÓÉa±äµ½bʱ, µãM(x,y)´ÓLµÄÆðµãAÑØLÔ˶¯µ½ÖÕµãB, Ôò

,

.

ÌÖÂÛ:=£¿

Ìáʾ:.

¶¨Àí: ÈôP(x,y)ÊǶ¨ÒåÔڹ⻬ÓÐÏòÇúÏßL:x=j(t),y=y(t)(a£t£b)ÉϵÄÁ¬Ðøº¯Êý,LµÄ·½ÏòÓëtµÄÔö¼Ó·½ÏòÒ»ÖÂ, Ôò

.

¼òÒªÖ¤Ã÷: ²»·ÁÉèa£b. ¶ÔÓ¦ÓÚtµãÓëÇúÏßLµÄ·½ÏòÒ»ÖµÄÇÐÏòÁ¿Îª{j¢(t),y¢(t)},

ËùÒÔ,

´Ó¶ø

.

ӦעÒâµÄÎÊÌâ:

ÏÂÏÞa¶ÔÓ¦ÓÚLµÄÆðµã, ÉÏÏÞb¶ÔÓ¦ÓÚLµÄÖÕµã,a²»Ò»¶¨Ð¡ÓÚb.

ÌÖÂÛ:

Èô¿Õ¼äÇúÏßGÓɲÎÊý·½³Ìx=j?t),y=y(t),z=w(t)¸ø³ö, ÄÇôÇúÏß»ý·Ö

=£¿

ÈçºÎ¼ÆË㣿?

Ìáʾ:

,

ÆäÖÐa¶ÔÓ¦ÓÚGµÄÆðµã,b¶ÔÓ¦ÓÚGµÄÖÕµã.

ÀýÌâ:

Àý1.¼ÆËã, ÆäÖÐLΪÅ×ÎïÏßy2=xÉÏ´ÓµãA(1, -1)µ½µãB(1, 1)µÄÒ»¶Î»¡.

Àý2.¼ÆËã.

(1)LΪ°´ÄæʱÕë·½ÏòÈÆÐеÄÉÏ°ëÔ²ÖÜx2+y2=a2;

(2)´ÓµãA(a, 0)ÑØxÖáµ½µãB(-a, 0)µÄÖ±Ï߶Î.

Àý3¼ÆËã. (1)Å×ÎïÏßy=x2ÉÏ´ÓO(0, 0)µ½B(1, 1)µÄÒ»¶Î»¡; (2)Å×ÎïÏßx=y2ÉÏ´ÓO(0, 0)µ½B(1, 1)µÄÒ»¶Î»¡; (3)´ÓO(0, 0)µ½A(1, 0), ÔÙµ½R(1, 1)µÄÓÐÏòÕÛÏßOAB.

Àý4.¼ÆËã, ÆäÖÐGÊÇ´ÓµãA(3, 2, 1)µ½µãB(0, 0, 0)µÄÖ±Ï߶Î.

Àý5.ÉèÒ»¸öÖʵãÔÚM(x,y)´¦Êܵ½Á¦FµÄ×÷ÓÃ,FµÄ´óСÓëMµ½Ô­µãOµÄ¾àÀë³ÉÕý±È,FµÄ·½ÏòºãÖ¸ÏòÔ­µã. ´ËÖʵãÓɵãA(a, 0)ÑØÍÖÔ²°´ÄæʱÕë·½ÏòÒƶ¯µ½µãB(0,b), ÇóÁ¦FËù×÷µÄ¹¦W.

Èý¡¢Á½ÀàÇúÏß»ý·ÖÖ®¼äµÄÁªÏµ

Óɶ¨Òå, µÃ

,

ÆäÖÐF={P,Q},T={cost, sint}ΪÓÐÏòÇúÏß»¡LÉϵã(x,y)´¦µ¥Î»ÇÐÏòÁ¿,dr=Tds={dx,dy}.

ÀàËƵØÓÐ

.

ÆäÖÐF={P,Q,R},T={cosa, cosb, cosg}ΪÓÐÏòÇúÏß»¡GÉϵã(x,y,z)´¦µ¥ÃÇÇÐÏòÁ¿,dr=Tds={dx,dy,dz}.

С½á

1.µÚ¶þÀàÇúÏß»ý·ÖµÄ¶¨Ò壻

2. µÚ¶þÀàÇúÏß»ý·ÖµÄ¼ÆËã·½·¨¡£

½Ìѧ·½Ê½¼°½Ìѧ¹ý³ÌÖÐӦעÒâµÄÎÊÌâ

ÔÚ½Ìѧ¹ý³ÌÖÐҪעÒâµÚ¶þÀàÇúÏß»ý·ÖµÄ¶¨ÒåºÍ¼ÆËã·½·¨£¬Òª½áºÏʵÀý£¬·´¸´½²½â¡£

ʦÉú»î¶¯Éè¼Æ

1. ÒÑ֪ΪÕÛÏßABCOA£¬¼ÆËã

½²¿ÎÌá¸Ù¡¢°åÊéÉè¼Æ

×÷ÒµP200: 3£¨1£©£¨3£©£¨5£©£¨7£©£¬4

¡ì11.3¸ñÁÖ¹«Ê½¼°ÆäÓ¦ÓÃ

Ò»¡¢¸ñÁÖ¹«Ê½

µ¥Á¬Í¨Ó븴Á¬Í¨ÇøÓò:

ÉèDΪƽÃæÇøÓò, Èç¹ûDÄÚÈÎÒ»±ÕÇúÏßËùΧµÄ²¿·Ö¶¼ÊôÓÚD, Ôò³ÆDΪƽÃæµ¥Á¬Í¨ÇøÓò, ·ñÔò³ÆΪ¸´Á¬Í¨ÇøÓò.

¶ÔƽÃæÇøÓòDµÄ±ß½çÇúÏßL, ÎÒÃǹ涨LµÄÕýÏòÈçÏÂ:µ±¹Û²ìÕßÑØLµÄÕâ¸ö·½ÏòÐÐ×ßʱ,DÄÚÔÚËû½ü´¦µÄÄÇÒ»²¿·Ö×ÜÔÚËûµÄ×ó±ß.

ÇøÓòDµÄ±ß½çÇúÏߵķ½Ïò:

¶¨Àí1Éè±ÕÇøÓòDÓɷֶι⻬µÄÇúÏßΧ³É, º¯ÊýP(x,y)¼°Q(x,y)ÔÚDÉϾßÓÐÒ»½×Á¬ÐøÆ«µ¼Êý, ÔòÓÐ

,

ÆäÖÐLÊÇDµÄÈ¡ÕýÏòµÄ±ß½çÇúÏß.

¼òÒªÖ¤Ã÷:½ö¾ÍD¼´ÊÇX£­Ð͵ÄÓÖÊÇY£­Ð͵ÄÇøÓòÇéÐνøÐÐÖ¤Ã÷.

ÉèD={(x,y)|j1(xy£j2(x),a£x£b}. ÒòΪÁ¬Ðø, ËùÒÔÓɶþÖØ»ý·ÖµÄ¼ÆËã·¨ÓÐ

.

ÁíÒ»·½Ãæ, ÓɶÔ×ø±êµÄÇúÏß»ý·ÖµÄÐÔÖʼ°¼ÆËã·¨ÓÐ

.

Òò´Ë

.

ÉèD={(x,y)|y1(yx£y2(y),c£y£d}. ÀàËƵؿÉÖ¤

.

ÓÉÓÚD¼´ÊÇX£­Ð͵ÄÓÖÊÇY£­Ð͵Ä, ËùÒÔÒÔÉÏÁ½Ê½Í¬Ê±³ÉÁ¢, Á½Ê½ºÏ²¢¼´µÃ

.

ӦעÒâµÄÎÊÌâ:

¶Ô¸´Á¬Í¨ÇøÓòD, ¸ñÁÖ¹«Ê½ÓÒ¶ËÓ¦°üÀ¨ÑØÇøÓòDµÄÈ«²¿±ß½çµÄÇúÏß»ý·Ö, Çұ߽çµÄ·½Ïò¶ÔÇøÓòDÀ´Ëµ¶¼ÊÇÕýÏò.

ÉèÇøÓòDµÄ±ß½çÇúÏßΪL, È¡P=-y,Q=x, ÔòÓɸñÁÖ¹«Ê½µÃ

, »ò.

Àý1.ÍÖÔ²x=acosq,y=bsinqËùΧ³ÉͼÐεÄÃæ»ýA.

·ÖÎö:Ö»Òª, ¾ÍÓÐ.

Àý2ÉèLÊÇÈÎÒâÒ»Ìõ·Ö¶Î¹â»¬µÄ±ÕÇúÏß, Ö¤Ã÷

.

Àý3.¼ÆËã, ÆäÖÐDÊÇÒÔO(0, 0),A(1, 1),B(0, 1)Ϊ¶¥µãµÄÈý½ÇÐαÕÇøÓò.

·ÖÎö:Ҫʹ, Ö»ÐèP=0,.

Àý4¼ÆËã, ÆäÖÐLΪһÌõÎÞÖص㡢·Ö¶Î¹â»¬ÇÒ²»¾­¹ýÔ­µãµÄÁ¬Ðø±ÕÇúÏß,LµÄ·½ÏòΪÄæʱÕë·½Ïò.

½â: Áî,. Ôòµ±x2+y2¹0ʱ, ÓÐ.

¼ÇLËùΧ³ÉµÄ±ÕÇøÓòΪD. µ±(0, 0)ÏDʱ, ÓɸñÁÖ¹«Ê½µÃ;

µ±(0, 0)ÎDʱ, ÔÚDÄÚÈ¡Ò»Ô²ÖÜl:x2+y2=r2(r>0). ÓÉL¼°lΧ³ÉÁËÒ»¸ö¸´Á¬Í¨ÇøÓòD1, Ó¦ÓøñÁÖ¹«Ê½µÃ

,

ÆäÖÐlµÄ·½ÏòÈ¡ÄæʱÕë·½Ïò.

ÓÚÊÇ=2p.

¼ÇLËùΧ³ÉµÄ±ÕÇøÓòΪD.

µ±(0, 0)ÏDʱ, ÓɸñÁÖ¹«Ê½µÃ

.

·ÖÎö:ÕâÀï,, µ±x2+y2¹0ʱ, ÓÐ.

¶þ¡¢Æ½ÃæÉÏÇúÏß»ý·ÖÓë·¾¶Î޹صÄÌõ¼þ

ÇúÏß»ý·ÖÓë·¾¶ÎÞ¹Ø:

ÉèGÊÇÒ»¸ö¿ªÇøÓò,P(x,y)¡¢Q(x,y)ÔÚÇøÓòGÄÚ¾ßÓÐÒ»½×Á¬ÐøÆ«µ¼Êý. Èç¹û¶ÔÓÚGÄÚÈÎÒâÖ¸¶¨µÄÁ½¸öµãA¡¢BÒÔ¼°GÄÚ´ÓµãAµ½µãBµÄÈÎÒâÁ½ÌõÇúÏßL1¡¢L2, µÈʽ

ºã³ÉÁ¢, ¾Í˵ÇúÏß»ý·ÖÔÚGÄÚÓë·¾¶ÎÞ¹Ø, ·ñÔò˵Óë·¾¶ÓйØ.

ÉèÇúÏß»ý·ÖÔÚGÄÚÓë·¾¶ÎÞ¹Ø,L1ºÍL2ÊÇGÄÚÈÎÒâÁ½Ìõ´ÓµãAµ½µãBµÄÇúÏß, ÔòÓÐ

,

ÒòΪ

Û

ÛÛ,

ËùÒÔÓÐÒÔϽáÂÛ:

ÇúÏß»ý·ÖÔÚGÄÚÓë·¾¶ÎÞ¹ØÏ൱ÓÚÑØGÄÚÈÎÒâ

±ÕÇúÏßCµÄÇúÏß»ý·ÖµÈÓÚÁã.

¶¨Àí2É迪ÇøÓòGÊÇÒ»¸öµ¥Á¬Í¨Óò, º¯ÊýP(x,y)¼°Q(x,y)ÔÚGÄÚ¾ßÓÐÒ»½×Á¬ÐøÆ«µ¼Êý, ÔòÇúÏß»ý·ÖÔÚGÄÚÓë·¾¶Î޹أ¨»òÑØGÄÚÈÎÒâ±ÕÇúÏßµÄÇúÏß»ý·ÖΪÁ㣩µÄ³ä·Ö±ØÒªÌõ¼þÊǵÈʽ

ÔÚGÄÚºã³ÉÁ¢.

³ä·ÖÐÔÒ×Ö¤:

Èô, Ôò, ÓɸñÁÖ¹«Ê½, ¶ÔÈÎÒâ±ÕÇúÏßL, ÓÐ.

±ØÒªÐÔ:

¼ÙÉè´æÔÚÒ»µãM0ÎG, ʹ, ²»·ÁÉèh>0, ÔòÓɵÄÁ¬ÐøÐÔ, ´æÔÚM0µÄÒ»¸ödÁÚÓòU(M0,d), ʹÔÚ´ËÁÚÓòÄÚÓÐ. ÓÚÊÇÑØÁÚÓòU(M0,d)±ß½çlµÄ±ÕÇúÏß»ý·Ö

,

ÕâÓë±ÕÇúÏß»ý·ÖΪÁãÏàì¶Ü, Òò´ËÔÚGÄÚ.

ӦעÒâµÄÎÊÌâ:

¶¨ÀíÒªÇó, ÇøÓòGÊǵ¥Á¬Í¨ÇøÓò, ÇÒº¯ÊýP(x,y)¼°Q(x,y)ÔÚGÄÚ¾ßÓÐÒ»½×Á¬ÐøÆ«µ¼Êý. Èç¹ûÕâÁ½¸öÌõ¼þÖ®Ò»²»ÄÜÂú×ã, ÄÇô¶¨ÀíµÄ½áÂÛ²»Äܱ£Ö¤³ÉÁ¢.

ÆÆ»µº¯ÊýP¡¢Q¼°¡¢Á¬ÐøÐԵĵã³ÆΪÆæµã.

Àý5¼ÆËã, ÆäÖÐLΪÅ×ÎïÏßy=x2ÉÏ´ÓO(0, 0)µ½B(1, 1)µÄÒ»¶Î»¡.

½â: ÒòΪÔÚÕû¸öxOyÃæÄÚ¶¼³ÉÁ¢,

ËùÒÔÔÚÕû¸öxOyÃæÄÚ, »ý·ÖÓë·¾¶ÎÞ¹Ø.

.

ÌÖÂÛ:ÉèLΪһÌõÎÞÖص㡢·Ö¶Î¹â»¬ÇÒ²»¾­¹ýÔ­µãµÄÁ¬Ðø±ÕÇúÏß,LµÄ·½ÏòΪÄæʱÕë·½Ïò, ÎÊÊÇ·ñÒ»¶¨³ÉÁ¢£¿

Ìáʾ:ÕâÀïºÍÔÚµã(0, 0)²»Á¬Ðø.

ÒòΪµ±x2+y2¹0ʱ,, ËùÒÔÈç¹û(0, 0)²»ÔÚLËùΧ³ÉµÄÇøÓòÄÚ, Ôò½áÂÛ³ÉÁ¢, ¶øµ±(0, 0)ÔÚLËùΧ³ÉµÄÇøÓòÄÚʱ, ½áÂÛδ±Ø³ÉÁ¢.

Èý¡¢¶þÔªº¯ÊýµÄȫ΢·ÖÇó»ý

ÇúÏß»ý·ÖÔÚGÄÚÓë·¾¶ÎÞ¹Ø, ±íÃ÷ÇúÏß»ý·ÖµÄÖµÖ»ÓëÆðµã´Óµã(x0,y0)ÓëÖÕµã(x,y)ÓйØ.

Èç¹ûÓë·¾¶ÎÞ¹Ø, Ôò°ÑËü¼ÇΪ

¼´.

ÈôÆðµã(x0,y0)ΪGÄÚµÄÒ»¶¨µã, ÖÕµã(x,y)ΪGÄڵĶ¯µã, Ôò

u(x,y)

ΪGÄڵĵĺ¯Êý.

¶þÔªº¯Êýu(x,y)µÄȫ΢·ÖΪdu(x,y)=ux(x,y)dx+uy(x,y)dy.

±í´ïʽP(x,y)dx+Q(x,y)dyÓ뺯ÊýµÄȫ΢·ÖÓÐÏàͬµÄ½á¹¹, µ«Ëüδ±Ø¾ÍÊÇij¸öº¯ÊýµÄȫ΢·Ö. ÄÇôÔÚʲôÌõ¼þϱí´ïʽP(x,y)dx+Q(x,y)dyÊÇij¸ö¶þÔªº¯Êýu(x,y)µÄȫ΢·ÖÄØ£¿µ±ÕâÑùµÄ¶þÔªº¯Êý´æÔÚʱÔõÑùÇó³öÕâ¸ö¶þÔªº¯ÊýÄØ£¿

¶¨Àí3É迪ÇøÓòGÊÇÒ»¸öµ¥Á¬Í¨Óò, º¯ÊýP(x,y)¼°Q(x,y)ÔÚGÄÚ¾ßÓÐÒ»½×Á¬ÐøÆ«µ¼Êý, ÔòP(x,y)dx+Q(x,y)dyÔÚGÄÚΪijһº¯Êýu(x,y)µÄȫ΢·ÖµÄ³ä·Ö±ØÒªÌõ¼þÊǵÈʽ

ÔÚGÄÚºã³ÉÁ¢.

¼òÒªÖ¤Ã÷:

±ØÒªÐÔ: ¼ÙÉè´æÔÚijһº¯Êýu(x,y), ʹµÃdu=P(x,y)dx+Q(x,y)dy,

ÔòÓÐ,. ÒòΪ¡¢Á¬Ðø, ËùÒÔ, ¼´.

³ä·ÖÐÔ: ÒòΪÔÚGÄÚ, ËùÒÔ»ý·ÖÔÚGÄÚÓë·¾¶ÎÞ¹Ø. ÔÚGÄÚ´Óµã(x0,y0)µ½µã(x,y)µÄÇúÏß»ý·Ö¿É±íʾΪu(x,y).

ÒòΪu(x,y)

,

ËùÒÔ.

ÀàËƵØÓÐ, ´Ó¶ødu=P(x,y)dx+Q(x,y)dy. ¼´P(x,y)dx+Q(x,y)dyÊÇijһº¯ÊýµÄȫ΢·Ö.

ÇóÔ­º¯ÊýµÄ¹«Ê½:

,

,

.

Àý6ÑéÖ¤:ÔÚÓÒ°ëƽÃæ(x>0)ÄÚÊÇij¸öº¯ÊýµÄȫ΢·Ö, ²¢Çó³öÒ»¸öÕâÑùµÄº¯Êý.

½â: ÕâÀï,.

ÒòΪP¡¢QÔÚÓÒ°ëƽÃæÄÚ¾ßÓÐÒ»½×Á¬ÐøÆ«µ¼Êý, ÇÒÓÐ

,

ËùÒÔÔÚÓÒ°ëƽÃæÄÚ,ÊÇij¸öº¯ÊýµÄȫ΢·Ö.

È¡»ý·Ö·ÏßΪ´ÓA(1, 0)µ½B(x, 0)ÔÙµ½C(x,y)µÄÕÛÏß, ÔòËùÇóº¯ÊýΪ

.

ÎÊ:Ϊʲô(x0,y0)²»È¡(0, 0)?

Àý7ÑéÖ¤: ÔÚÕû¸öxOyÃæÄÚ,xy2dx+x2ydyÊÇij¸öº¯ÊýµÄȫ΢·Ö, ²¢Çó³öÒ»¸öÕâÑùµÄº¯Êý.

½â ÕâÀïP=xy2,Q=x2y.

ÒòΪP¡¢QÔÚÕû¸öxOyÃæÄÚ¾ßÓÐÒ»½×Á¬ÐøÆ«µ¼Êý, ÇÒÓÐ

,

ËùÒÔÔÚÕû¸öxOyÃæÄÚ,xy2dx+x2ydyÊÇij¸öº¯ÊýµÄȫ΢·Ö.

È¡»ý·Ö·ÏßΪ´ÓO(0, 0)µ½A(x, 0)ÔÙµ½B(x,y)µÄÕÛÏß, ÔòËùÇóº¯ÊýΪ

.

˼¿¼ÓëÁ·Ï°:

1.ÔÚµ¥Á¬Í¨ÇøÓòGÄÚ, Èç¹ûP(x,y)ºÍQ(x,y)¾ßÓÐÒ»½×Á¬ÐøÆ«µ¼Êý, ÇÒºãÓÐ, ÄÇô

(1)ÔÚGÄÚµÄÇúÏß»ý·ÖÊÇ·ñÓë·¾¶ÎÞ¹Ø?

(2)ÔÚGÄڵıÕÇúÏß»ý·ÖÊÇ·ñΪÁã?

(3) ÔÚGÄÚP(x,y)dx+Q(x,y)dyÊÇ·ñÊÇijһº¯Êýu(x,y)µÄȫ΢·Ö?

2.ÔÚÇøÓòGÄÚ³ýM0µãÍâ, Èç¹ûP(x,y)ºÍQ(x,y)¾ßÓÐÒ»½×Á¬ÐøÆ«µ¼Êý, ÇÒºãÓÐ,G1ÊÇGÄÚ²»º¬M0µÄµ¥Á¬Í¨ÇøÓò, ÄÇô

(1)ÔÚG1ÄÚµÄÇúÏß»ý·ÖÊÇ·ñÓë·¾¶ÎÞ¹Ø?

(2)ÔÚG1ÄڵıÕÇúÏß»ý·ÖÊÇ·ñΪÁã?

(3) ÔÚG1ÄÚP(x,y)dx+Q(x,y)dyÊÇ·ñÊÇijһº¯Êýu(x,y)µÄȫ΢·Ö?

3. ÔÚµ¥Á¬Í¨ÇøÓòGÄÚ, Èç¹ûP(x,y)ºÍQ(x,y)¾ßÓÐÒ»½×Á¬ÐøÆ«

µ¼Êý,, µ«·Ç³£¼òµ¥, ÄÇô

(1)ÈçºÎ¼ÆËãGÄڵıÕÇúÏß»ý·Ö?

(2)ÈçºÎ¼ÆËãGÄڵķDZÕÇúÏß»ý·Ö?

(3)¼ÆËã, ÆäÖÐLΪÄæʱÕë·½ÏòµÄ

ÉÏ°ëÔ²ÖÜ(x-a)2+y2=a2,y³0,

С½á

1.¸ñÁÖ¹«Ê½

2. ¸ñÁÖ¹«Ê½ÖеĵȼÛÌõ¼þ¡£

½Ìѧ·½Ê½¼°½Ìѧ¹ý³ÌÖÐӦעÒâµÄÎÊÌâ

ÔÚ½Ìѧ¹ý³ÌÖÐҪעÒâ¸ñÁÖ¹«Ê½ºÍÆäÖеĵȼÛÌõ¼þ£¬Òª½áºÏʵÀý£¬·´¸´½²½â¡£

ʦÉú»î¶¯Éè¼Æ

½²¿ÎÌá¸Ù¡¢°åÊéÉè¼Æ

×÷ÒµP214: 2 (1); 3 ; 4 (3) ;

5 (1) , (4) ; 6 (2) , (5)

¡ì11.4¶ÔÃæ»ýµÄÇúÃæ»ý·Ö

Ò»¡¢¶ÔÃæ»ýµÄÇúÃæ»ý·ÖµÄ¸ÅÄîÓëÐÔÖÊ

ÎïÖÊÇúÃæµÄÖÊÁ¿ÎÊÌâ:ÉèSΪÃæÃܶȷǾùÔȵÄÎïÖÊÇúÃæ, ÆäÃæÃܶÈΪr(x,y,z), ÇóÆäÖÊÁ¿: °ÑÇúÃæ·Ö³Én¸öС¿é: DS1, DS2, ¡Á ¡Á ¡Á, DSn(DSiÒ²´ú±íÇúÃæµÄÃæ»ý);ÇóÖÊÁ¿µÄ½üËÆÖµ:((xi,hi,zi)ÊÇDSiÉÏÈÎÒâÒ»µã); È¡¼«ÏÞÇó¾«È·Öµ:(lΪ¸÷С¿éÇúÃæÖ±¾¶µÄ×î´óÖµ).

¶¨ÒåÉèÇúÃæSÊǹ⻬µÄ, º¯Êýf(x,y,z)ÔÚSÉÏÓнç. °ÑSÈÎÒâ·Ö³ÉnС¿é: DS1, DS2, ¡Á ¡Á ¡Á, DSn(DSiÒ²´ú±íÇúÃæµÄÃæ»ý), ÔÚDSiÉÏÈÎÈ¡Ò»µã(xi,hi,zi), Èç¹ûµ±¸÷С¿éÇúÃæµÄÖ±¾¶µÄ×î´óÖµl®0ʱ, ¼«ÏÞ×Ü´æÔÚ, Ôò³Æ´Ë¼«ÏÞΪº¯Êýf(x,y,z)ÔÚÇúÃæSÉ϶ÔÃæ»ýµÄÇúÃæ»ý·Ö»òµÚÒ»ÀàÇúÃæ»ý·Ö, ¼Ç×÷, ¼´

.

ÆäÖÐf(x,y,z)½Ð×ö±»»ýº¯Êý, S½Ð×ö»ý·ÖÇúÃæ.

¶ÔÃæ»ýµÄÇúÃæ»ý·ÖµÄ´æÔÚÐÔ:

ÎÒÃÇÖ¸³öµ±f(x,y,z)Ôڹ⻬ÇúÃæSÉÏÁ¬Ðøʱ¶ÔÃæ»ýµÄÇúÃæ»ý·ÖÊÇ´æÔÚµÄ. ½ñºó×ܼٶ¨f(x,y,z)ÔÚSÉÏÁ¬Ðø.

¸ù¾ÝÉÏÊö¶¨ÒåÃæÃܶÈΪÁ¬Ðøº¯Êýr(x,y,z)µÄ¹â»¬ÇúÃæSµÄÖÊÁ¿M¿É±íʾΪr(x,y,z)ÔÚSÉ϶ÔÃæ»ýµÄÇúÃæ»ý·Ö:

Èç¹ûSÊÇ·ÖƬ¹â»¬µÄÎÒÃǹ涨º¯ÊýÔÚSÉ϶ÔÃæ»ýµÄÇúÃæ»ý·ÖµÈÓÚº¯ÊýÔڹ⻬µÄ

¸÷ƬÇúÃæÉ϶ÔÃæ»ýµÄÇúÃæ»ý·ÖÖ®ºÍ. ÀýÈçÉèS¿É·Ö³ÉÁ½Æ¬¹â»¬ÇúÃæS1¼°S2(¼Ç×÷S=S1+S2)¾Í¹æ¶¨

.

¶ÔÃæ»ýµÄÇúÃæ»ý·ÖµÄÐÔÖÊ:

(1)Éèc1¡¢c2Ϊ³£Êý, Ôò

;

(2)ÈôÇúÃæS¿É·Ö³ÉÁ½Æ¬¹â»¬ÇúÃæS1¼°S2, Ôò

;

(3)ÉèÔÚÇúÃæSÉÏf(x,y,zg(x,y,z), Ôò

;

(4), ÆäÖÐAΪÇúÃæSµÄÃæ»ý.

¶þ¡¢¶ÔÃæ»ýµÄÇúÃæ»ý·ÖµÄ¼ÆËã

ÃæÃܶÈΪf(x,y,z)µÄÎïÖÊÇúÃæµÄÖÊÁ¿Îª.

ÁíÒ»·½Ãæ, Èç¹ûSÓÉ·½³Ìz=z(x,y)¸ø³ö, SÔÚxOyÃæÉϵÄͶӰÇøÓòΪD, ÄÇô ÇúÃæµÄÃæ»ýÔªËØΪ

,

ÖÊÁ¿ÔªËØΪ

.

¸ù¾ÝÔªËØ·¨, ÇúÃæµÄÖÊÁ¿Îª

.

Òò´Ë.

»¯ÇúÃæ»ý·ÖΪ¶þÖØ»ý·Ö:ÉèÇúÃæSÓÉ·½³Ìz=z(x,y)¸ø³ö, SÔÚxOyÃæÉϵÄͶӰÇøÓòΪDxy, º¯Êýz=z(x,y)ÔÚDxyÉϾßÓÐÁ¬ÐøÆ«µ¼Êý, ±»»ýº¯Êýf(x,y,z)ÔÚSÉÏÁ¬Ðø, Ôò

.

Èç¹û»ý·ÖÇúÃæSµÄ·½³ÌΪy=y(z,x),DzxΪSÔÚzOxÃæÉϵÄͶӰÇøÓò, Ôòº¯Êýf(x,y,z)ÔÚSÉ϶ÔÃæ»ýµÄÇúÃæ»ý·ÖΪ

.

Èç¹û»ý·ÖÇúÃæSµÄ·½³ÌΪx=x(y,z),DyzΪSÔÚyOzÃæÉϵÄͶӰÇøÓò, Ôòº¯Êýf(x,y,z)ÔÚSÉ϶ÔÃæ»ýµÄÇúÃæ»ý·ÖΪ

.

Àý1¼ÆËãÇúÃæ»ý·Ö, ÆäÖÐSÊÇÇòÃæx2+y2+z2=a2±»Æ½Ãæ

z=h(0<h<a)½Ø³öµÄ¶¥²¿.

½â SµÄ·½³ÌΪ,Dxy:x2+y2£a2-h2.

ÒòΪ,,

,

ËùÒÔ

.

Ìáʾ:.

Àý2¼ÆËã, ÆäÖÐSÊÇÓÉƽÃæx=0,y=0,z=0¼°x+y+z=1ËùΧ³ÉµÄËÄÃæÌåµÄÕû¸ö±ß½çÇúÃæ.

½â Õû¸ö±ß½çÇúÃæSÔÚƽÃæx=0¡¢y=0¡¢z=0¼°x+y+z=1ÉϵIJ¿·ÖÒÀ´Î¼ÇΪS1¡¢S2¡¢S3¼°S4, ÓÚÊÇ

.

Ìáʾ: S4:z=1-x-y,

.

С½á

1. ¶ÔÃæ»ýµÄÇúÃæ»ý·ÖµÄ¶¨ÒåºÍ¼ÆËã

2. ¸ñÁÖ¹«Ê½ÖеĵȼÛÌõ¼þ¡£

½Ìѧ·½Ê½¼°½Ìѧ¹ý³ÌÖÐӦעÒâµÄÎÊÌâ

ÔÚ½Ìѧ¹ý³ÌÖÐҪעÒâÀûÓÃÇòÃæ×ø±ê¡¢ÖùÃæ×ø±ê¡¢¶Ô³ÆÐÔ¡¢ÖØÐĹ«Ê½£¬¼ò»¯¼ÆËãµÄ¼¼ÇÉ. £¬Òª½áºÏʵÀý£¬·´¸´½²½â¡£

ʦÉú»î¶¯Éè¼Æ

¿ÎºóÏ°Ì⣺1£¬3£¬7

½²¿ÎÌá¸Ù¡¢°åÊéÉè¼Æ

×÷ÒµP218: 4(3); 5(2);6(1), (3), (4);8

¡ì11.5¶Ô×ø±êµÄÇúÃæ»ý·Ö

Ò»¡¢¶Ô×ø±êµÄÇúÃæ»ý·ÖµÄ¸ÅÄîÓëÐÔÖÊ

ÓÐÏòÇúÃæ: ͨ³£ÎÒÃÇÓöµ½µÄÇúÃ涼ÊÇË«²àµÄ. ÀýÈçÓÉ·½³Ìz=z(x,y) ±íʾµÄÇúÃæ·ÖΪÉϲàÓëϲà. Éèn=(cosa,cosb, cosg)ΪÇúÃæÉϵķ¨ÏòÁ¿, ÔÚÇúÃæµÄÉϲàcosg>0, ÔÚÇúÃæµÄϲàcosg<0. ±ÕÇúÃæÓÐÄÚ²àÓëÍâ²àÖ®·Ö.

ÀàËƵØ, Èç¹ûÇúÃæµÄ·½³ÌΪy=y(z,x),ÔòÇúÃæ·ÖΪ×ó²àÓëÓÒ²à, ÔÚÇúÃæµÄÓÒ²àcosb>0, ÔÚÇúÃæµÄ×ó²àcosb<0. Èç¹ûÇúÃæµÄ·½³ÌΪx=x(y,z), ÔòÇúÃæ·ÖΪǰ²àÓëºó²à, ÔÚÇúÃæµÄÇ°²àcosa>0, ÔÚÇúÃæµÄºó²àcosa<0.

ÉèSÊÇÓÐÏòÇúÃæ. ÔÚSÉÏȡһС¿éÇúÃæDS, °ÑDSͶӰµ½xOyÃæÉϵÃһͶӰÇøÓò, ÕâͶӰÇøÓòµÄÃæ»ý¼ÇΪ(Ds)xy.¼Ù¶¨DSÉϸ÷µã´¦µÄ·¨ÏòÁ¿ÓëzÖáµÄ¼Ð½ÇgµÄÓàÏÒcosgÓÐÏàͬµÄ·ûºÅ(¼´cosg¶¼ÊÇÕýµÄ»ò¶¼ÊǸºµÄ). ÎÒÃǹ涨DSÔÚxOyÃæÉϵÄͶӰ(DS)xyΪ

,

ÆäÖÐcosgº0Ò²¾ÍÊÇ(Ds)xy=0µÄÇéÐÎ. ÀàËƵؿÉÒÔ¶¨ÒåDSÔÚyOzÃæ¼°ÔÚzOxÃæÉϵÄͶӰ(DS)yz¼°(DS)zx.

Á÷ÏòÇúÃæÒ»²àµÄÁ÷Á¿: ÉèÎȶ¨Á÷¶¯µÄ²»¿ÉѹËõÁ÷ÌåµÄËٶȳ¡ÓÉ

v(x,y,z)=(P(x,y,z) ,Q(x,y,z) ,R(x,y,z))

¸ø³ö, SÊÇËٶȳ¡ÖеÄһƬÓÐÏòÇúÃæ, º¯ÊýP(x,y,z)¡¢Q(x,y,z)¡¢R(x,y,z)¶¼ÔÚSÉÏÁ¬Ðø, ÇóÔÚµ¥Î»Ê±¼äÄÚÁ÷ÏòSÖ¸¶¨²àµÄÁ÷ÌåµÄÖÊÁ¿, ¼´Á÷Á¿F.

Èç¹ûÁ÷ÌåÁ÷¹ýƽÃæÉÏÃæ»ýΪAµÄÒ»¸ö±ÕÇøÓò, ÇÒÁ÷ÌåÔÚÕâ±ÕÇøÓòÉϸ÷µã´¦µÄÁ÷ËÙΪ(³£ÏòÁ¿)v, ÓÖÉènΪ¸ÃƽÃæµÄµ¥Î»·¨ÏòÁ¿, ÄÇôÔÚµ¥Î»Ê±¼äÄÚÁ÷¹ýÕâ±ÕÇøÓòµÄÁ÷Ìå×é³ÉÒ»¸öµ×Ãæ»ýΪA¡¢Ð±¸ßΪ|v|µÄбÖùÌå.

µ±(v,^n)ʱ, ÕâбÖùÌåµÄÌå»ýΪ

A|v|cosq=Av¡Án.

µ±(v,^n)ʱ, ÏÔÈ»Á÷Ìåͨ¹ý±ÕÇøÓòAµÄÁ÷ÏònËùÖ¸Ò»²àµÄÁ÷Á¿FΪÁã, ¶øAv¡Án=0, ¹ÊF=Av¡Án;

µ±(v,^n)ʱ,Av¡Án<0, ÕâʱÎÒÃÇÈÔ°ÑAv¡Án³ÆΪÁ÷Ìåͨ¹ý±ÕÇøÓòAÁ÷ÏònËùÖ¸Ò»²àµÄÁ÷Á¿, Ëü±íʾÁ÷Ìåͨ¹ý±ÕÇøÓòAʵ¼ÊÉÏÁ÷Ïò-nËùÖ¸Ò»²à, ÇÒÁ÷Ïò-nËùÖ¸Ò»²àµÄÁ÷Á¿Îª-Av¡Án. Òò´Ë, ²»ÂÛ(v,^n)ΪºÎÖµ, Á÷Ìåͨ¹ý±ÕÇøÓòAÁ÷ÏònËùÖ¸Ò»²àµÄÁ÷Á¿¾ùΪAv¡Án.

°ÑÇúÃæS·Ö³ÉnС¿é: DS1, DS2, ¡Á ¡Á ¡Á, DSn(DSiͬʱҲ´ú±íµÚiС¿éÇúÃæµÄÃæ»ý). ÔÚSÊǹ⻬µÄºÍvÊÇÁ¬ÐøµÄÇ°ÌáÏÂ, Ö»ÒªDSiµÄÖ±¾¶ºÜС, ÎÒÃǾͿÉÒÔÓÃDSiÉÏÈÎÒ»µã(xi,hi,zi)´¦µÄÁ÷ËÙ

vi=v(xi,hi,zi)=P(xi,hi,zi)i+Q(xi,hi,zi)j+R(xi,hi,zi)k

´úÌæDSiÉÏÆäËü¸÷µã´¦µÄÁ÷ËÙ, ÒԸõã(xi,hi,zi)´¦ÇúÃæSµÄµ¥Î»·¨ÏòÁ¿

ni=cosaii+cosbij+ cosgik

´úÌæDSiÉÏÆäËü¸÷µã´¦µÄµ¥Î»·¨ÏòÁ¿. ´Ó¶øµÃµ½Í¨¹ýDSiÁ÷ÏòÖ¸¶¨²àµÄÁ÷Á¿µÄ½üËÆֵΪ

vi¡ÁniDSi(i=1, 2, ¡Á ¡Á ¡Á ,n)

ÓÚÊÇ, ͨ¹ýSÁ÷ÏòÖ¸¶¨²àµÄÁ÷Á¿

,

µ« cosai¡ÁDSi»(DSi)yz, cosbi¡ÁDSi»(DSi)zx, cosgi¡ÁDSi»(DSi)xy,

Òò´ËÉÏʽ¿ÉÒÔд³É

;

Áîl®0È¡ÉÏÊöºÍµÄ¼«ÏÞ, ¾ÍµÃµ½Á÷Á¿FµÄ¾«È·Öµ. ÕâÑùµÄ¼«ÏÞ»¹»áÔÚÆäËüÎÊÌâÖÐÓöµ½. ³éÈ¥ËüÃǵľßÌåÒâÒå, ¾ÍµÃ³öÏÂÁжÔ×ø±êµÄÇúÃæ»ý·ÖµÄ¸ÅÄî.

Ìáʾ: °ÑDSi¿´³ÉÊÇһС¿éƽÃæ, Æä·¨ÏßÏòÁ¿Îªni, Ôòͨ¹ýDSiÁ÷ÏòÖ¸¶¨²àµÄÁ÷Á¿½üËƵصÈÓÚÒ»¸öбÖùÌåµÄÌå»ý.

´ËбÖùÌåµÄб¸ßΪ|vi|, ¸ßΪ|vi|cos(vi,^ni)=vi¡Áni, Ìå»ýΪvi¡ÁniDSi.

ÒòΪni=cosaii+cosbij+ cosgik,

vi=v(xi,hi,zi)=P(xi,hi,zi)i+Q(xi,hi,zi)j+R(xi,hi,zi)k,

vi¡ÁniDSi=[P(xi,hi,zi)cosai+Q(xi,hi,zi)cosbi+R(xi,hi,zi)cosgi]DSi,

¶ø cosai¡ÁDSi»(DSi)yz, cosbi¡ÁDSi»(DSi)zx, cosgi¡ÁDSi»(DSi)xy,

ËùÒÔvi¡ÁniDSi»P(xi,hi,zi)(DSi)yz+Q(xi,hi,zi)(DSi)zx+R(xi,hi,zi)(DSi)xy.

¶ÔÓÚSÉϵÄÒ»¸öС¿és, ÏÔÈ»ÔÚDtʱ¼äÄÚÁ÷¹ýsµÄÊÇÒ»¸öÍäÇúµÄÖùÌå. ËüµÄÌå»ý½üËÆÓÚÒÔsΪµ×, ¶ø¸ßΪ

(|V|Dt)cos(V,^n)=V¡ÁnDt

µÄÖùÌåµÄÌå»ý:V¡ÁnDtDS, ÕâÀïn=(cosa,cosb, cosg)ÊÇsÉϵĵ¥Î»·¨ÏòÁ¿, DS±íʾsµÄÃæ»ý. ËùÒÔµ¥Î»Ê±¼äÄÚÁ÷ÏòsÖ¸¶¨²àµÄÁ÷ÌåµÄÖÊÁ¿½üËÆÓÚ

V¡ÁnDS»(P(x,y,z)cosa+Q(x,y,z)cosb+R(x,y,z)cosg)DS.

Èç¹û°ÑÇúÃæS·Ö³ÉnС¿ési(i=1, 2, ¡¤ ¡¤ ¡¤ ,n), µ¥Î»Ê±¼äÄÚÁ÷ÏòSÖ¸¶¨²àµÄÁ÷ÌåµÄÖÊÁ¿½üËÆÓÚ

m.

°´¶ÔÃæ»ýµÄÇúÃæ»ý·ÖµÄ¶¨Òå,

.

ÉáÈ¥Á÷ÌåÕâ¸ö¾ßÌåµÄÎïÀíÄÚÈÝ, ÎÒÃǾͳéÏó³öÈç϶Ô×ø±êµÄÇúÃæ»ý·ÖµÄ¸ÅÄî.

¶¨ÒåÉèSΪ¹â»¬µÄÓÐÏòÇúÃæ, º¯ÊýR(x,y,z)ÔÚSÉÏÓнç. °ÑSÈÎÒâ·Ö³Én¿éСÇúÃæDSi(DSiͬʱҲ´ú±íµÚiС¿éÇúÃæµÄÃæ»ý). ÔÚxOyÃæÉϵÄͶӰΪ(DSi)xy, (xi,hi,zi)ÊÇDSiÉÏÈÎÒâÈ¡¶¨µÄÒ»µã. Èç¹ûµ±¸÷С¿éÇúÃæµÄÖ±¾¶µÄ×î´óÖµl®0ʱ,

×Ü´æÔÚ, Ôò³Æ´Ë¼«ÏÞΪº¯ÊýR(x,y,z)ÔÚÓÐÏòÇúÃæSÉ϶Ô×ø±êx¡¢yµÄÇúÃæ»ý·Ö:,

¼Ç×÷,

¼´.

ÀàËƵØÓÐ

.

.

ÆäÖÐR(x,y,z)½Ð×ö±»»ýº¯Êý, S½Ð×ö»ý·ÖÇúÃæ.

¶¨ÒåÉèSÊÇ¿Õ¼äÄÚÒ»¸ö¹â»¬µÄÇúÃæ,n=(cosa,cosb, cosg)ÊÇÆäÉϵĵ¥Î»·¨ÏòÁ¿,V(x,y,z)=(P(x,y,z),Q(x,y,z),R(x,y,z))ÊÇÈ·ÔÚSÉϵÄÏòÁ¿³¡. Èç¹ûÏÂÁи÷ʽÓҶ˵Ļý·Ö´æÔÚ, ÎÒÃǶ¨Òå

,

,

.

²¢³ÆΪPÔÚÇúÃæSÉ϶Ô×ø±êy¡¢zµÄÇúÃæ»ý·Ö,ΪQÔÚÇúÃæSÉ϶Ô×ø±êz¡¢xµÄÇúÃæ»ý·Ö,ΪRÔÚÇúÃæSÉ϶Ô×ø±êy¡¢zµÄÇúÃæ»ý·Ö. ÆäÖÐP¡¢Q¡¢R½Ð×ö±»»ýº¯Êý, S½Ð×ö»ý·ÖÇúÃæ.

ÒÔÉÏÈý¸öÇúÃæ»ý·ÖÒ²³ÆΪµÚ¶þÀàÇúÃæ»ý·Ö.

¶Ô×ø±êµÄÇúÃæ»ý·ÖµÄ´æÔÚÐÔ:

¶Ô×ø±êµÄÇúÃæ»ý·ÖµÄ¼ò¼ÇÐÎʽ:

ÔÚÓ¦ÓÃÉϳöÏֽ϶àµÄÊÇ

.

Á÷ÏòSÖ¸¶¨²àµÄÁ÷Á¿F¿É±íʾΪ

F.

Ò»¸ö¹æ¶¨:Èç¹ûÊÇ·ÖƬ¹â»¬µÄÓÐÏòÇúÃæ, ÎÒÃǹ涨º¯ÊýÔÚSÉ϶Ô×ø±êµÄÇúÃæ»ý·ÖµÈÓÚº¯ÊýÔÚ¸÷Ƭ¹â»¬ÇúÃæÉ϶Ô×ø±êµÄÇúÃæ»ý·ÖÖ®ºÍ.

¶Ô×ø±êµÄÇúÃæ»ý·ÖµÄÐÔÖÊ:

¶Ô×ø±êµÄÇúÃæ»ý·Ö¾ßÓÐÓë¶Ô×ø±êµÄÇúÏß»ý·ÖÀàËƵÄһЩÐÔÖÊ. ÀýÈç

(1)Èç¹û°ÑS·Ö³ÉS1ºÍS2, Ôò

.

(2)ÉèSÊÇÓÐÏòÇúÃæ, -S±íʾÓëSÈ¡Ïà·´²àµÄÓÐÏòÇúÃæ, Ôò

.

ÕâÊÇÒòΪÈç¹ûn=(cosa,cosb, cosg)ÊÇSµÄµ¥Î»·¨ÏòÁ¿, Ôò-SÉϵĵ¥Î»·¨ÏòÁ¿ÊÇ

-n=(- cosa,-cosb, -cosg).

¶þ¡¢¶Ô×ø±êµÄÇúÃæ»ý·ÖµÄ¼ÆËã·¨

½«ÇúÃæ»ý·Ö»¯Îª¶þÖØ»ý·Ö: Éè»ý·ÖÇúÃæSÓÉ·½³Ìz=z(x,y)¸ø³öµÄ, SÔÚxOyÃæÉϵÄͶӰÇøÓòΪDxy, º¯Êýz=z(x,y)ÔÚDxyÉϾßÓÐÒ»½×Á¬ÐøÆ«µ¼Êý, ±»»ýº¯ÊýR(x,y,z)ÔÚSÉÏÁ¬Ðø, ÔòÓÐ

,

ÆäÖе±SÈ¡Éϲàʱ, »ý·ÖÇ°È¡¡°+¡±; µ±Sȡϲàʱ, »ý·ÖÇ°È¡¡°-¡±.

ÕâÊÇÒòΪ, °´¶Ô×ø±êµÄÇúÃæ»ý·ÖµÄ¶¨Òå, ÓÐ

=.

µ±SÈ¡Éϲàʱ, cosg>0, ËùÒÔ(DSi)xy=(Dsi)xy.ÓÖÒò(xi,hi,zi)ÊÇSÉϵÄÒ»µã, ¹Êzi=z(xi,hi).

´Ó¶øÓÐ

.

Áîl®0È¡ÉÏʽÁ½¶ËµÄ¼«ÏÞ, ¾ÍµÃµ½

.

ͬÀíµ±Sȡϲàʱ, ÓÐ

.

ÒòΪµ±SÈ¡Éϲàʱ, cosg>0, (DSi)xy=(Dsi)xy. µ±(xi,hi,zi)ÎSʱ,zi=z(xi,hi). ´Ó¶øÓÐ

.

ͬÀíµ±Sȡϲàʱ, ÓÐ

.

ÕâÊÇÒòΪn=(cosa,cosb, cosg),,

,

.

ÀàËƵØ, Èç¹ûSÓÉx=x(y,z)¸ø³ö, ÔòÓÐ

.

Èç¹ûSÓÉy=y(z,x)¸ø³ö, ÔòÓÐ

.

ӦעÒâµÄÎÊÌâ: ӦעÒâ·ûºÅµÄÈ·¶¨.

Àý1.¼ÆËãÇúÃæ»ý·Ö, ÆäÖÐSÊdz¤·½ÌåWµÄÕû¸ö±íÃæµÄÍâ²à, W=((x,y,z) |0£x£a, 0£y£b, 0£z£c).

½â: °ÑWµÄÉÏÏÂÃæ·Ö±ð¼ÇΪS1ºÍS2; Ç°ºóÃæ·Ö±ð¼ÇΪS3ºÍS4; ×óÓÒÃæ·Ö±ð¼ÇΪS5ºÍS6.

S1:z=c(0£x£a, 0£y£b)µÄÉϲà; S2:z=0 (0£x£a, 0£y£b)µÄϲà;

S3:x=a(0£y£b, 0£z£c)µÄÇ°²à; S4:x=0 (0£y£b, 0£z£c)µÄºó²à;

S5:y=0 (0£x£a, 0£z£c)µÄ×ó²à. S6:y=b(0£x£a, 0£z£c)µÄÓÒ²à;

³ýS3¡¢S4Íâ, ÆäÓàËÄƬÇúÃæÔÚyOzÃæÉϵÄͶӰΪÁã, Òò´Ë

=a2bc.

ÀàËƵؿɵÃ

,.

ÓÚÊÇËùÇóÇúÃæ»ý·ÖΪ(a+b+c)abc.

Àý2¼ÆËãÇúÃæ»ý·Ö, ÆäÖÐSÊÇÇòÃæx2+y2+z2=1Íâ²àÔÚx³0,y³0µÄ²¿·Ö.

½â °ÑÓÐÏòÇúÃæS·Ö³ÉÒÔÏÂÁ½²¿·Ö:

:(x³0,y³0)µÄÉϲà,

:(x³0,y³0)µÄϲà.

S1ºÍS2ÔÚxOyÃæÉϵÄͶӰÇøÓò¶¼ÊÇDxy:x2+y2£1(x³0,y³0).

ÓÚÊÇ

.

Èý¡¢Á½ÀàÇúÃæ»ý·ÖÖ®¼äµÄÁªÏµ

Éè»ý·ÖÇúÃæSÓÉ·½³Ìz=z(x,y)¸ø³öµÄ, SÔÚxOyÃæÉϵÄͶӰÇøÓòΪDxy, º¯Êýz=z(x,y)ÔÚDxyÉϾßÓÐÒ»½×Á¬ÐøÆ«µ¼Êý, ±»»ýº¯ÊýR(x,y,z)ÔÚSÉÏÁ¬Ðø.

Èç¹ûSÈ¡Éϲà, ÔòÓÐ

.

ÁíÒ»·½Ãæ, ÒòÉÏÊöÓÐÏòÇúÃæSµÄ·¨ÏòÁ¿µÄ·½ÏòÓàÏÒΪ

,,,

¹ÊÓɶÔÃæ»ýµÄÇúÃæ»ý·Ö¼ÆË㹫ʽÓÐ

.

Óɴ˿ɼû, ÓÐ

.

Èç¹ûSȡϲà, ÔòÓÐ

.

µ«Õâʱ, Òò´ËÈÔÓÐ

,

ÀàËƵؿÉÍƵÃ

,

.

×ÛºÏÆðÀ´ÓÐ

,

ÆäÖÐcosa¡¢cosb¡¢cosgÊÇÓÐÏòÇúÃæSÉϵã(x,y,z)´¦µÄ·¨ÏòÁ¿µÄ·½ÏòÓàÏÒ.

Á½ÀàÇúÃæ»ý·ÖÖ®¼äµÄÁªÏµÒ²¿Éд³ÉÈçÏÂÏòÁ¿µÄÐÎʽ:

, »ò,

ÆäÖÐA=(P,Q,R),n=(cosa, cosb, cosg)ÊÇÓÐÏòÇúÃæSÉϵã(x,y,z)´¦µÄµ¥Î»·¨ÏòÁ¿,

dS=ndS=(dydz,dzdx,dxdy), ³ÆΪÓÐÏòÇúÃæÔª,AnΪÏòÁ¿AÔÚÏòÁ¿nÉϵÄͶӰ.

Àý3¼ÆËãÇúÃæ»ý·Ö, ÆäÖÐSÊÇ

ÇúÃæ½éÓÚƽÃæz=0¼°z=2Ö®¼äµÄ²¿·ÖµÄϲà.

?Ìáʾ:ÇúÃæÉÏÏòϵķ¨ÏòÁ¿Îª(x,y, -1) )

,,.

¹Ê

=8p.

.

С½á

1.Á½ÀàÇúÃæ»ý·Ö¼°ÆäÁªÏµ£»

2.³£ÓüÆË㹫ʽ¼°·½·¨£¬×¢Ò⣺¶þÖØ»ý·ÖÊǵÚÒ»ÀàÇúÃæ»ý·ÖµÄÌØÊâÇé¿ö¡£

½Ìѧ·½Ê½¼°½Ìѧ¹ý³ÌÖÐӦעÒâµÄÎÊÌâ

ÔÚ½Ìѧ¹ý³ÌÖÐҪעÒâ¶þÖØ»ý·ÖÊǵÚÒ»ÀàÇúÃæ»ý·ÖµÄÌØÊâÇé¿ö£¬Á½ÀàÇúÃæ»ý·Ö¼°ÆäÁªÏµ£¬Òª½áºÏʵÀý£¬·´¸´½²½â¡£

ʦÉú»î¶¯Éè¼Æ

1.Á½ÀàÇúÏß»ý·ÖµÄ¶¨ÒåÒ»¸öÓë S µÄ·½ÏòÎÞ¹Ø, Ò»¸öÓë SµÄ·½ÏòÓйØ,ÓëÊéÖÐÁªÏµ¹«Ê½ÊÇ·ñì¶Ü ?

2.¿ÎºóÏ°Ì⣺2£¬3

½²¿ÎÌá¸Ù¡¢°åÊéÉè¼Æ

×÷ÒµP228: 3 (1) ,(2) ,(4) ;4

¡ì11.6¸ß˹¹«Ê½

¸ß˹¹«Ê½

¶¨Àí1Éè¿Õ¼ä±ÕÇøÓòWÊÇÓÉ·ÖƬ¹â»¬µÄ±ÕÇúÃæSËùΧ³É, º¯ÊýP(x,y,z)¡¢Q(x,y,z)¡¢R(x,y,z)ÔÚWÉϾßÓÐÒ»½×Á¬ÐøÆ«µ¼Êý, ÔòÓÐ

,

»ò,

¼òÒªÖ¤Ã÷ ÉèWÊÇÒ»ÖùÌå, Éϱ߽çÇúÃæΪS1:z=z2(x,y), ϱ߽çÇúÃæΪS2:z=z1(x,y), ²àÃæΪÖùÃæS3, S1ȡϲà, S2È¡Éϲà; S3È¡Íâ²à.

¸ù¾ÝÈýÖØ»ý·ÖµÄ¼ÆËã·¨, ÓÐ

.

ÁíÒ»·½Ãæ, ÓÐ

,

,

,

ÒÔÉÏÈýʽÏà¼Ó, µÃ

.

ËùÒÔ.

ÀàËƵØÓÐ

,

,

°ÑÒÔÉÏÈýʽÁ½¶Ë·Ö±ðÏà¼Ó, ¼´µÃ¸ß˹¹«Ê½.

Àý1ÀûÓøß˹¹«Ê½¼ÆËãÇúÃæ»ý·Ö, ÆäÖÐSΪÖùÃæx2+y2=1¼°Æ½Ãæz=0,z=3ËùΧ³ÉµÄ¿Õ¼ä±ÕÇøÓòWµÄÕû¸ö±ß½çÇúÃæµÄÍâ²à.

½â ÕâÀïP=(y-z)x,Q=0,R=x-y,

,,.

Óɸß˹¹«Ê½, ÓÐ

.

Àý2¼ÆËãÇúÃæ»ý·Ö, ÆäÖÐSΪ׶Ãæx2+y2=z2½éÓÚƽÃæz=0¼°z=h(h>0)Ö®¼äµÄ²¿·ÖµÄϲà, cosa¡¢cosb¡¢cosgÊÇSÉϵã(x,y,z)´¦µÄ·¨ÏòÁ¿µÄ·½ÏòÓàÏÒ.

½â ÉèS1Ϊz=h(x2+y2£h2)µÄÉϲà, ÔòSÓëS1Ò»Æð¹¹³ÉÒ»¸ö±ÕÇúÃæ, ¼ÇËüÃÇΧ³ÉµÄ¿Õ¼ä±ÕÇøÓòΪW, Óɸß˹¹«Ê½µÃ

Ìáʾ:.

¶ø,

Òò´Ë.

Àý3É躯Êýu(x,y,z)ºÍv(x,y,z)ÔÚ±ÕÇøÓòWÉϾßÓÐÒ»½×¼°¶þ½×Á¬ÐøÆ«µ¼Êý, Ö¤Ã÷

,

ÆäÖÐSÊDZÕÇøÓòWµÄÕû¸ö±ß½çÇúÃæ,Ϊº¯Êýv(x,y,z)ÑØSµÄÍâ·¨Ïß·½ÏòµÄ·½Ïòµ¼Êý, ·ûºÅ, ³ÆΪÀ­ÆÕÀ­Ë¹Ëã×Ó. Õâ¸ö¹«Ê½½Ð×ö¸ñÁÖµÚÒ»¹«Ê½.

Ö¤: ÒòΪ·½Ïòµ¼Êý

,

ÆäÖÐcosa¡¢cosb¡¢cosgÊÇSÔÚµã(x,y,z)´¦µÄÍâ·¨ÏßÏòÁ¿µÄ·½ÏòÓàÏÒ. ÓÚÊÇÇúÃæ»ý·Ö

.

ÀûÓøß˹¹«Ê½, ¼´µÃ

,

½«ÉÏʽÓҶ˵ڶþ¸ö»ý·ÖÒÆÖÁ×ó¶Ë±ãµÃËùÒªÖ¤Ã÷µÄµÈʽ.

С½á

1.¸ß˹¹«Ê½¼°ÆäÓ¦ÓÃÌõ¼þ£»

2. ¸ß˹¹«Ê½Ó¦ÓõĶÔÏó¡£

½Ìѧ·½Ê½¼°½Ìѧ¹ý³ÌÖÐӦעÒâµÄÎÊÌâ

ÔÚ½Ìѧ¹ý³ÌÖÐҪעÒâ¸ß˹¹«Ê½¼°ÆäÓ¦ÓÃÌõ¼þºÍ¶ÔÏó£¬Òª½áºÏʵÀý£¬·´¸´½²½â¡£

ʦÉú»î¶¯Éè¼Æ

1. Éè å ÊÇÒ»¹â»¬±ÕÇúÃæ,ËùΧÁ¢Ìå W µÄÌå»ýΪV,qÊÇ å Íâ·¨ÏßÏòÁ¿Óëµã (x , y , z) µÄÏò¾¶µÄ¼Ð½Ç,£¬ÊÔÖ¤

½²¿ÎÌá¸Ù¡¢°åÊéÉè¼Æ

×÷ÒµP236: 1£¨1£©£¨4£©£¨5£©

¡ì11.7˹ÍпË˹¹«Ê½

˹ÍпË˹¹«Ê½

¶¨Àí1ÉèGΪ·Ö¶Î¹â»¬µÄ¿Õ¼äÓÐÏò±ÕÇúÏß, SÊÇÒÔGΪ±ß½çµÄ·ÖƬ¹â»¬µÄÓÐÏòÇúÃæ, GµÄÕýÏòÓëS µÄ²à·ûºÏÓÒÊÖ¹æÔò, º¯ÊýP(x,y,z)¡¢Q(x,y,z)¡¢R(x,y,z)ÔÚÇúÃæS(Á¬Í¬±ß½ç)ÉϾßÓÐÒ»½×Á¬ÐøÆ«µ¼Êý, ÔòÓÐ

.

¼ÇÒ䷽ʽ:

,

»ò,

ÆäÖÐn=(cosa, cosb, cosg)ΪÓÐÏòÇúÃæSµÄµ¥Î»·¨ÏòÁ¿.

ÌÖÂÛ: Èç¹ûSÊÇxOyÃæÉϵÄÒ»¿éƽÃæ±ÕÇøÓò, ˹ÍпË˹¹«Ê½½«±ä³Éʲô?

Àý1ÀûÓÃ˹ÍпË˹¹«Ê½¼ÆËãÇúÏß»ý·Ö, ÆäÖÐGΪƽÃæx+y+z=1±»Èý¸ö×ø±êÃæËù½Ø³ÉµÄÈý½ÇÐεÄÕû¸ö±ß½ç, ËüµÄÕýÏòÓëÕâ¸öÈý½ÇÐÎÉϲàµÄ·¨ÏòÁ¿Ö®¼ä·ûºÏÓÒÊÖ¹æÔò.

½â ÉèSΪ±ÕÇúÏßGËùΧ³ÉµÄÈý½ÇÐÎƽÃæ, SÔÚyOzÃæ¡¢zOxÃæºÍxOyÃæÉϵÄͶӰÇøÓò·Ö±ðΪDyz¡¢DzxºÍDxy, °´Ë¹ÍпË˹¹«Ê½, ÓÐ

.

Àý2ÀûÓÃ˹ÍпË˹¹«Ê½¼ÆËãÇúÏß»ý·Ö

,

ÆäÖÐGÊÇÓÃƽÃæ½ØÁ¢·½Ìå: 0£x£1, 0£y£1, 0£z£1µÄ±íÃæËùµÃµÄ½ØºÛ, Èô´ÓxÖáµÄÕýÏò¿´È¥È¡ÄæʱÕë·½Ïò.

½â È¡SΪƽÃæµÄÉϲ౻GËùΧ³ÉµÄ²¿·Ö, SµÄµ¥Î»·¨ÏòÁ¿, ¼´. °´Ë¹ÍпË˹¹«Ê½, ÓÐ

.

,

ÆäÖÐDxyΪSÔÚxOyƽÃæÉϵÄͶӰÇøÓò, ÓÚÊÇ

.

Ìáʾ :..

.

С½á

1.˹ÍпË˹¹«Ê½¼°ÆäÓ¦ÓÃÌõ¼þ£»

2. ˹ÍпË˹¹«Ê½Ó¦ÓõĶÔÏó¡£

½Ìѧ·½Ê½¼°½Ìѧ¹ý³ÌÖÐӦעÒâµÄÎÊÌâ

ÔÚ½Ìѧ¹ý³ÌÖÐҪעÒâ˹ÍпË˹¹«Ê½¼°ÆäÓ¦ÓÃÌõ¼þºÍ¶ÔÏó£¬Òª½áºÏʵÀý£¬·´¸´½²½â¡£

ʦÉú»î¶¯Éè¼Æ

¿ÎºóÏ°Ì⣺1

½²¿ÎÌá¸Ù¡¢°åÊéÉè¼Æ

×÷ÒµP245: 2£¨1£©£¨3£©£¨4£©

Ï°Ìâ¿Î

Ò»¡¢ÇúÏß»ý·ÖµÄ¼ÆËã·¨

1. »ù±¾·½·¨

ÇúÏß»ý·Ö

(1) ͳһ»ý·Ö±äÁ¿

(2) È·¶¨»ý·ÖÉÏÏÂÏÞ

2.»ù±¾¼¼ÇÉ

(1) ÀûÓöԳÆÐÔ¼°ÖØÐĹ«Ê½¼ò»¯¼ÆËã ;

(2) ÀûÓûý·ÖÓë·¾¶Î޹صĵȼÛÌõ¼þ;

(3) ÀûÓøñÁÖ¹«Ê½ (×¢Òâ¼Ó¸¨ÖúÏߵļ¼ÇÉ) ;

(4) ÀûÓÃ˹ÍпË˹¹«Ê½ ;

(5) ÀûÓÃÁ½ÀàÇúÏß»ý·ÖµÄÁªÏµ¹«Ê½ .

¶þ¡¢ÇúÃæ»ý·ÖµÄ¼ÆËã·¨

1. »ù±¾·½·¨

ÇúÃæ»ý·Ö

(1) ͳһ»ý·Ö±äÁ¿ ¡ª ´úÈëÇúÃæ·½³Ì

(2) »ý·ÖÔªËØͶӰ

(3) È·¶¨¶þÖØ»ý·ÖÓò

¡ª °ÑÇúÃæ»ý·ÖÓòͶӰµ½Ïà¹Ø×ø±êÃæ

2.»ù±¾¼¼ÇÉ

(1) ÀûÓöԳÆÐÔ¼°ÖØÐĹ«Ê½¼ò»¯¼ÆËã

(2) ÀûÓøß˹¹«Ê½

(¸¨ÖúÃæÒ»°ãȡƽÐÐ×ø±êÃæµÄƽÃæ)

(3) Á½ÀàÇúÃæ»ý·ÖµÄת»¯

Èý£®ÀýÌâ·ÖÎö

1.P246 Ìâ 3 (1), (3), (6)£¬4£¨1£©£¬£¨3£©£¬£¨4£©£¬6£¬7

2.¼ÆË㣬ÆäÖÐΪÇúÏß¡£

3.¼ÆË㣬ÆäÖÐLÊÇÑØÄæʱÕë·½ÏòÒÔÔ­µãΪÖÐÐÄ£¬aΪ°ë¾¶µÄÉÏ°ëÔ²ÖÜ.

×÷Òµ£º

P246£º 3 (2) , (4) ; 4 (2)

5 ; 8

¸ü¶àÏà¹ØÍƼö£º
¸ßÊýϹ«Ê½×ܽá

¸ßµÈÊýѧϲṫʽ×ܽá1¡¢Nά¿Õ¼äÖÐÁ½µãÖ®¼äµÄ¾àÀ빫ʽ£ºp(x1,x2,...,xn),Q(y1,y2,...,yn)µÄ¾àÀëPQ?2¡¢¶àÔªº¯Êýz?f(x,y)ÇóÆ«µ¼Ê±£¬¶ÔË­ÇóÆ«µ¼£¬¾ÍÒâζ×ÅÆäËüµÄ±äÁ¿¶¼ÔÝʱ¿´×÷³£Á¿¡­

¸ßÊý×ܽá

¸ßµÈÊýѧ¸´Ï°½Ì³ÌµÚÒ»½²º¯ÊýÁ¬ÐøÓ뼫ÏÞÒ»ÀíÂÛÒªÇó1º¯Êý¸ÅÄîÓëÐÔÖÊ2¼«ÏÞ3Á¬Ðø¶þÌâÐÍÓë½â·¨A¼«ÏÞµÄÇ󷨺¯ÊýµÄ»ù±¾ÐÔÖʵ¥µ÷ÓнçÆæżÖÜÆÚ¼¸Àà³£¼ûº¯Êý¸´ºÏ·Ö¶Î·´Òþ³õµÈº¯Êý¼«ÏÞ´æÔÚÐÔÓë×óÓÒ¼«ÏÞÖ®¼äµÄ¹Øϵ¼Ð±Æ¶¨ÀíºÍµ¥µ÷Óн綨...

¸ßÊý×ܽá

¸ßµÈÊýѧ¸´Ï°½Ì³ÌµÚÒ»½²º¯ÊýÁ¬ÐøÓ뼫ÏÞÒ»ÀíÂÛÒªÇó1º¯Êý¸ÅÄîÓëÐÔÖÊ2¼«ÏÞ3Á¬Ðø¶þÌâÐÍÓë½â·¨A¼«ÏÞµÄÇ󷨺¯ÊýµÄ»ù±¾ÐÔÖʵ¥µ÷ÓнçÆæżÖÜÆÚ¼¸Àà³£¼ûº¯Êý¸´ºÏ·Ö¶Î·´Òþ³õµÈº¯Êý¼«ÏÞ´æÔÚÐÔÓë×óÓÒ¼«ÏÞÖ®¼äµÄ¹Øϵ¼Ð±Æ¶¨ÀíºÍµ¥µ÷Óн綨...

¸ßÊýϲá×ܽá

¸ßÊýÏÂС½áһ΢·Ö·½³Ì¸´Ï°Òªµã½â΢·Ö·½³ÌʱÏÈÒªÅжÏһϷ½³ÌÊÇÊôÓÚʲôÀàÐÍÈ»ºó°´ËùÊôÀàÐ͵ÄÏàÓ¦½â·¨Çó³öÆäͨ½âÒ»½×΢·Ö·½³ÌµÄ½â·¨Ð¡½á¶þ½×΢·Ö·½³ÌµÄ½â·¨Ð¡½á1·ÇÆë´Î·½³ÌypyqyfxµÄÌؽâyµÄÐÎʽΪÖ÷ÒªÒ»½×1¿É·ÖÀë±äÁ¿...

¸ßÊýϲá×ܽá(ͬ¼ÃµÚÁù°æ)

¸ßÊýͬ¼Ã°æϸßÊýÏÂС½áһ΢·Ö·½³Ì¸´Ï°Òªµã½â΢·Ö·½³ÌʱÏÈÒªÅжÏһϷ½³ÌÊÇÊôÓÚʲôÀàÐÍÈ»ºó°´ËùÊôÀàÐ͵ÄÏàÓ¦½â·¨Çó³öÆäͨ½âÒ»½×΢·Ö·½³ÌµÄ½â·¨Ð¡½á1¸ßÊýͬ¼Ã°æ϶þ½×΢·Ö·½³ÌµÄ½â·¨Ð¡½á·ÇÆë´Î·½³ÌypyqyfxµÄÌؽâyµÄÐÎ...

¸ßÊýϲṫʽ×ܽá

¸ßµÈÊýѧһ½Ì°¸ÆÚÄ©×ܸ´Ï°µÚ°ËÕÂÏòÁ¿Óë½âÎö¼¸ºÎ2¸ßµÈÊýѧһ½Ì°¸ÆÚÄ©×ܸ´Ï°3¸ßµÈÊýѧһ½Ì°¸ÆÚÄ©×ܸ´Ï°µÚÊ®ÕÂÖØ»ý·Ö4¸ßµÈÊýѧһ½Ì°¸ÆÚÄ©×ܸ´Ï°5¸ßµÈÊýѧһ½Ì°¸ÆÚÄ©×ܸ´Ï°µÚʮһÕÂÇúÏß»ý·ÖÓëÇúÃæ»ý·Ö6¸ßµÈÊýѧһ½Ì°¸ÆÚÄ©×ܸ´...

¸ßÊýϲá×ܽá

µÚËĽ²ÏòÁ¿´úÊý¡¢¶àÔªº¯Êý΢·ÖÓë¿Õ¼ä½âÎö¼¸ºÎÒ»¡¢ÀíÂÛÒªÇó1.ÏòÁ¿´úÊýÀí½âÏòÁ¿µÄ¸ÅÄµ¥Î»ÏòÁ¿¡¢·½ÏòÓàÏÒ¡¢Ä££©Á˽âÁ½¸öÏòÁ¿Æ½ÐС¢´¹Ö±µÄÌõ¼þÏòÁ¿¼ÆËãµÄ¼¸ºÎÒâÒåÓë×ø±ê±íʾÀí½â¶þÔªº¯ÊýµÄ¼¸ºÎÒâÒå¡¢Á¬Ðø¡¢¼«ÏÞ¸ÅÄ±ÕÓòÐÔÖÊ¡­

±±Ê¦´ó°æÆßÄ꼶Êýѧϲá֪ʶµã×ܽá

±±Ê¦´ó°æÊýѧÆßÄ꼶ϲá֪ʶµã×ܽáµÚÒ»ÕÂÕûʽµÄÔËËãÒ»µ¥Ïîʽµ¥ÏîʽµÄ´ÎÊýÖ»º¬ÓÐÊý×ÖÓë×ÖĸµÄ»ýµÄ´úÊýʽ½Ð×öµ¥Ïîʽµ¥¶ÀµÄÒ»¸öÊý»òÒ»¸ö×ÖĸҲÊǵ¥Ïîʽһ¸öµ¥ÏîʽÖÐËùÓÐ×ÖĸµÄÖ¸ÊýµÄºÍ½Ð×öÕâ¸öµ¥ÏîʽµÄ´ÎÊý¶þ¶àÏîʽ1¶àÏîʽ¶àÏîʽ...

±±Ê¦´ó°æÆßÄ꼶Êýѧϲá֪ʶµã×ܽá

µÚÒ»ÕÂÕûʽÔËËã֪ʶµãÒ»¸ÅÄîÓ¦ÓÃ1µ¥ÏîʽºÍ¶àÏîʽͳ³ÆΪÕûʽµ¥ÏîʽÓÐÈýÖÖµ¥¶ÀµÄ×ÖĸawµÈµ¥¶ÀµÄÊý×Ö125Êý×ÖÓë×Öĸ³Ë»ýµÄÒ»°ãÐÎʽ2s332514562µÈ725xaµÈ32µ¥ÏîʽµÄϵÊýÊÇÖ¸Êý×Ö²¿·ÖÈç23abcµÄϵÊýÊÇ2...

³õ¶þÊýѧϲá֪ʶµã×ܽá,³¬¾­µä!

³õ¶þÊýѧÏÂ֪ʶµã×ܽắÊý¼°ÆäÏà¹Ø¸ÅÄî1±äÁ¿Óë³£Á¿ÔÚijһ±ä»¯¹ý³ÌÖпÉÒÔÈ¡²»Í¬ÊýÖµµÄÁ¿½Ð×ö±äÁ¿ÊýÖµ±£³Ö²»±äµÄÁ¿½Ð×ö³£Á¿Ò»°ãµØÔÚijһ±ä»¯¹ý³ÌÖÐÓÐÁ½¸ö±äÁ¿xÓëyÈç¹û¶ÔÓÚxµÄÿһ¸öÖµy¶¼ÓÐΨһȷ¶¨µÄÖµÓëËü¶ÔÓ¦ÄÇô¾Í˵xÊÇ...

¸ßÊý¹«Ê½×ܽá

¸ßµÈÊýѧ¹«Ê½µ¼Êý¹«Ê½2tgxsecx2arcsinx11x2ctgxcscxarccosx11x21x2secxsecxtgxcscxcscxctgxaalnalogaxxarctgxx1xlnaarcctgx...

¸ßÊý×ܽá Îĵµ

¸ßÊý×ܽá Îĵµ£¬ÄÚÈݸ½Í¼¡£

¸ßÊýÏÂ×ܽᣨ39ƪ£©