Mecanique Des Solides - Cours

K. HWX IN INV V — Tg Tgng ngst strrg ; & 2 ;.

Nãdk`cqug hgs sjochgs - DJWRT

@jtc @j tcj` j`ss vgd vgdtj tjrc rcgo goog ogss hg bk bksg sg

 

   



Xjrsgur  < jutco nktmãnktcqug djnpjsãg h–u` vgdtgur kppgoã rãsuotk`tg gt h–u` dmknp  hg vgdtgurs kppgoã njng`t gt vãrceck`t ok rgoktcj` <

@jtktcj` <

 >  +  ∩ 

   >    

gt

 Ok rãsuotk`tg gst c`hãpg`hk`tg hu pjc`t dj`schãrã. •

Tjct hgux tjrsgurs

 Og njn jng` g`tt hã hãp pg`h g`h hu pjc` jc`t dj`schãrã. •

      >      

   >     

- Xjrsgur dc`ãnktcqug gst gst

u`

  

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HWX IN INV V — Tg Tgng ngst strrg ; & 2 2.

Nãdk`cqug hgs sjochgs - DJWRT

@jtc @j tcj` j`ss hg hg Dc` Dc`ãn ãnkt ktcq cqug ug

[u– [u–gst gst dg qug qug ok Dc`ã Dc`ãnk nktc tcqu qug g:

- Rgpírgs, hãrcvktcj` hãrcvktcj` vgdtjrcgoog

 .      > ⃙.   +  . 

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- Djnpjsctcj` hgs njuvgng`ts

;6 / 36

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HW HWX X INV INV — Tg Tgng ngsstr trg g ;& 2 2.

Tjct hgux rgpírgs R(J,x,y,z) gt R;(J,x;,y;,z;). J` suppjsg qug R;  tjur`g kutjur hg R (vjcr eciurg dc-dj`trg), gt qug sk pjsctcj` c`stk`tk`ãg gst hãec`cg pkr og pkrknítrg pkrknítr g k`iuokcrg ν(t).

Tjct u` rgpírg R(J,x,y,z).

N(t)

•

 

 K ok hktg t, ok pjsctcj` h–u` pjc`t N gst hj``ã pkr og vgdtgur hg R.

 /

, J ãtk`t o–jrcic`g

J` kpp kppgoog goog vgd vgdtgu tgurr rjtk rjtktcj tcj` ` og vgd vgdtgu tgurr quc hãec`ct ok vctgssg hg rjtktcj` hg R; pkr

 J` hãec`ct ok vctgssg hg N pkr rkppjrt æ R

/ >      

rkp kppj pjrt rt æ R, `j `jtã tã

pkr ok hãrcvktcj` hu vgdtgur pjsctcj` hg N <

•

 J` hãec`ct o–kddãoãrktcj` hg N pkr rkppjrt æ R pkr ok hãrcvktcj` hu vgdtgur vctgssg hg N <

  /  / >    

Nãdk`cqug hgs sjochgs - DJWRT

@jtc @j tcj` j`ss hg hg Dc` Dc`ãn ãnkt ktcq cqug ug

Ugdtgur gdtgur Rjtk Rjtktcj tcj` `<

Ugdt gdtgur gurss < Vj Vjsct sctcj` cj`,, Uctgss Uctgssg, g, Kddãoã Kddãoãrk rktcj tcj` `

•

Nãdk`cqug hgs sjochgs - DJWRT

@jtc @j tcj` j`ss hg hg Dc` Dc`ãn ãnkt ktcq cqug ug

- Irk`hgurs Irk`hgurs ãtuhcãgs < pjsctcj`, vctgssg, kddãoãrktcj`

.

Ogur gur pr prjh jhu uct tjrsjr jrcg cgoo sdkokcrg <



2.

Gt Gtuhg uhg hu nju njuvgn vgng`t g`t hgs dj djrps rps,, c`hãpg c`hãpg`hk `hknng nng`t `t hgs ejrd ejrdgs gs quc ogs prjhucsg`t.

Vr Vrjhu jhuct ct hg hgux hgux tjrs tjrsgur gurss<

W` tjrsgur hãpg`h hu pjc`t jó co gst ãdrct <

K. HW HWX X INV INV — Tg Tgng ngsstr trg g ;& 2

⃙ >   Σ/ >  .⃙Σ/ 

  gt tgo qug

 <

Hãrcvktcj` Hãrcv ktcj` vgdt vgdtjrcgo jrcgoog og <

 

Tjct Tjct u` vgd vgdtgu tgurr   gt u` rgpírg R;(J;,x;,y;,z;) njbcog njbcog hk` hk`ss u` rgpír rgpírg g R(J,x,y,z) R(J,x,y ,z) ecxg. J` hãec`ct ok hãrcvktcj` hg  hk`s R pkr <

    >     + / ∩ 

;; / 36

Rkpmkáo UGOKTDJ < CWX h Kncg`s — Hãpkrtgng`t INV

;2 / 36

    +   ∩ 

Rkpmkáo UGOKTDJ < CWX h Kncg`s — Hãpkrtgng`t INV

 

K. HWX IN INV V — Tg Tgng ngst strrg ; & 2 4.

Nãdk`cqug hgs sjochgs - DJWRT

@jttcj @j cj`s `s hg Ttktc tcq qug

K. HW HWX X INV INV — Tg Tgng ngsstr trg g ;& 2 4.

Nãdk`cqug hgs sjochgs - DJWRT

@jttcj @j cj`s `s hg Ttktc tcq qug

 

Xjrsgur jrsgur hgs kdtcj` kdtcj`ss nãd nãdk`c k`cqug qugss <

[u [u––gst gst dg qug qug ok Tt Ttkt ktcq cqug ug :

Gtuh Gtuhg g hg o–ãq o–ãquc ucoc ocbr brg g hgs hgs djrp djrpss sj sjus us o–kd o–kdtc tcj` j` hg hgss ejrd ejrdgs gs qu qucc ou oucc sj`t sj`t kppocquãgs. - Xjrsgur hgs kdtcj`s nãdk`cqugs

Tjct Tjct u` sys ysttín íng g hg sj sjoc ochg hgss T su surr og ogqu qugo go s– s–gx gxgr grdg dg` `t hg hgss kd kdtc tcj` j`ss nãdk`cqugs. J` hãdrct ogs kdtcj`s sur T ku pjc`t V pkr og tjrsgur hgs kdtcj`s nãdk`cqugs <

  ̇

rgprãsg`tg og ncocgu gxtãrcgur æ T. rgprãsg`tg

  ⟵⟵̇    >    ̇⟵⟵  ̇⟵⟵   

Ok rgoktcj` ej`hkng`tkog hgs tjrsgurs s–ãdrct hk`s og dks hu tjrsgur hgs kdtcj`s nãdk`cqugs <

- Ojc hg Djuojnb < erjttgng`t, khm khmãrg`dg ãrg`dg

- Xmãjríngs iã`ãrkux < rrãsuotk`tg, ãsuotk`tg, njng`t, kdtcj`s kdtcj`s nutugoogs

2; / 36

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HWX IN INV V — Tg Tgng ngst strrg ; & 2 4.

 >  +  ∩  ⇒    ⟵̇⟵   >    ⟵̇⟵  +  + ∩    ⟵⟵̇  ⟵   >    ⟵  ⟵   

Tc j` ãtuhcg ogs c`tgrkdtcj`s hg hgux sjochgs T;   gt T2  g`trg gux, co gst pjsscbog h–utcocsgr og tjrsgur hgs kdtcj`s nãdk`cqugs pjur hãdrcrg o–kdtcj` h–u` sjochg sur o–kutrg. O–kdtcj` hg T2 sur T; s–ãdrct kojrs <

- Vrc`dcpg ej`hkng` ej`hkng`tko tko hg ok stktcqug

@jttcj @j cj`s `s hg Ttktc tcq qug

Nãdk`cqug hgs sjochgs - DJWRT

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Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HW HWX X INV INV — Tg Tgng ngsstr trg g ;& 2 4.

Nãdk`cqug hgs sjochgs - DJWRT

@jttcj @j cj`s `s hg Ttktc tcq qug

Ojc hg Dju Djuojn ojnb b (er (erjtt jttgng gng`t, `t, khm khmãr ãrg`d g`dg) g) <

Ojc hg Dju Djuojn ojnb b (erjt (erjttgn tgng` g`t, t, khm khmãr ãrg`d g`dg) g) <

Tjct T; gt T2 hgux sjochgs g` dj`tkdt surekdcqug.

Kec` h–ã`j`dgr ogs ojcs hg Djuojnb ku `cvgku hg ok surekdg (T) hg dj`tkdt g`trg T; gt T2, j` hãec`ct < •

∍/

 ∍ ()

 ok vctgssg hg iocssgng`t hg T 2 pkr rkppjrt æ T; g` V

Ogs ojcs hg Djuojnb hj``g`t hgs c`ejrnktcj`s c`ejrnktcj`s sur ogs geejrts geejrts `jrnko gt tk`ig`tcgo, s–co y k khmãrg`dg ju erjttgng`t g`trg T; gt T2 <

•

•

•

•

Φ   ⟵  ⟵

gst og pok` tk`ig`t djnnu` æ T; gt T2 ku `cvgku hu dj`tkdt   gst ok ejrdg ejrdg gxgrd gxgrdãg ãg pkr T; sur T2   gst ok prjlgdtcj` `jrnkog hg ok ejrdg ejrdg gxgrdãg pkr T; sur T2

  gst ok prjlgdtcj` tk`ig`tcgoog tk`ig`tcgoog hg ok ejrdg gxgrdãg gxgrdãg pkr T; sur T2

 Erjttgng`t <

•

 Khmãrg`dg <

•

∍/ ≦ 6 ∍/

 >6

  (iocssgng`t rrgoktce goktce ku pjc`t V)

  (pks hg i iocssgng`t ocssgng`t ku pjc`t V)

24 / 36



Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HWX IN INV V — Tg Tgng ngst strrg ; & 2 4.

Nãdk`cqug hgs sjochgs - DJWRT

@jttcj @j cj`s `s hg Ttktc tcq qug

  6

28 / 36

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HW HWX X INV INV — Tg Tgng ngsstr trg g ;& 2 4.

Nãdk`cqug hgs sjochgs - DJWRT

@jttcj @j cj`s `s hg Ttktc tcq qug

 

Ojc hg Dju Djuojn ojnb b (er (erjtt jttgng gng`t, `t, khm khmãr ãrg`d g`dg) g) <

Ojc hg Dju Djuojn ojnb b (erjt (erjttgn tgng` g`t, t, khm khmãr ãrg`d g`dg) g) <

∍/ ≦ 6  ⟵  > .  ⟵

∍/ > 6  ⟵   ≨ .  ⟵  ⟵  ≨ .  ⟵

Dks hu erjttgng`t <

Dks hg o–khmãrg`dg <

O–geejrt hg dj`tkdt ku pjc`t V gst tgo qug <

O–geejrt hg dj`tkdt ku pjc`t V gst tgo qug < ju pkr kpprjx kpprjxcnktcj` cnktcj` <

•

 e>tk`(ϗ) gst kppgoã djgeecdcg`t hg erjttgng`t hgs nktãrckux ; gt 2.

 ϗ gs gstt o–k` o–k`io iog g hg erjt erjttg tgng ng`t `t gt pgrn pgrngt gt hg dj`s dj`str truc ucrg rg og dø`g dø`g hg

  ⟵

•

erjttgng`t sur ogqugo sg trjuvg trjuvg o– o–geejrt geejrt hg dj`tkdt dj`tkdt

 .

Rgnkrqug  < O–geejrt tk`ig`tcgo gst jppjsã ku vgdtgur hg iocssgng`t hu pjc`t V hg T2 pkr rkppjrt æ T;. 27 / 36

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HWX IN INV V — Tg Tgng ngst strrg ; & 2 4.

@jttcj @j cj`s `s hg Ttktc tcq qug

 e 6>tk`(ϗ6) gst kp kppgo pgoã ã djgee djgeecdc cdcg`t g`t h– h–khm khmãr ãrg`d g`dg g hgs nk nktãr tãrcku ckuxx ; gt 2, sjuvg`t ksscncoã æ e pkr scnpocecdktcj`.

•

Nãdk`cqug hgs sjochgs - DJWRT

 ϗ6 gst o–k`iog h–khmãrg`dg gt pgrngt hg dj`strucrg og dø`g h–khmãrg`dg æ

•

  ⟵

25 / 36

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HW HWX X INV INV — Tg Tgng ngsstr trg g ;& 2 4.

Nãdk`cqug hgs sjochgs - DJWRT

@jttcj @j cj`s `s hg Ttktc tcq qug

Vrc`dc Vrc`dcpg pg ej` ej`hkn hkng` g`tk tkoo hg ok st stkt ktcqu cqug g <

Xmãjrí Xmã jríngs ngs iã`ãr iã`ãrkux kux hg ok stktc stktcqug qug <

Tjct u` systíng hg sjochgs T. Vjur qug T sjct g` ãqucocbrg pkr rkppjrt æ u` rgpírg ikocoãg`, co ekut qug og tjrsgur hgs kdtcj`s nãdk`cqugs gxtãrcgurgs æ T sjct `uo <

Xm Xmãj ãjrí ríng ng hg ok rãsu rãsuot otk` k`tg tg stkt stktcq cqug ug <

    ⟵̇⟵   > 6

Dj`hct Dj` hctcj` cj` `ãd `ãdgss gsskcr kcrg g gt sueecs sueecsk`t k`tg g< Vjur qu–u` systíng hg sjochgs sjct g` ãqucocbrg pkr rkppjrt æ u` rgpírg ikocoãg`, co ekut gt co sueect qug <  co sjct g` ãqucocbrg ku hãbut hg o–ãtuhg

•

 pjur tjut sjus-systíng hg sjochgs s hg T <

•

Xm Xmãj ãjrí ríng ng hu nj njng ng`t `t stkt stktcq cqug ug <

O–kdtcj` nãdk`cqug h–u` systíng hg sjochgs T;  sur u` systíng hg sjochgs T  (kvgd T  gxtãrcgur æ T ) gst jppjsãg æ o–kdtcj` nãdk`cqug hg T  sur T  <

⟵

;

⟵

2

Vj Vjur ur u` sy syst stíng íng hg sjo sjochg chgss T g` ãqucoc ãqucocbr brg g pkr rkpp rkppjrt jrt æ u` rg rgpír pírg g ikocoãg`, ok rãsuotk`tg hu tjrsgur hgs kdtcj`s nãdk`cqugs gxtãrcgurgs æ T gst `uoog <

  ̇→→ > 6

    ⟵⟵̇    > 6

Xmãj Xmãjrí ríng ng hg hgss kdtc kdtcj` j`ss nu nutu tugo goog ogss <

2

 .

o–c`tãrcgur huqugo sg trjuvg o– o–geejrt geejrt hg dj`tk dj`tkdt dt

2

;

Vj Vjur ur u` sy syst stíng íng hg sjo sjochg chgss T g` ãqucoc ãqucocbr brg g pkr rkpp rkppjrt jrt æ u` rg rgpír pírg g ikocoãg`, og njng`t hu tjrsgur hgs kdtcj`s nãdk`cqugs gxtãrcgurgs æ T g` V qugodj`qug gst `uo <

   →→̇ 

   

29 / 36

 

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HWX IN INV V — Tg Tgng ngst strrg ; & 2 4.

Nãdk`cqug hgs sjochgs - DJWRT

@jttcj @j cj`s `s hg Ttktc tcq qug

  6

2= / 36

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HW HWX X INV INV — Tg Tgng ngsstr trg g ;& 2 4.

Nãdk`cqug hgs sjochgs - DJWRT

@jttcj @j cj`s `s hg Ttktc tcq qug

 

Dks pkrtc pkrtcduocg duocgrs rs h–ãquc h–ãqucocbr ocbrg g <

Njhãoc Njh ãocsk sktcj tcj` ` nãd nãdk`c k`cqu qug g gt dc`ãnk dc`ãnktcq tcqug ug hgs nãd nãdk`c k`csng sngss < Ockcsj`s 2H<

Tj Tjoc ochg hg g` ãquc ãqucoc ocbr brg g sj sjus us o– o–kd kdtc tcj` j` hg hg hgux ux ej ejrd rdgs gs < Tc u` sjochg gst g` ãqucocbrg pkr rkppjrt æ u` rgpírg ikocoãg` sjus o–kdtcj` hg hgux ejrdgs, dgs ejrdgs sj`t hcrgdtgng`t jppjsãgs.

Hgirãs hg ocbgrtã  

6

Tjo joch chg g g` ãq ãqu uco cocb cbrrg sj sju us o–kd o–kdtc tcj` j` hg trjc jcss ejrd ejrdgs gs <

 ,,⃙ ,,⃙

 djpok`kcrgs (hk`s og nìng pok`)

•

 dj` j`ddju jurrk` k`ttgs g` u` nìng nìng pj pjc` c`tt ju pkrkooíogs

2

 

Kdtcj`s trk`sncsscbogs

Xjrsgur dc`ãnktcqug hg ; / 6

66 66 66 6 6Σ   66  66 6 Σ   6

Ockcsj` g`dkstrgng`t

Ockcsj` krtcduoktcj` h–kxg

;

•

Tdmã Td mãnk nktc tcsk sktc tcj` j` pok`g

Ockcsj` iocsscírg h–kxg

;

Tc u` sjochg gst g` ãqucocbrg pkr rkppjrt æ u` rgpí rgpírg rg ikocoãg` ikocoãg` sjus o– o–kdtcj kdtcj` ` hg trjc trjcss ejrdgs, dgs ejrdgs sj`t <

Hãsc Hã sci` i`kt ktcj cj` `

   

Ockcsj` kppuc scnpog

•

 hg sjnng vgdtjrcgoog `uoog

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Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HW HWX X INV INV — Tg Tgng ngsstr trg g ;& 2 4.

Nãdk`cqug hgs ssjochgs jochgs - DJWRT

@jt jtcj cj` `s hg hg Tt Ttkt ktcq cqu ug

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K. HW HWX X INV INV — Tg Tgng ngsstr trg g ;& 2 4.

Nãdk`cqug hgs hgs sjochgs - DJWRT

@jt jtcj cj`s `s hg Ttkt ktcq cqu ug

Gxgrdcdg Gxgrdc dg h–kppocdktcj` h–kppocdktcj` <

Hã Hãtg tgrn rnc` c`kt ktcj cj` ` hgs hgs ej ejrd rdgs gs g` J gt K

Hã Hãttgr grn nc`k c`ktcj tcj` hgs ejrdg jrdgss g` J gt K <

 

;)

 <

     6 6 →    >  66   →    >  666     6 6 ∛  >   6   +   66   +  −66   > 6 1

J` kppocqug og VET æ T; g ` J <

Tur Tur

  

(;) (2)

  66 66 6  66 66 6 66 66

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

Gxgrdcdg Gxgrd cdg h–kppocdktcj` h–kppocdktcj` <

Hk`s (J 1 x; 1

Xjrsgur hgs kdt. nãd. hg 6 sur ;

∛       >       +   ⃙   > 6 2.2.6  ∩   6   + 6   ∩  −6   > 6  6 >  ∵ 2.6. − 6.  > 6 6 Tur

(4)

H–kp H–kprí ríss (2 (2)) <



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 →        →       

  ∵   + 6    6

4; / 36

   

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HWX IN INV V — Tg Tgng ngst strrg ; & 2 4.

Nãdk`cqug hgs sjochgs - DJWRT

@jttcj @j cj`s `s hg Ttktc tcq qug

Njhãoc Njh ãocsk sktcj tcj` ` nãd nãdk`c k`cqu qug g gt dc`ãnk dc`ãnktcq tcqug ug hgs nãdk`c nãdk`csn sngs gs <

  

6

Hãsc Hã sci` i`kt ktcj cj` `

 

Tdmã Td mãnk nktc tcsk sktc tcj` j` spktckog

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Xjrsgur dc`ãnktcqug hg 2 / ;

Ockcsj` g`dkstrgng`t

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Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HWX IN INV V — Tg Tgng ngst strrg ; & 2 4. Hgirãs hg ocbgrtã

8

Nãdk`cqug hgs sjochgs - DJWRT

@jttcj @j cj`s `s hg Ttktc tcq qug Hãsc Hã sci` i`kt ktcj cj` `

Ockcsj` oc`ãkcrg k``uokcrg hg dg`trg J gt hg hcrgdtcj`  x

Tdmã Td mãnk nktc tcsk sktc tcj` j` spktckog

Tdmãnktcsktcj` pok`g

Xjrsgur dc`ãnktcqug hg 2 / ;

Xjrsgur hgs kdt. nãd. hg ; sur 2

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Ockcsj` mãocdjêhkog

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;

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Nãdk`cqug hgs sjochgs - DJWRT

@jttcj @j cj`s `s hg Ttktc tcq qug

Hgirãs hg ocbgrtã

6

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HW HWX X INV INV — Tg Tgng ngsstr trg g ;& 2

J` rkppgoog hk`s og tkbogku sucvk`t ogs njhãocsktcj`s spktckog gt pok`g hgs oc ockc kcsj sj`s `s 4H pkrekctgs usugoogs g`trg 2 sjochgs, kc`sc qug ogs tjrsgurs dc`ãnk dc`ãnktcq tcqug ug gt hgs kdt kdtcj` cj`ss nãdk` nãdk`cqu cqugs gs khncs khncsscb scbogs ogs g`trg g`trg ogs hgux hgux sjochgs dj`schãrãgs hk`s ogs ockcsj`s. Hgirãs hg ocbgrtã

       

42 / 36

Ockcsj` kppuc pok` hg `jrnkog  x

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48 / 36

j

j

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HW HWX X INV INV — Tg Tgng ngsstr trg g ;& 2 8.

@jttcj @j cj`s `s hg Dc` c`ã ãtc tcq qug

Nãdk`cqug hgs sjochgs - DJWRT

[u– [u–gst gst dg qug qug ok Dc`ã Dc`ãtc tcqu qug: g:

6

Gtuhg hgs djrps g` prg`k`t g` djnptg ok `jtcj` hg nkssg gt h–c`grtcg.

r

8

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℥ ℥

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℥

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- Irk`hgurs ãtuhcãgs < nkssg, dg`trg h–c`grtcg, njng`ts h–c`grtcg, prjhucts h–c`grtcg

 ]  6

6

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- @jtcj` ej`hkng`tkog ej`hkng`tkog < Jpãrktgur h–c`grtcg >0 Nktrcdg h–c`grtcg

r

j

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j

- Xjrsgur dc`ãtcqug Ok bj``g dj``kcssk`dg hg dgs ockcsj`s `jrnkocsãgs gt dj`schãrãgs djnng pkre pk rekc kctg tgss (s (sk` k`ss erjt erjttg tgng ng`t `t)) pgrn pgrngt gt hg hã hãtg tgrn rnc` c`gr gr rkpc rkpchg hgng ng`t `t og ogss pjsscbcoctãs hg njuvgng`t h–u` nãdk`csng pous djnpogxg, nkcs ãikogng`t h–ãtuhcgr h–ãtuhcgr pous ekdcogng`t ogs geejrts trk`sncs ku sgc` hg dgs ockcsj`s.

47 / 36

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HWX IN INV V — Tg Tgng ngst strrg ; & 2 8.

Nãdk`cqug hgs sjochgs - DJWRT

@jttcj @j cj`s `s hg Dc` c`ã ãtc tcqu qug g

45 / 36

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HW HWX X INV INV — Tg Tgng ngsstr trg g ;& 2 8.

Nãdk`cqug hgs sjochgs - DJWRT

@jttcj @j cj`s `s hg Dc` c`ã ãtc tcq qug

   

Nkssg Nks sg h–u` h–u` sjochg sjochg<

Dg`tr Dg` trg g h–c`gr h–c`grtcg tcg h–u` h–u` sjochg sjochg <

Ok nkss nkssg g h–u` h–u` sj sjoc ochg hg T, djnp djnpjs jsã ã h–u` h–u`g g c` c`ec ec`c `ctã tã hg pj pjc` c`ts ts N kutj kutjur ur

Og dg`trg h–c`grtcg h–u` sjochg T hg nkssg n gst og pjc`t I hãec`c pkr ok

hgsqugos j` hãec`ct hgs vjoungs ãoãng`tkcrgs, gst hj``ãg pkr ok rgoktcj` <

rgoktcj` <

 >      >      .

Rgnkr Rgn krqug qugss <

. > ∍   . 

Vrjprcãt Vrjp rcãtãs ãs < 

O–c –c` `tã tãir irko kog g k ok hc hcng ng`s `scj cj` ` hu sj sjoc ochg hg 1 goog gst trcpog sc T gst u` vjoung, hjubog sc T gst u`g surekdg gt scnpog sc T gst u`g oci`g.

 I gst u`cqug 1 sc T gst c`hãejrnkbog, sk pjsctcj` hk`s u` rgpírg kttkdmã

•

æ T gst ecxg.    Ok ej ejrn rnuo uog g hj`` hj``ãg ãg gs gstt vk vkok okbo bog g hk`s hk`s og dks dks iã`ã iã`ãrrko h– h–u` u` sj sjoc ochg hg T

  >

mãtãrjií`g (hg nkssg vjouncqug vkrckbog sgoj` N). Hk`s og dks h–u` sjochg sjoch g mjnj mjnjií`g ií`g, `g hãpg`h pous hu pjc`t N dj`schãrã.

49 / 36

 >       . > ..    

K. HW HWX X INV INV — Tg Tgng ngsstr trg g ;& 2 @jt jtcj cj` `s hg hg Dc` Dc`ã ãtcq cqu ug

;))

Gxgrdcdg Gxgrd cdg h–kppocdktcj` h–kppocdktcj` < (hk`s (J 1 x; 1 Hãtgrnc`ktcj` hg ok nkssg <

 >   .   > .     > .      > . . 

. > >     . > >   66  .  .. > ..    . .  > . .     . 

Hãtg Hãtgrn rnc` c`kt ktcj cj` ` hu dg dg`t `trg rg h–c` h–c`gr grtc tcg gI<

  .      ..

∍  .  > 6

 Tc T khngt u` ãoãng`t hg synãtrcg nktãrcgoog (pok`, kxg, dg`trg), kojrs sj` dg`trg h–c`grtcg I kppkrtcg`t æ dgt ãoãng`t.

•

Rnq   < ok synãtrcg nktãrcgoog rgirjupg ok synãtrcg iãjnãtrcqug gt ok synãtrcg hg rãpkrtctcj` hg ok nkssg. Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

8.

 I gst tgo qug <

•

 

Nãdk`cqug hgs ssjochgs jochgs - DJWRT

4= / 36

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HW HWX X INV INV — Tg Tgng ngsstr trg g ;& 2 8.

Nãdk`cqug hgs sjochgs - DJWRT

@jttcj @j cj`s `s hg Dc` c`ã ãtc tcq qug

Nk Nktr trcd cdg g h–c` h–c`gr grtc tcg g h–u` h–u` sjoc sjochg hg T < Ok `jtcj` hg nktrcdg h–c`grtcg gst u` ãoãng`t gssg`tcgo hg ok dc`ãtcqug. Ok dj``kcssk`dg hg sk hãec`ctcj` gt hgs nãtmjhgs pgrngttk`t sj` dkoduo gst `ãdgss `ãdgsskcr kcrg g pjur pjur hãt hãtgrn grnc`g c`grr ogs jut jutcos cos hg ok rã rãsjo sjoutc utcj` j` hg prjbo prjboíngs íngs hy`kncqugs djnprg`k`t ku njc`s u`g rjtktcj`.

Hãec`c Hãec`ctcj tcj` ` hg o–jpã o–jpãrk rktg tgur ur h–c`gr h–c`grtcg tcg < T u` sjochg dj`tc`u 1 R hg dg`trg J u` rgpírg qugodj`qug. N u` pjc`t hg T.

    

J` hãec`ct u`g kppocdktcj` oc`ãkcrg hg R hk`s R quc k tjut vgdtgur  < kssjdc kssjdcg g og vgd vgdtg tgur ur





 

hg R

  >   >    > 

43 / 36

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HWX IN INV V — Tg Tgng ngst strrg ; & 2 8.

Nãdk`cqug hgs sjochgs - DJWRT

@jttcj @j cj`s `s hg Dc` c`ã ãtc tcqu qug g

∆ →      >      ∩  ∩  . 

86 / 36

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HW HWX X INV INV — Tg Tgng ngsstr trg g ;& 2 8.

Nãdk`cqug hgs sjochgs - DJWRT

@jttcj @j cj`s `s hg Dc` c`ã ãtc tcq qug

 

Hãec Hãec`c `ctc tcj` j` hg ok nk nktr trcd cdg g h–c` h–c`gr grtc tcg g<

Hã Hãec ec`c `ctc tcj` j` hg ok nk nktr trcd cdg g h–c` h–c`gr grtc tcg g<

T u` sjochg dj`tc`u 1 R(J, , , ) u` rgpírg 1 N(x,y,z) u` pjc`t hg T.

⃙ ⃙ ⃙    >  −     −    −− >   − − −− −   −       − −   ⃙     ⃙        

K pkrt pkrtcr cr hg o– o–jp jpãr ãrkt ktgu gurr h–c` h–c`gr grtc tcg, g, j` hãec hãec`c `ctt u`g u`g nk nktr trcd cdg g kppg kppgoã oãg g nk nktr trcd cdg g h– h–c` c`gr grtc tcg g g` J hk`s hk`s R (h (hgu gux x ãd ãdrc rctu turg rgss usug usugoo oogs gs)) <



Ok pr prgn gncí círg rg oc oci` i`g g hg ok nk nktr trcd cdg g dj djrr rrgs gspj pj`h `h ku vg vgdt dtgu gurr



Ok hg hgux uxcí cíng ng oc oci` i`g g hg ok nk nktr trcd cdg g dj djrr rrgs gspj pj`h `h ku vg vgdt dtgu gurr



Ok trjc trjcsc scín íng g oc oci` i`g g hg ok nk nktr trcd cdg g dj djrr rrgs gspj pj`h `h ku vg vgdt dtgu gurr

8; / 36

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

K. HWX IN INV V — Tg Tgng ngst strrg ; & 2 8.

@jttcj @j cj`s `s hg Dc` c`ã ãtc tcqu qug g

Nãdk`cqug hgs sjochgs - DJWRT

Xmãj Xmãjrí ríng ngss oc ocãs ãs kux kux nk nktr trcd cdgs gs h–c` h–c`gr grtc tcg g<

C`tgr C`tgrpr prãt ãtktc ktcj` j` hgs djgeec djgeecdcg dcg`ts `ts hg ok nk nktrc trcdg dg

  >  >     +    .   >  >     +    .     >  > >     .  + .  .   >  >   . .

<

  

⃙ ⃙ ⃙ ⃙ ⃙ ⃙ ⃙

< Njn jng` g`tt h–c` h–c`gr grttcg hg T / kxg (J, (J, )

< Njn jng` g`tt h–c` –c`gr grtc tcg g hg T / kxg (J, (J, ) < Njn Njng`t h–c`g –c`grt rtcg cg hg T / kxg (J, )

< Vrj Vrjhuct huct h–c` h–c`gr grttcg hg T / kxg (J, )&(J, )

< Vrjhu jhuct h–c`g –c`grt rtcg cg hg T / kxg (J, )&(J, ) < Vrjhuc jhuctt h–c` –c`gr grtc tcg g hg T / kxg (J, (J, )&(J, )

 >  >   . .

82 / 36

K. HW HWX X INV INV — Tg Tgng ngsstr trg g ;& 2 8.

@jttcj @j cj`s `s hg Dc` c`ã ãtc tcq qug

⃙ ⃙

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

Nãdk`cqug hgs sjochgs - DJWRT

Xmãj Xmãjrí ríng ngss ocãs ocãs kux kux nk nktr trcd cdgs gs h–c` h–c`gr grtc tcg g<

Hgs Hgs tmãj tmãjrí ríng ngsspjc`t pgrn pgrngt tg`t `t hg dkod dkoduo uogr nktr nktrcd cdg g h– h–c` c`gr grtc tcg gT g` hgI.T g` `–cnpjrtg qugo gtgttg rgpírg æ pkrtcr hggrok ok nktrcdg h–c`grtcg hg

   >  −−     −−     −−   ,⃙,,⃙ ⇒    >  −−     −−     −−   ,⃙,,⃙

   >  −     −     −   ,⃙,,⃙ ⇒    >  −     −     −   ,⃙,,⃙ Xmãjrí Xmã jríng ng h–Muy h–Muyig` ig`ss (g (gxtg xtg`sc `scj`) j`) <

Xmãjrí Xmã jríng ng h–M h–Muyi uyig`s g`s <

W`g gxtg`scj` ku tmãjríng h–Muyig`s pgrngt hg hãtgrnc`gr ogs tgrngs

Og tmãjríng h–Muyig`s pgrngt hg dkoduogr ogs tgrngs hckij`kux hg ok nktrcdg g` V g` ej`dtcj` hgs tgrngs hckij`kux hg ok nktrcdg g` I <

mjrs hg ok hckij`kog hg ok nktrcdg hg T g` V (prjhucts h–c`grtcg hg T pkr rkppjrt kux kxgs ocãs æ V) g` ej`dtcj` hgs tgrngs mjrs hg ok hckij`kog hg ok nktrcdg g` I <

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 >  + .  

kvgd hx ok hcstk`dg g`trg ogs kxgs kvgd

 



kvgd hy ok hcstk`dg g`trg ogs kxgs

kvgd hz ok hcstk`dg g`trg ogs kxgs

 ,⃙  ,⃙   gt

  gt

  gt

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•

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 >  + . .   

 kvgd yI gt zI ogs djjrhj``ãgs hg I hk`s (V, x, y, z)



kvgd xI gt zI ogs djjrhj``ãgs hg I hk`s (V, x, y, z)  kvgd xI gt yI ogs djjrhj``ãgs hg I hk`s (V, x, y, z)

   >    + .   

84 / 36

 ,  ,

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

  >   + .  .   

88 / 36

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

D.

HWX IN INV V - G` G`gr griã iãtc tcqu qug g 2.

Nãdk`cqug hgs sjochgs - DJWRT

Ogss hgux Og hgux ã`g ã`gri ricg cgss hg hg ok nã nãdk dk`c `cqu qug g

D.

HWX HW X INV INV - G` G`gr griã iãtc tcqu qug g 2.

Nãdk`cqug hgs sjochgs - DJWRT

Ogss hgux Og hgux ã`g ã`gri ricg cgss hg hg ok nã nãdk dk`c `cqu qug g

 

Rgoktcj` g`trg g`trg trkvkco gt ã`gr ã`gricg icg pjtg`tcgoog <

G`gricg pjtg`tcgoog pjtg`tcgoog hg ok pgsk`t pgsk`tgur gur <

J` k chg`tcecã ogs rgoktcj`s sucvk` sucvk`tgs tgs <

Og tjrsgur hg o–kdtcj` hg ok pgsk`tgur gt og tjrsgur dc`ãnktcqug h–u` sjochg T g` njuvgng`t hk`s u` rgpírg R s–gxprcng`t g` sj` dg`trg hg irkvctã I pkr <

  →/ →̇ / > −      →/ →̇ / ̇→/   >     →/ ̇→/.       →/        →→/   > −       →/ ̇ / .   → ̇ /     →/ ̇ /   −      →/ ̇→/ ̇→/   > −      →→/

→   >   > 6.⃙     ξ/   >   Σ//    →/ > →   . ξ/ →/ > .⃙. / >   .   .⃙.   

J` c`tíirg o–ã`gricg o–ã`gricg pjtg`tcgoog hk`s ok prgncírg rgoktcj` pjur trjuvgr qug <

Og tr trkv kvkc kcoo g`tr g`trg g hg hgux ux c` c`st stk` k`ts ts t; gt t2 h–u` h–u`g g kdtc kdtcj` j` nã nãdk dk`c `cqu qug g gst gst hj hj`d `d ãiko ãiko (ku (ku sc sci` i`g g prís ís)) æ ok vkrc rck ktcj cj` ` h–ã`gr gric icg g pjtg` g`tc tcgo goog og ks kssj sjdc dcã ãg æ dgt dgttg kdtc kdtcj` j` nã nãdk dk`c `cqu qug g su surr T. •

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 O–ã`gricg pjtg`tcgoog gst mjnjií`g æ u` trkvkco gt s–gxprcng hj`d g` Ljuogs.

Ok pucssk`dg hg o–kdtcj` nãdk`cqug hg ok pgsk`tgur sur T s–gxprcng kojrs <

J` pgut hj`d kssjdcgr u`g ã`gricg pjtg`tcgoog æ o– o–kd kdtc tcj` j` hg ok pgsk pgsk`t `tgu gurr su surr T, hk`s hk`s sj sj` ` njuvgng`t pkr rkppjrt æ R <

  [uk`h [uk`h co gxcs gxcstg tg u`g ã`gricg ã`gricg pjtg pjtg`tcgo `tcgoog, og, og trkvkco gst c`hãpg`hk`t hu dmgnc` dmgnc` sucvc. sucvc.

kvgd zI o–kotctuhg hg I gt (J,z) o–kxg ksdg`hk`t.

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99 / 36

D.

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

HWX IN INV V - G` G`gr griã iãtc tcqu qug g 2.

Ogss hgux Og hgux ã`g ã`gri ricg cgss hg hg ok nã nãdk dk`c `cqu qug g

Nãdk`cqug hgs sjochgs - DJWRT

9= / 36

D.

 .→/   > −−.. . > ... 

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

HWX HW X INV INV - G` G`gr griã iãtc tcqu qug g 2.

Ogss hgux Og hgux ã`g ã`gri ricg cgss hg hg ok nã nãdk dk`c `cqu qug g

G`gricg pjtg`tcgoog pjtg`tcgoog h–u` rgss rgssjrt jrt trkdtcj`-tjrscj` trkdtcj`-tjrscj` <

G`gricg pjtg`tcgoog pjtg`tcgoog h–u` rgss rgssjrt jrt trkdtcj`-tjrscj` trkdtcj`-tjrscj` <

Tjct hgux sjochgs ;  gt 2  g` ockcsj` pcvjt iocssk`t h–kxg (J,x). J` c`tgrdkog u` rgssjrt r hg trkdtcj`-tjr trkdtcj`-tjrscj` scj` hg nkssg `uoog hg dkrkdtãrcstcqug dkrkdtãrcstcqug <

Og tjrsgur hg o–kdtcj` hg r sur 2 s–gxprcng g` K <

•

 A < rkchgur g` trkdtcj`-djn trkdtcj`-djnprgsscj` prgsscj`

•

 D < rkch rkchgur gur g` tjrscj` tjrscj`

•

  O6   < oj`iugur æ vchg

Nãdk`cqug hgs sjochgs - DJWRT

→   >  −.−.−.⃙   .⃙   →   > − →

Og VEH kppocquã æ r (hg nkssg `uoog) hj``g < sjct, h–kprís h–kprís og prc`dcpg hgs kdtcj`s nutugoogs <

J` hãec`ct ogs pkrknítrgs hg pjsctcj` hu sjochg 2 pkr rkppjrt ku sjochg ; < •

•

 >>.⃙.⃙

Ojrsqug og rgssjrt `–gst pks sjoocdctã < O>O 6 gt ν>6.

→   > − → ↖→ > →  . ξ /   > →  . ξ/

J` pgut kojrs dj`schãrgr qu–co gxcstg u`g kdtcj` nutugoog g`trg ; gt 2 pkr og bckcs hu rgssjrt. Ok pucssk`dg hg dgttg kdtcj` nutugoog s–gxp s–gxprcng rcng kojrs pkr <



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Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV

=6 / 36

↖→ >   −.−. −.⃙   .⃙   .     .⃙    > −.  −    .   − .  .  

Rkpmkáo UGOKTDJ < CWX h–Kncg`s h–Kncg`s — Hãpkrtgng`t INV