co hoc moi truong lien tuc

7/25/2019 Co Hoc Moi Truong Lien Tuc

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1

Thn gi cc anh, ch sinh vin Kho QH-2007-I/CQ-H

Ti gi mi ngi Bn tho Bi ging Chng 1 m ti s gii thiu vimc ch cc anh, ch nm c nhng ni dung ti trnh by

y hon ton cha phi lmt ti liu hon chnh vc th cn

nhng ch sai st in n. V vy, cc anh, ch cn c 2 ti liu chnhsau:

1. G. Thomas Mase & George E. Mase, Continuum Mechanics for

Engineers, CRS Press, 19992. o Huy Bch, Nguyn ng Bch, C hc Mi trng lin tc, NXB i

hc Quc gia Hni, 2003

Ti s chun b vgi tip bn tho bi ging.


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2

NI DUNGChng 1

ng hc cc Mi tr ng lin tc

I. M uII. Khi nim v Tenx (Tensors)

1. Cci lng v hng, vectv tenxtrong h to Descartes2. i s tenx, k hiu tng trng v quyc tng

3. K hiu theo chs4. Cc php tnh vectqua cch k hiu tng trngv k hiu theo chs5. Ma trn v nh thc. Biu din tenxtheo ma trn6. Bin i ca tenxtrong h to Descartes7. Gi trchnh v hng chnh ca tenxbc hai ixng

8. Trng tenx, Php tnh vi phn ca tenx9. Cc nh l tch phn Gauss & Octrogradsky vStokes


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3

Chng 1 (tip theo)

III.ng hc cc Mi trng lin tc

1. Phng php biu din chuyn ng ca Mi trnglin tc

2.Phng php Euler v phng php Lagrange

nghin cu chuyn ng Mi trng lin tc3.Vn tc v gia tc

4.o hm vt cht theo thi gian

5.Gradient bin dng. Tenxbin dng hu hn

6.Tenxbin dng tuyn tnh (nh)


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4

ChngChng 22ngng llcc hhcc cccc MiMi trtrngng linlin ttcc

I.I. TrTrngng ththii ngng susutt

1.1. MMtt khkhii llngng,, llcc khkhii vv llcc mmtt2.2. NguynNguyn llngng susutt CauchyCauchy vv VectVectngng susutt3.3. TrTrngng ththiingng susutt ttii 11 iimm vv TenxTenxngng susutt,,

4.4. CnCn bbngng llcc vv mm men,men, ttnhnhii xxngng ccaa tenxtenxngngsusutt..5.5. QuyQuy lulutt bibinnii cccc ththnhnh phphnn tenxtenxngng susutt6.6. ngng susutt chchnhnh vv hhngngngng susutt chchnhnh7.7. GiGi trtrngng susutt cccc trtr,, ccccngng trntrn MohrMohr8.8. TenxTenxngng susutt llchch vv tenxtenxngng susutt ccuu


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5

Chng 2 (tip theo)

II. Ccnh lut v phng trnh c bn ca C hc Mitrng lin tc

1. o hm vt cht cc tch phn th tch, tch phn mtv tch phnng2. Bo ton khi lng v phng trnh lin tc3. Nguyn l bining lng v phng trnh chuyn

ng4. Nguyn l bini m men ng lng5. nh lut bo ton nng lng v phng trnh nng

lng6. Entrpi (Entropy), nh lut thhai ca nhitng lc

hc v btng thc Clausius- Duhem

7. ng kn h phng trnh v xy dng cc biu thcxcnh


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6

Chng 3Mt s m hnh ca Mi tr ng lin tc

1. ng kn h phng trnh, xy dng cc biu thc xcnh v chc cc mi tr ng lin tc khc nhau

2. Tenx ng hng3. L thuytn hi tuyn tnh4. L thuyt cht lng c in (nht tuyn tnh)5. L thuytn nht tuyn tnh6. L thuyt do


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7

Bi ging 1

M u: C hc Mi trng lin tc trong s phttrin ca C hc hin i

1. C hc trng thnh nhmt khoa hc c lp mit th k 17,mc d cc kin thc c hc xut hin t lu. Tuy nhin,khc vi nhiu ngnh khoa hc c bn khc, s pht trin cac hc lun b nhiu bin ng

2.Trong thi k c i, trung th k v giai on pht trin ca c hcc in (th k 17 19) c ba xu hng pht trin chnh cac hc:

- Xu hng vt l (cn gi l xu hng trit hc) i su vocc hc thuyt v khng gian, thi gian, vt cht, v chuyn

ng v

ngun gc ca n.


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8

Bi ging 1

Xu hng ton hc ha: Nhng ngi theo xu hng ny, nghin cu s chuyn ng ca cc vt th trong v tr cgng a ra cc m hnh hnh + ng hc thun ty ton

hc. Xu hng k thut ho: H c gng a ra nhng nguyn l

thc tin gii thch s hot ng ca cc cng c, thit b

khc nhau nhm tha mn mt s mc ch no . Ba xu hng trn pht trin c lp vi nhau, trong ni bt

hn c l trng phi ton hc ha cc khi nim c bn ca

c hc. C th ni, lc ny quan h gia ton v c rng buckhng kht n ni u th k 20 c rt nhiu kin cho rngc hc c in mt ht vai tr, v n tr thnh mt lnh

vc ca ton hc thun ty.


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Bi ging 1

Nhvy c th ni rng hin nay c nhng phng tinthc nghim v tnh ton c hiu lc hu nhcho php giiquyt mt cch thc t mi bi ton c hc c thit lpmt cch ng n.

V vy nhng ngi lm c hc phi nhn thc c rngnhng thnh tu nghin cu tip tc ca mnh ph thuc trctip vo vic xy dng cc m hnh mi vo vic thit lp ccbi ton mi. Ngha l trng tm ca s suy ngh sng to vsc lc ca cc nh c hc phi, v thc t chuyn sang vn m hnh ha cc i tng c hc v thit lp mt cchng n cc bi ton mi. y l c im quan trng, c

th ba ca s pht trin c hc hin i.Cc ph

ng php nghin cu ca C hc Mi tr

ng lin tc

ng vai tr quan trng v nhiu khi l quyt nh trong vn

m hnh ha cc i t

ng c hc v

thit lp mt cch ng

n cc b

i ton mi


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11

Bi ging 1

II. Pht trin ca C hc Mi tr

ng lin tc

Chc mi trng lin tc l mt nhnh c hcni chung. Mn khoa hc ny thng nghin cu

cc chuyn ng vm ca mi trng th rn,lng, kh, ngoi ra cn nghin cu cc mi trngc bit khc nhcc trng in t, bc x,

trng trng,...y l mt mn khoa hc kh r ngv phn nhnh, n c ng dng kh rng ritrong ch to my, luyn kim, tnh ton m, nghincu cu to ca tri t v v tr, v nhiu lnhvc khc.

C hc cho tng Mi trng lin tc pht trin mt cchring bit t lu i, song song vi C hc h cht im v

h vt rn tuyt i.
http://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dchttp://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dchttp://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dchttp://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dchttp://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dchttp://vi.wikipedia.org/w/index.php?title=Chuy%E1%BB%83n_%C4%91%E1%BB%99ng_v%C4%A9_m%C3%B4&action=edithttp://vi.wikipedia.org/w/index.php?title=Chuy%E1%BB%83n_%C4%91%E1%BB%99ng_v%C4%A9_m%C3%B4&action=edithttp://vi.wikipedia.org/w/index.php?title=Chuy%E1%BB%83n_%C4%91%E1%BB%99ng_v%C4%A9_m%C3%B4&action=edithttp://vi.wikipedia.org/w/index.php?title=Chuy%E1%BB%83n_%C4%91%E1%BB%99ng_v%C4%A9_m%C3%B4&action=edithttp://vi.wikipedia.org/w/index.php?title=Chuy%E1%BB%83n_%C4%91%E1%BB%99ng_v%C4%A9_m%C3%B4&action=edithttp://vi.wikipedia.org/w/index.php?title=Chuy%E1%BB%83n_%C4%91%E1%BB%99ng_v%C4%A9_m%C3%B4&action=edithttp://vi.wikipedia.org/w/index.php?title=Chuy%E1%BB%83n_%C4%91%E1%BB%99ng_v%C4%A9_m%C3%B4&action=edithttp://vi.wikipedia.org/wiki/Th%E1%BB%83_r%E1%BA%AFnhttp://vi.wikipedia.org/wiki/Th%E1%BB%83_r%E1%BA%AFnhttp://vi.wikipedia.org/wiki/Th%E1%BB%83_r%E1%BA%AFnhttp://vi.wikipedia.org/wiki/Th%E1%BB%83_r%E1%BA%AFnhttp://vi.wikipedia.org/wiki/Th%E1%BB%83_r%E1%BA%AFnhttp://vi.wikipedia.org/w/index.php?title=Th%E1%BB%83_l%E1%BB%8Fng&action=edithttp://vi.wikipedia.org/w/index.php?title=Th%E1%BB%83_l%E1%BB%8Fng&action=edithttp://vi.wikipedia.org/w/index.php?title=Th%E1%BB%83_l%E1%BB%8Fng&action=edithttp://vi.wikipedia.org/wiki/Th%E1%BB%83_kh%C3%ADhttp://vi.wikipedia.org/wiki/Tr%C6%B0%E1%BB%9Dng_%C4%91i%E1%BB%87n_t%E1%BB%ABhttp://vi.wikipedia.org/wiki/Tr%C6%B0%E1%BB%9Dng_%C4%91i%E1%BB%87n_t%E1%BB%ABhttp://vi.wikipedia.org/wiki/Tr%C6%B0%E1%BB%9Dng_%C4%91i%E1%BB%87n_t%E1%BB%ABhttp://vi.wikipedia.org/wiki/Tr%C6%B0%E1%BB%9Dng_%C4%91i%E1%BB%87n_t%E1%BB%ABhttp://vi.wikipedia.org/wiki/Tr%C6%B0%E1%BB%9Dng_%C4%91i%E1%BB%87n_t%E1%BB%ABhttp://vi.wikipedia.org/wiki/Tr%C6%B0%E1%BB%9Dng_%C4%91i%E1%BB%87n_t%E1%BB%ABhttp://vi.wikipedia.org/wiki/Tr%C6%B0%E1%BB%9Dng_%C4%91i%E1%BB%87n_t%E1%BB%ABhttp://vi.wikipedia.org/wiki/Tr%C6%B0%E1%BB%9Dng_%C4%91i%E1%BB%87n_t%E1%BB%ABhttp://vi.wikipedia.org/wiki/B%E1%BB%A9c_x%E1%BA%A1http://vi.wikipedia.org/wiki/B%E1%BB%A9c_x%E1%BA%A1http://vi.wikipedia.org/wiki/B%E1%BB%A9c_x%E1%BA%A1http://vi.wikipedia.org/wiki/B%E1%BB%A9c_x%E1%BA%A1http://vi.wikipedia.org/wiki/Tr%E1%BB%8Dng_tr%C6%B0%E1%BB%9Dnghttp://vi.wikipedia.org/wiki/Tr%E1%BB%8Dng_tr%C6%B0%E1%BB%9Dnghttp://vi.wikipedia.org/wiki/Tr%E1%BB%8Dng_tr%C6%B0%E1%BB%9Dnghttp://vi.wikipedia.org/wiki/Tr%E1%BB%8Dng_tr%C6%B0%E1%BB%9Dnghttp://vi.wikipedia.org/wiki/Tr%E1%BB%8Dng_tr%C6%B0%E1%BB%9Dnghttp://vi.wikipedia.org/w/index.php?title=Ch%E1%BA%BF_t%E1%BA%A1o_m%C3%A1y&action=edithttp://vi.wikipedia.org/w/index.php?title=Ch%E1%BA%BF_t%E1%BA%A1o_m%C3%A1y&action=edithttp://vi.wikipedia.org/w/index.php?title=Ch%E1%BA%BF_t%E1%BA%A1o_m%C3%A1y&action=edithttp://vi.wikipedia.org/w/index.php?title=Ch%E1%BA%BF_t%E1%BA%A1o_m%C3%A1y&action=edithttp://vi.wikipedia.org/w/index.php?title=Ch%E1%BA%BF_t%E1%BA%A1o_m%C3%A1y&action=edithttp://vi.wikipedia.org/wiki/Luy%E1%BB%87n_kimhttp://vi.wikipedia.org/wiki/Luy%E1%BB%87n_kimhttp://vi.wikipedia.org/wiki/Luy%E1%BB%87n_kimhttp://vi.wikipedia.org/w/index.php?title=M%E1%BB%8F&action=edithttp://vi.wikipedia.org/w/index.php?title=M%E1%BB%8F&action=edithttp://vi.wikipedia.org/wiki/Tr%C3%A1i_%C4%91%E1%BA%A5thttp://vi.wikipedia.org/wiki/Tr%C3%A1i_%C4%91%E1%BA%A5thttp://vi.wikipedia.org/wiki/Tr%C3%A1i_%C4%91%E1%BA%A5thttp://vi.wikipedia.org/wiki/V%C5%A9_tr%E1%BB%A5http://vi.wikipedia.org/wiki/V%C5%A9_tr%E1%BB%A5http://vi.wikipedia.org/wiki/V%C5%A9_tr%E1%BB%A5http://vi.wikipedia.org/wiki/V%C5%A9_tr%E1%BB%A5http://vi.wikipedia.org/wiki/V%C5%A9_tr%E1%BB%A5http://vi.wikipedia.org/wiki/Tr%C3%A1i_%C4%91%E1%BA%A5thttp://vi.wikipedia.org/w/index.php?title=M%E1%BB%8F&action=edithttp://vi.wikipedia.org/wiki/Luy%E1%BB%87n_kimhttp://vi.wikipedia.org/w/index.php?title=Ch%E1%BA%BF_t%E1%BA%A1o_m%C3%A1y&action=edithttp://vi.wikipedia.org/wiki/Tr%E1%BB%8Dng_tr%C6%B0%E1%BB%9Dnghttp://vi.wikipedia.org/wiki/B%E1%BB%A9c_x%E1%BA%A1http://vi.wikipedia.org/wiki/Tr%C6%B0%E1%BB%9Dng_%C4%91i%E1%BB%87n_t%E1%BB%ABhttp://vi.wikipedia.org/wiki/Th%E1%BB%83_kh%C3%ADhttp://vi.wikipedia.org/w/index.php?title=Th%E1%BB%83_l%E1%BB%8Fng&action=edithttp://vi.wikipedia.org/wiki/Th%E1%BB%83_r%E1%BA%AFnhttp://vi.wikipedia.org/w/index.php?title=Chuy%E1%BB%83n_%C4%91%E1%BB%99ng_v%C4%A9_m%C3%B4&action=edithttp://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc


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Bi ging 1

Da vo tnh cht ca vt th ta c th phn loi thnh:

+ Chc vt rn bin dng, i khi c bit n nhl

thuyt n hi hoc sc bn vt liu... Chc cht rnnghin cu s cn bng v chuyn ng ca vt cht bbin dng bi ngoi lc.

+ Chc cht lng, nghin cu qu trnh vt l ca dngchy cc phn t vt cht. Cc phn t vt cht ny cth d dng chuyn ng trong khng gian.
http://vi.wikipedia.org/w/index.php?title=C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_r%E1%BA%AFn&action=edithttp://vi.wikipedia.org/w/index.php?title=C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_r%E1%BA%AFn&action=edithttp://vi.wikipedia.org/w/index.php?title=C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_r%E1%BA%AFn&action=edithttp://vi.wikipedia.org/w/index.php?title=C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_r%E1%BA%AFn&action=edithttp://vi.wikipedia.org/w/index.php?title=C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_r%E1%BA%AFn&action=edithttp://vi.wikipedia.org/w/index.php?title=C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_r%E1%BA%AFn&action=edithttp://vi.wikipedia.org/w/index.php?title=C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_r%E1%BA%AFn&action=edithttp://vi.wikipedia.org/w/index.php?title=C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_r%E1%BA%AFn&action=edithttp://vi.wikipedia.org/w/index.php?title=C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_r%E1%BA%AFn&action=edithttp://vi.wikipedia.org/w/index.php?title=C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_r%E1%BA%AFn&action=edithttp://vi.wikipedia.org/w/index.php?title=C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_r%E1%BA%AFn&action=edithttp://vi.wikipedia.org/w/index.php?title=C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_r%E1%BA%AFn&action=edithttp://vi.wikipedia.org/w/index.php?title=C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_r%E1%BA%AFn&action=edithttp://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_l%C6%B0uhttp://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_l%C6%B0uhttp://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_l%C6%B0uhttp://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_l%C6%B0uhttp://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_l%C6%B0uhttp://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_l%C6%B0uhttp://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_l%C6%B0uhttp://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_l%C6%B0uhttp://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_l%C6%B0uhttp://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_l%C6%B0uhttp://vi.wikipedia.org/w/index.php?title=C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_r%E1%BA%AFn&action=edit


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14

Bi ging 1

L thuytn hiChc Vt rn

bin dngL thuyt do

Cht lng phiNewton

Lu binhc

Chc Chtlng

Cht lng Newton

Chc Mitrng lintc


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II. Khi nim v Tenx Tensor)

1.Cc i lng v hng, vect v tenx trong h to Descartes

2. i s tenx, k hiu tng trng vquyc tng3. K hiu theo chs4. Ma trn vnh thc5. Bini ca tenxtrong h to Descartes

6. Gi trchnh vhng chnh ca tenxbc haii xng7. Trng tenx, Php tnh vi phn ca tenx8. Ccnh l tch phn Gauss & Octrogradsky vStokes

B

i gi

i

ng

g

1


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16

Bi ging 1

1. Cci lng v hng, vect v tenx trong hto Descartes

- H ta Decartes l l h ta vung gc nhm xcnh v tr ca cc vt th (2, 3 hoc nhiu chiu). Sauy ta chxt khng gian 3 chiu.

-i lngv hng li lng c gi tr khng phthuc vo h ta. Th d nh mt, di, thtch

- Vect li lngc xcnh bi gi tr tuyti vhng trong khng gian. Th d nh vect tc, vect

gia tc, lcVect li lng bt bini vi php bini h ta, tuy nhin cc thnh phn li thayi ph thuc vos la chn h ta.

- Ten x l khi nim m rng cai lng v hng vi lng vect.


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2.i s tenx, k hiu tng tr ng v quyc tng2.1 Vectcstrong h ta Decartes

Trong l cc vect n v vung gc lnnhau vc gi l vect c s ca h ta Decarteseee 321 ,,


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2.2 Biu din tng trng ca vectv ten x

Biu din tng trng ca vect

Biu din tng trng ca ten x (th d cho ten xbc 2)

=

=++=3

1332211

iii evevevevv

=

= 3

1,

jijiij eeTT


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2.3 Quyc tng n gin cch vit ngi t a t hng b du

tng v thay th bng cch vit sau

y s dng Quyc tng: ly tng theocc cp chs ging nhau.

eeT

ev

jiij

ii

T

v

=

=


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2.4 K hiu Kronecker Ta nh ngha va ra k hiu Kronecker

nh sau

Suy ra

ee jiij .=

=

= ji

ji

ij ;0

;1


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2.5 K hiu Levi-Civita

Ta nh ngha va ra k hiu Levi-Civita nh sau

0 khi 2 chs bt k bng nhau1 khi cc chs lp thnh hon v chn ca 1,2,3

-1 khi cc chs lp thnh hon v l ca 1,2,3

2.6 Mt s hngng thc cn nh

=ijk

mjkjmkjkqjkq

jmmkqmjq

ijmkikmjjkqmiq

==

=

=

;6

;2

;


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23

Bi ging 1

3. K hiu theo ch s

Bn cnh cch k hiu tng trng, ta cn thng dngcch k hiu theo chs.

i lng v hng:

Vect: (3 thnh phn) Dyad: (9 thnh phn) :

Tenx bc 2 (dyadic): (9 thnh phn)

Tenx bc 3 (triadic): (27 thnh phn) Tenx bc 4 (treadic): (81 thnh phn)

uivu ji

Tij

Qijk

Cijkm


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24

Bi ging 1

4. Cc php tnh vect qua cch k hiu tng tr ng vk hiu theo ch s

Cng vect hoc ten x:T V + S = Q

Nhn v hng (nhn trong) 2 vect:w = U.V

Nhn vect 2 vect:W = U x V

ii VUW=

VUW kjijki =

ijijijij QSVT =+

Bi i 1


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25

Bi ging 1

Nhn v hng vect vi tenx:

W = U . T

Nhn vect ca vect vi tenx

W = U x T

Nhn v hng tenx vi tenx mt ln (cho tenx bc 2):

Q = T . S

Nhn v hng tenx vi tenx hai ln (cho tenx bc 2):

q = T .. S

TUW jiji=

TUW ljkiklij =

STQ kjikij

=

ST ijijq=

2 Ch i h S l i A l h


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2.Chng minh rng nu S l ten x i xng,A l ten x phni xng th tch S..A = 0

-Nu

-Th

Chng minh:ASASASAS jijijiijjiijijij ===

0=AS ijij


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Th d tm cc biu thc sau:

Tm gi tr sau:


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Tm gi tr sau:

Li gii


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Tch v hng 2 vect c xc nh nhsau


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-Nhn v hng hai vect

Trong :u, v l gi tr tuyt i ca u v v

l gc gia 2 vect

H qu:

Tch v hng ca 2 vect vung gc bng khng

Tch v h

ng ca 2 vect cng h

ng bng tch 2 gi trtuyt i


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-Tch vect ca 2 vect

nh ngha:

Trong l gc gia 2 vect ( 0 ),

l vect n v vung gc vi mt phng cha 2 vectu, vtheo hng quay phi tu sang v

Bi ging 2


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32

Bi ging 2

5. Ma trn vnh thc. Biu din tenxtheo ma trn

nh ngha: Ma trn [MxN] A l s sp xp cc phn t mt cchtrt t theo hnh ch nht c ng li bi du mc vung (M hng, Nct). Chsu chhng (i = 1,M), chs th hai chct(j = 1,N).

Ma tr n chuyn v l ma trn c chuyn i hng thnh ct

[ ]

==

MNMM

N

N

ij

AAA

AAA

AAA

AA

...

:::

...

...

21

22221

11211

ji

T

ij AA =

Bi ging 2


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33

Bi ging 2

Nu N = M ta c ma trn vung.

Ma tr n vung c =0 khi i khc j, ta c ma tr nng cho.

Ma tr n n v l ma trn ng cho vi cc phn tc gi tr = 1

Vi M=N, ma trn i xng khi

Phni xng khi

Aij

TAA =

TAA =

Cng ma trn (cho cc ma trn cng s hng v s ct)


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34

-Cng ma trn (cho cc ma trn cng s hng v s ct)

-Bt k mt ma trn vung nou c th phn tch thnh

tng ca mt ma trni xng

v mt ma trn phni xng

A=B+C

Trong B li xng, C phni xng

ijijij CBA +=

( )TAAB +=2

1

( )T

AAC =

2

1

-Nhn ma trn vi ma trn ch c thc hin khi s ct ca


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35

ma trnu = s hng ca ma trn sau v c gi tr

-nh thc ca ma trn vung (th d bc ba)

l i lng c gi trc xc nh sau

kjikij BAC =

333231

232221

131211

det

AAA

AAA

AAA

AAij==

kjiijkkjiijk AAAAAAA 321321det ==


36/129

36

Cch n gin

-Ma trn o

-Ma trn vung gc

AAAA 11 =

TQQ =1


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37

-Ma trn k d

-Ma trn phni xng l ma trn k d (chng minh?)

-nh thc ca tch ma trn

-nh thc ca ma trn chuyn v

C th chng minhc

Ma trn vung gc


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38

C thchng minhc rng, i vi ma trn vunggc, ta c

5. Bini ca tenxtrong hto


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39

g Descartes

-Vect, ten xl cci lng khng phthuc vohquy chiu hay hto bt k.

-Tuy nhin cc thnh phn ca chng li hon tonphthuc

Bin i h to


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40

Bini hto

Ta c


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41

Bini ca vect

Suy ra

Bini ca tenx


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42

C thchng minhc rng

Chng minh thng qua tch ca 2 vect

(Bi ging 4)


43/129

43

6. Gi trchnh v hng chnh ca ten xbc haii xng

Hng chnh: ta gi nl hng chnh ca ten xbc

2 i xng Tnu nh(vectring)hay

(vect T.ncng hng vi n)

Cch tm

Phng trnhi s ng nht bc 3, iu kin


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44

tn ti nghim

Thu c phng trnhc trng sau

Trong cc bt bin chnh ca ten x T cdng

*C thchng minhc rng vi ten xbc 2 i


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45

xng vi cc phn tc gi trthc, nghim ca

phng trnhc trng cng c gi trthc*Bitc

Gii hphng trnh sau ta s c hng chnh

Cng viiu kin trc chun sau

*Nu khc nhau, cc hng chnh s


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46

, gduy nht v vung gc ln nhau (htrc chnh),

*Nu th chc mt hng chnh duy

nhtng vi , hai hng chnh cn li sl btk2 hng vung gc ln nhau nm trong mt

phng vung gc vi

*Nu th bt kmt tp hp no ca 3

hng vung gc vi nhauu l cc hng chnh*Trong htrc chnh, ma trn biu din ca ten xstrthnh ma trnng cho c cc gi trbng

cc gi trring

( ) ( )pq

q

i

p

i nn =

*Chng minh hai hng chnh tngng vung gc vih


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47

nhau

Githit l

Chng minh

p n:Ta c

Nhn phng trnhu vi , phng trnh sau viri trvi nhau:

Ch tnhi xng ca T v khc nhau ca gi trrin

( ) ( ) 21 ( ) ( )

021

=nn ii

( ) ( )0

11= nT jijij

( ) ( )0

22= nT jijij

( ) ( ) ( ) ( ) ( ) ( )0

122211= nnTnnT ijijijijijij

( ) ( )

( ) ( )

pqq

i

p

i

ii

nnnn

=

=021

( )

ni2

( )

ni1

Chng minh: Trong htrc chnh, ma trn biu dinca ten x s tr thnh ma trn ng cho c cc


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48

ca ten xstrthnh ma trnng cho c cc

gi trbng cc gi trringPhp bini gia htrc

Chnh v hto X:xi

( )

na p

jpj=

( ) ( )

pqq

j

p

jqjpj nnaa ==

T nh ngha gi trchnh ta c:


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49

g g

Nhn 2 vvi ta c

Suy ra

( ) ( ) ( )

nnT p

i

pp

jij =( )

n

q

i

( ) ( ) ( ) ( ) ( )

nnnTn q

i

p

i

pp

jij

q

i =

( ) ( ) ( )

pqpp

jij

q

ipq nTnT ==

Th d:


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50

Xcnh gi trchnh v hng

chnh ca tenxc ma trn sau

p n:

Phng trnhc trng c dng

( ) ( ) ( )

1,6,3 321

===

0672

=+

ng vi ta c phng trnh xcnh hngchnh th nht:

( )3

1=


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51

chnh thnht:

ng vi ta c phng trnh xcnh hngchnh thhai:

021

==

nn

1=nn ii 13 =n( )

62

=

ng vi ta c phng trnh xcnhh h h h b

( )1

3=


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52

hng chnh thba:

Ma trn bini vhtrc chnhc dng

Trong htrc chnh, ten xcma trn sau

[ ] [ ][ ][ ]

==

100

060

003

aTaT ijijT

ijij

1

02

02

24

3

21

21

=

=

=+

=+

nnn

nn

nn

ii

o

05

2

51

3

2

1

=

=

=

n

n

n

7. Trng tenx, Php tnh vi phn ca tenx


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53

Ta c trng ten xkhi ti miim ca x ca khnggian Euclide xcnh cc gi trsau

Chng l hm ca x v thi gian t, v c thly cco hm:

o hm ring bc 1 theo thi gian

o hm ring bc 1 theo khng gian

o hm ring bc cao

( )txT kij

,K

t

=

q

qx

= qmmq xx

2

Mt sk hiu quyc

i h it h TTij


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54

n gin cch vito hm:

Ton t

Vi ton tny, ta c cco hm sau

-Gradient v hng-Divergence (div)

-Rotor (curl)

-Gradient vector

Txx

klij

lk

ij

,=

x

ei

i=

iiii exe , ==

u iiu ,. =

euu ijkijkkjijku , == eeu jijiu ,=

Php tnh vi phn ca tenx


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55

eT jiijT ,. =

eeT lijklijkT

,=

eeeT kjkij iT ,=

8. Ccnh l tch phn Gauss & Octrogradskyv Stokes


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56

v Stokes

Xt trng tenx T

khvi lin tc bt k

trong vng khng gian

c thtch V, gii hn

bi din tch bmt S,vect n vca

php tuyn ngoi l n

Nhc li mt s nh l tch phn Gauss cho vect


57/129

57

1.

2.

3.

Nhc linh l tch phn Stokes cho vect


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58

p

Mrng cho tenxbc 2


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59

g

==

==

vs v

vs v

curlTdVTdVTdSn

divTdVTdVdSnT

..

=

=

S V

jklijkkljijk

S V

iijij

dVdS

dVdSi

TTn

TTn

,

,

Bi ging 4


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60

III. ng hc cc Mi tr

ng lin tc

1. Phng php biu din chuyn ng ca Mi trnglin tc2. Phng php Euler v phng php Lagrange nghincu chuyn ng Mi trng lin tc3. Vn tc v gia tc

4.o hm vt cht theo thi gian5. Gradient bin dng. Tenxbin dng hu hn6. Tenxbin dng tuyn tnh (nh)


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61

1. Phng php biu din chuyn ng ca mi trng lin tc

Ht hay cht im X:

Vt cht thc t khng lin tc. Gi thit lin tc: Vt th c th chia rathnh cc phn t (cht im) nh bao nhiu tu m vn gi nguyncc tnh cht v m ca vt th. Cc phn t vt cht ny lp y mtmin hoc ton khng gian mt cch lin tc. Khong cch gia chngthay i trong qu trnh chuyn ng v bin dng

Vt th B: tp hp ca cc ht hay cht im X

Vd 1 Vd 2

Vd 3 Vd 4B

X2

X3 X

4

X1

v

T

dV nh

n

u?

Cu hnh hay hnh thi: php nh x cc v tr ca cc chti t th B kh i l i


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62

im X ca vt th B vo khng gian x v ngc li

Chuyn v: thay i v tr ca cc hnh thi

Chuyn v vt th cng: ch c tnh tin v quay

Bin dng: thay i kch thc, hnh dng

Xx = ( )xX 1

=

To khng gian v to vt cht


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63

To khng gian x

To vt cht X

B k h t i P t i thi i t


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64

Bn knh vect ca im P ti thi im to

Bn knh vect ca im p ti thi im t

Vect chuyn v:

Chuyn ng c xc nh khi:

Trong hm kh vi lin tc theo cc bin

ex iix =

i=1,2,3

Xxu =

tXx , =

tXA ,

iu kin tng thch mt-mt gia x v X:

Jacobien khc khng


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65

Jacobien khc khng

Khi y c th tm ngc to vt cht:

Trong cng l cc hm kh vi lin tc ca x,t

0

3

3

2

3

1

3

3

2

2

2

1

2

3

1

2

1

1

1

=

=

X

x

X

x

X

xXxXxXx

Xx

Xx

Xx

Xx AiDetJ

( )txxxX AA ,,, 3211

=

A

2. Phng php Euler v phng php Lagrangenghin cu chuyn ng Mi trng lin tc


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66

nghin cu chuyn ng Mi trng lin tc

- Ph

ng php Euler

Nghin cu s bin i ca cc i lng c trng ca C

hc MTLT (mt , tc , nhit , ng sut, bin dng.)ti mt v tr c nh ca ngi quan st l ni dung caphng php Euler nghin cu C hc MTLT

l cc bin Euler

Ti v tr c nh nhng bin i c gy ra bi s thay ica cc ht vt cht khc nhau i qua im c nh cho trc.

Th d: o vn tc, mc nc ti mt trm quan trc trn sng.

txxx ,,, 321


67/129

67

Xi1

Xi2

Xi3

Xi4

Xi1

Xi2

Xi3

Xi4

e1

e2

x1

x2

x

X

u

Minh ho trn khng gian 2 chiu

Xi1

Xi2


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68

Xi1 Xi2

Xi

3 Xi4

Xi3

Xi4

x1

x2


Born 15 April 1707Basel, Switzerland
http://en.wikipedia.org/wiki/Baselhttp://en.wikipedia.org/wiki/Switzerlandhttp://en.wikipedia.org/wiki/Switzerlandhttp://en.wikipedia.org/wiki/Basel


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69

Died18 September 1783(aged 76)St. Petersburg, Russia

Residence Prussia, RussiaSwitzerland

Nationality Swiss

Fields Mathematician and Physicist

Institutions Imperial Russian Academy ofSciencesBerlin Academy

Alma mater University of Basel

Doctoral

advisorJohann Bernoulli

Known for See full listEuler

- Phng php Lagrange:
http://en.wikipedia.org/wiki/St._Petersburghttp://en.wikipedia.org/wiki/Russiahttp://en.wikipedia.org/wiki/Kingdom_of_Prussiahttp://en.wikipedia.org/wiki/Russian_Empirehttp://en.wikipedia.org/wiki/Old_Swiss_Confederacyhttp://en.wikipedia.org/wiki/Old_Swiss_Confederacyhttp://en.wikipedia.org/wiki/Mathematicianhttp://en.wikipedia.org/wiki/Physicisthttp://en.wikipedia.org/wiki/Russian_Academy_of_Scienceshttp://en.wikipedia.org/wiki/Russian_Academy_of_Scienceshttp://en.wikipedia.org/wiki/Prussian_Academy_of_Scienceshttp://en.wikipedia.org/wiki/Alma_materhttp://en.wikipedia.org/wiki/University_of_Baselhttp://en.wikipedia.org/wiki/Doctoratehttp://en.wikipedia.org/wiki/Johann_Bernoullihttp://en.wikipedia.org/wiki/List_of_topics_named_after_Leonhard_Eulerhttp://en.wikipedia.org/wiki/Switzerlandhttp://en.wikipedia.org/wiki/Baselhttp://en.wikipedia.org/wiki/Russiahttp://en.wikipedia.org/wiki/St._Petersburghttp://en.wikipedia.org/wiki/Old_Swiss_Confederacyhttp://en.wikipedia.org/wiki/Russian_Empirehttp://en.wikipedia.org/wiki/Kingdom_of_Prussiahttp://en.wikipedia.org/wiki/Old_Swiss_Confederacyhttp://en.wikipedia.org/wiki/Physicisthttp://en.wikipedia.org/wiki/Mathematicianhttp://en.wikipedia.org/wiki/Prussian_Academy_of_Scienceshttp://en.wikipedia.org/wiki/Russian_Academy_of_Scienceshttp://en.wikipedia.org/wiki/Russian_Academy_of_Scienceshttp://en.wikipedia.org/wiki/Alma_materhttp://en.wikipedia.org/wiki/University_of_Baselhttp://en.wikipedia.org/wiki/Doctoratehttp://en.wikipedia.org/wiki/Johann_Bernoullihttp://en.wikipedia.org/wiki/List_of_topics_named_after_Leonhard_Euler


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70

Phng php nghin cu chuyn ng qua vic theo ri qu ochuyn ng ca tng ht vt cht (cht im)

Trong :

l to ban u ca ht vt cht

l to ca ht vt cht ti thi im t(So snh vi C hc L thuyt)

Quan st cc bin i ca cc i lng c trng cho C hcMTLT trong h to gn vi chuyn ng ca tng ht vt chtl phng php Lagrange

l cc bin Lagrange

XXX 321 ,,

xxx 321 ,,

tXXX ,,, 321

( )tXXXx ii ,,, 321=


71/129

71

Xi1

Xi2

Xi3

Xi4

Xi1

Xi2

Xi3

Xi4

e1

e2

x1

x2

x

X

u


Born January 25, 1736)Turin, Sardinia

Died April 10, 1813 (aged 77)Paris France
http://en.wikipedia.org/wiki/Turinhttp://en.wikipedia.org/wiki/Kingdom_of_Sardiniahttp://en.wikipedia.org/wiki/Parishttp://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/Kingdom_of_Sardiniahttp://en.wikipedia.org/wiki/Turinhttp://en.wikipedia.org/wiki/Parishttp://en.wikipedia.org/wiki/France


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72

Paris, France

Residence Sardinia

France, Prussia

Nationality ItalianFrench

Fields MathematicsMathematical physics

Institutions cole PolytechniqueDoctoral advisor Leonhard Euler

Doctoral students Joseph FourierGiovanni Plana

Simeon PoissonKnown for Analytical mechanics

Celestial mechanicsMathematical analysisNumber theory

Religious stance Roman Catholic

Lagrange

S dng khi no?

1. Phng php Lagrange nghin cu chuyn ng C hc MTLT cho
http://en.wikipedia.org/wiki/Parishttp://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/Sardiniahttp://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/Prussiahttp://en.wikipedia.org/wiki/Italyhttp://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematical_physicshttp://en.wikipedia.org/wiki/%C3%89cole_Polytechniquehttp://en.wikipedia.org/wiki/Doctoratehttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Joseph_Fourierhttp://en.wikipedia.org/wiki/Giovanni_Planahttp://en.wikipedia.org/wiki/Simeon_Poissonhttp://en.wikipedia.org/wiki/Analytical_mechanicshttp://en.wikipedia.org/wiki/Celestial_mechanicshttp://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Roman_Catholichttp://en.wikipedia.org/wiki/Parishttp://en.wikipedia.org/wiki/Prussiahttp://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/Sardiniahttp://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/Italyhttp://en.wikipedia.org/wiki/Mathematical_physicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/%C3%89cole_Polytechniquehttp://en.wikipedia.org/wiki/Doctoratehttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Simeon_Poissonhttp://en.wikipedia.org/wiki/Giovanni_Planahttp://en.wikipedia.org/wiki/Joseph_Fourierhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Celestial_mechanicshttp://en.wikipedia.org/wiki/Analytical_mechanicshttp://en.wikipedia.org/wiki/Roman_Catholic


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php ta tm hiu, theo ri s bin i cc b ca cc c tr

ng c-l-hogn vi cht im ca mi trng, loi tr cc bin i do chuyn ngthun tu gy ra. T s thun li hn trong vic xy dng cc quy lutng s vbin i ca mi trng.

Th d, vic xy dng quan h gia ng sut vbin dng s thun li khidng h to Lagrange.

(Minh ho bng vic o nhit ).

Phng php Lagrange khng thun li trong vic nghin cu chuynng vso snh vi kt qu o c thc nghim vquan st.

2. Phng php Euler nghin cu chuyn ng C hc MTLT cho php

ta tm hiu, theo ri s chuyn ng c hc thun tu mt cch thun lIhn.

3. Vn tc v gia tc:

-Vn tc


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74

Vn tc

-Gia tc

-Biu din vn tc qua vect chuyn v:

( )dt

udXu

dt

dv

=+=dt

du

v

i

i=

4. o h

m vt cht theo thi gian

Cho mt trng ten x trong to Lagrange:


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75

Chuyn v to Euler X-->x ta c

o hm vt cht theo thi gian c xc nh nhsau:

y l tc bin i theo thi gian ca tenx ti im X khng i.

Trong to Euler, o hm vt cht theo thi gian c dng:

Trong to Euler, o hm vt cht theo thi gian c dng:


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76

Nhvy, trong to Euler, o hm vt cht bc nht theo

thi gian c dng:

Lc ny vn tc c th vit:

x

uv

uv

k

i

k

i

i t

+

=

(tm o hm vt cht theo thi gianca i lng v hng vgia tc?)

Cng thc tng qut tnh o hm vt cht bc 1 theo thigian


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77

Lu : ly tng theo ch s k

(tm o hm vt cht theo thi gian ca ten x bc 2?)

Pv

t

P

dt

dP+

= .

( )[ ] ( )[ ] ( )[ ]txtxt

txdt

dP

xvPP ij

k

kijij ,,,

.........

+

=

(tm o hm vt cht theo thi gian ca i lng v hng vgia tc?)

Th d minh ho

Th d 1:


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78

Cho qu o chuyn ng ca vt ththeo phng php Lagrange

Tm:

a. Qu o ca ht vt cht xut pht t im X=(1,2,1)

b. Tc vgia tc ca ht vt cht ti thi im t=2 sc. Xc nhchuynng theophng php Euler (tm cc to

Lagrange) vxc nh tc & gia tc ti im x=(1,0,1), t = 2 s

d. Xc nh trng dch chuynu theo 2 phng phpe. Xc nh o hm vt cht theo thi gian ca nhit khi bit

p n:a. Thay 1,2,1

321 === XXX


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79

Tc chung Gia tc chung

b.Tc v

gia tc ca cht im X(1,2,1) ti thi im 2 s:

Trng tc & gia tc

trong to Lagrange

c. Bin i ngc tm to Lagrange (phng php Euler):


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80

Thay biu thc trn vo cng thc xc nh vn tc vgia tc

Vn tc vgia tc ti im x=(1,0,1), t=2 s

Trng tc & gia tc

trong to Euler

d. Trng dch chuyn

Ti t = 0 ta cx = X , ngha l2 h to khng gian vvt cht trngnhau.


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81

- Trng dch chuyn theo phng php Lagrange c dng

- Trng dch chuyn theo phng php Euler c dng (thay X -> x)

0333

12

222

2

2

111

==

==

==

XxuXtXxu

XtXxu

0

1

1

3

4

1

2

22

2

4

2

2

12

1

=

=

=

u

txtx

tu

t

xtxtu

e. o hm vt cht theo thi gian ca nhit

Ta c cng thc:d

+

+

+

=


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82

Vi

Tc c tnh trn

xvxvxvtdt 332211 +

+

+

( )

ex

ex

ex

xxxe

t

t

t

t

t

=

=

=

+=

3

2

32

3

2

1

321

Bi tp 4


83/129

83

1. Example 4.3-1

2. Example 4.3-2

3. Example 4.4-1

4. Example 4.5-1

5. Problem 4.3

6. Problem 4.7

7. Problem 4.8

Bi ging 5

5. Gradient bin dng. Tenx bin dng hu hn


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84

5.1 Gradient bin dng

Xt 2 im ln cn P, Q trong hnh thi ban u khng bin dng.

Cc im p,q lcc im tng ng trong hnh thI bin dng.

lkhong cch gia P vQ, lkhong cch gia p vqXAd xid


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85

Ta c

( ) AAdXdXdXdXdX == .2

AAIdXdX =


86/129

86

Vi gi thit tn ti php nh x 1-1 ta c

t - c gi l tenx gradient bin dng F xc nh mc

bin dng cc b ti cht imX.Vit li ta c

( ) iidxdxdxdxdx == .2

A

i

AiAA

i

i XxdXXdx

=

=

,

iAAi Fx ,

AiAi dXFdxdXFdx ==

ii edxdx =

Ngc li:

dxFdXdxXdXiiAA

== 1,


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87

5.2 Ten x bin dng hu hn Green vLagrange

Lc ny bin i bnh phng khong cch gia 2 im ln cn c xc

nh nh

sau

Khai trin ta c

( ) ( ) ( )( )

( )( ) BAABAB

BAABBiAi

BAABBBiAAi

dXdXC

dXdXxx

dXdXdXxdXxdXdx

=

=

=

,,

,,22

( ) ( ) AAii

dXdXdxdxdXdx = 22

Trong C l ten x i xng bc 2 , c gi l ten x bin dng Green

FFCxxC T

BiAiAB == ;,,


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88

Tenx bin dng hu hn Lagrange E

Lc ny:

Vi ten x E (c gi l ten x bin dng Lagrange, l ten x i xng) cth d dng xc nh s thay i di ca ca mt phn t ng.

ABABAB CEICE == 2;2

( ) ( ) dXEdXdXdXEdXdx BAAB == 2222


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89

George Green

Born July 14, 1793)

Sneinton, Nottinghamshire,England

Died May 31, 1841 (aged 47)

Nottingham,Nottinghamshire, England

Almamater

University of Cambridge

5.3 Tenx bin dng hu hn Cauchy vEulerTm php bin i di ca mt phn t ng trong to
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Euler

( ) ( ) ( )( )

( )( ) jiijij

jijAiAij

jjAiiAjiij

dxdxc

dxdxXX

dxXdxXdxdxdXdx

=

=

=

,,

,,22

Trong ta k hiuc

- ten x c cc thnh phn nhsau

C c gi l ten x bin dng Cauchy (i xng)( ) ( )

11,, ;

== FFcXXcT

jAiAij

Ta t

( )Iijijij ==

22


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Lc ny ta c

Ten x e c gi l ten x bin dng hu hn Euler, cho php ta xcnh bin i di phn t ng trong to Euler. y l ten x ixng

( )cIece ijijij 2;2

( ) ( ) dxedxdxdxedXdxjiij

== 2222

Born 21 August 1789)Paris, France

Died 23 May 1857 (aged 67)Sceaux, France
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Augustin Louis Cauchy

Residence France

Nationality French

Fields CalculusComplex analysis

Institutions

cole Nationale des Ponts et

Chaussescole polytechnique

Alma mater cole Nationale des Ponts etChausses

Doctoral

students

Known for Cauchy integral theorem

Religiousstance Catholic[1]

5.4. Biu din cc tenx bin dng qua vect chuyn v

1 Bi di t bi d L t h
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1. Biu din cc tenx bin dng Lagrange qua vect chuyn v

Trong to vt cht, vect chuyn v c xc nh nhsau

Lc ny, tenx bin dng Lagrange c dng

Rt gn li ta c

Bao nhiu thnh phn (?)

( ) ( ) iAiAi XXxXu =

ABiBBiiAAiABBiAiAB uuxxE ++== ,,,,2

BiAiABBAAB uuuuE ,,,,2 ++=

2. Biu din cc tenx bin dng Lagrange qua vect chuyn v

Trong to khng gian vect chuyn v c dng


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94

Trong to khng gian, vect chuyn v c dng

Tenx bin dng Euler s c xc nh nhsau

Rt gn li, ta c:

( ) ( )iAAiA xXxxu =

jAAjiAAiijjAiAijij uuXXe ,,,,2 ==

jAiAijjiij uuuue ,,,,2 +=

3 Tenx cu v tenx lch ca bin dng

Ta phn tch cc ten x bin dng trn (th d cho ten x bin dngEuler) thnh 2 thnh phn nhsau


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95

Tenx cu

Tenx lch

)3

1(

3

1 ijkkijijkkij eeee +=

ijkkeeij 310

= 1 thnh phn

031 11 =

= iiijkkijij eeee

So snh

Phng php Lagrange Phng php Euler


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96

Phng php Lagrange1.Bin Lagrange

2.Qu o


Phng php Euler1.Bin Euler

2. Qu o


tXXX ,,, 321

( )tXXXx ii ,,, 321=

( )[ ] ( )[ ]tXttXdtd

PP ijij ,,

=

txxx ,,, 321

( )txxxX AA ,,, 3211

=

( )[ ] ( )[ ]

( )[ ]tx

tx

t

tx

dt

d

Px

vPP

ij

k

k

ijij

,

,,

+

=

So snh



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97

Phng php Lagrange

4.Khong cch gia 2 im

5.Ten x gradient bin dng

Phng php Euler

4.Khong cch gia 2 im

5.Ten x gradient bin dng ngc

( ) AAdXdXdXdXdX == .2 ( ) iidxdxdxdxdx == .2

1

,

= iAiA

FX

dxFdX = 1

iAAi Fx ,

dXFdx =

So snh



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98

Phng php Lagrange6.Ten x bin dng Green

7.Ten x bin dng Lagrange

8.Dch chuyn

9. Biu din qua dch chuyn

Phng php Euler6.Ten x bin dng Cauchy

7.Ten x bin dng Euler

8. Dch chuyn

9. Biu din qua dch chuyn

FFCxxC T

BiAiAB

== ;,,

ABABAB CEICE == 2;2

( ) ( )11,, ;

== FFcXXc T

jAiAij

( )cIece

ijijij

== 2;2

( ) ( ) iAiAi XXxXu = ( ) ( )iAAiA xXxxu =

BiAiABBAAB uuuuE ,,,,2 ++= jAiAijjiij uuuue ,,,,2 +=

Th d:

Cho quy lut chuyn v sau:

;; XkXxXkXxXx ++


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99

Tm cc ten x bin dng hu hn Lagrange vEuler

p n:

1. Ten x bin dng Lagrange:

Tenx gradient bin dng

Tenx bin dng Green

Xx

FA

i

iA =

BiAiAB xxC ,,=

23332211 ;; XkXxXkXxXx +=+==

Tenx bin dng Lagrange

2 E = C - I


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100

Tenx cu bin dng Lagrange

Tenx lch bin dng Lagrange ijijkkij kEE

20

31

31 ==

=

=

6

0

60

003

31

2

2

2

1

k

k

k

EEE

k

kijkkijij 0

1

=Eii

2. Tenx bin dng Euler

Chuyn v to Euler ( )xxX

xX

k

11

1=

=


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101

Chuyn v to Euler

Tenx gradient bin dng ngc

( )

xkxkX

xxk

X

k

k

32223

3222

1

1

1

1

+

=

=

[ ]

=

kkkk

X

k

kiA

22

22,

1

1

10

11

10

001

Tenx bin dng Cauchy


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102

( ) ( )

( ) ( )

=

2222

2222,,

1

1

1

0

11

10

001

kk

kk

k

k

XXcjAiAij

Ten x bin dng Euler

( ) ( )

kk

k2210

000

11 122


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103

2 e = I c )

Tm tenx cu v tenx lch bin dng Euler ( ) ( )

( ) ( )

kk

kkk

21

20

1

1

1

11 1

22

( )

== kee ijkkij 213

2

3

1

11

2

0

( )

( ) ( )

( ) ( )

==

2222

2222

22

1

1

11

3

1

10

11

11

3

10

001

1132

3

1

kk

k

k

k

k

k

eeeijkkijij

Bi tp ti lp (Problem 4.21)

Cho trng bin dng sau


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104

1. Hy xc nh tenx bin dng Lagrange v tenx bin dng Euler

2. Chng minh rng ng trn trng thi cha bin dng

s bin dng thnh ng trn

p n


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105

Bi tp 5


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106

Example 4.6-1

Problem 4.16 Problem 4.21

Bi ging 6

6.Tenx bin dng tuyn tnh (nh)

-Biu din tenx bin dng Lagrange v

Euler qua vect chuyn vKhi cc chuyn v v gradient chuyn v l cc i lng nh c th b


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107

Khi cc chuyn v vgradient chuyn v lcc i lng nh c th bqua c tch ca chng, ta s c cc tenx bin dng nh sau:

Tenx bin dng nh (tuyn tnh) Lagrange

Tenx bin dng nh (tuyn tnh) Euler

Trong trng hp ny ta c

do ch ti

Suy rakAkiAi uu ,,

ABBAAB uuE ,,2 +=

ijjiij uue ,,2 +=

kA

k

i

kA

A

k

k

i

A

k

k

i

A

i

x

u

X

u

x

u

X

x

x

u

X

u

+

=

=

kkk Xux +=

Tng t vy, c th chng minh c

Nh vy trong l thuyt tuyn tnh gn ng bc 1 khng phn bit

jBjABA uu ,,


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108

Nhvy, trong l thuyt tuyn tnh, gn ng bc 1, khng phn bitgia o hm ca chuyn v theo to Lagrange hay Euler

C ngha lcc thnh phn ca ten x bin dng Lagrange vEuler sps bng nhau

Ten x bin dng nh c nh ngha nhsau:

( )uu ijjiij ,,2

1+=

jBiAijAB eE

ijji

i

j

j

i

Bi

B

j

Aj

A

i

ij uux

u

x

u

X

u

X

u,,2 +=

+

=

+

=

-Gi tr chnh vhng chnh ca tenx bin dng

Cc ten xE, e, lcc ten x i xng bc 2. C th tm c cc gitr chnh vhng chnh.

Cch tm cc i lng ny c trnh by trong chng v Tenx.


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109

g y y g gNhc li y l trn hng chnh, cc ten x bin dng c th a vdng ma trn ng cho.

Minh ho cho ten x bin dng nh

Trong l ten x chuyn v h trc chnh

[ ]

( )

( )

( )

=

=

III

II

I

ij

00

00

00

00

00

00

3

2

1

0)( =ijDet

lcc gi tr chnh lnghim ca phng trnh sau IIIIII ,,


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110

Cc bt bin ca tenx bin dng nh (xem chng Tenx)

0)(

2

)(

3

)(=+ IIIIII

( )[ ]

ijIIIIIIkjiijk

IIIIIIIIIIIIjiijjjii

IIIIIIii

DetIII

II

trI

===++==

++===

321

2

1

6.2.Y ngha vt l ca cc thnh phn ca tenx bin dng nh

-Xt ngha vt l cc thnh phn ten x bin dng nh nm trn ng cho

Xt bin i d

i ca 1 phn t

ng:

( ) ( )22


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111

Chia 2 v phng trnh trn cho

Ch ti iu kin bin dng nh ta c dx+dX=2 dX+(0)

Suy ra

Trong N

lvect n v trng vi vectdX

y chnh l dn t ng i ca phn t ng theo hngN

(cn gi lbin dng dc, hoc bin dng php tuyn)

( ) ( ) dXdXdXdXdXdxjiij == 22

22

( )2

dX

dX

dX

dX

dX

dX

dXdx

dX

dXdx jiij2=

+

( ) NNNN

dX

dXdxe

jiijN.. ==

=

Gi s rng, vect N trng vi trc

Ngha l0,0,1

321 === NNN

X1


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112

Lc ny, ta c bin dng dc theo hng chnh l

Tng t, ta c bin dng dc

theo hng : (?)

Nhvy, trong h to Descartes, cc thnh phn trn ng cho caten x bin dng nh chnh lbin dng dc theo cc trc tng ng

X1 ( ) 111 =e

( )

( )

333

222

=

=

e

e

XX 32

,

-Xt ngha vt l cc thnh phn ten x bin dng nh khng nm trnng cho.

l

2 vi vect khi cha

( ) ( )21 ; XX dd


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113

bin dng (vung gc).

lcc

vect trn trng thi

bin dng (khng vung

gc ln nhau).Ta s dng cc bin i sau

( ) ( )21 ; xx dd

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )2121

2121

212121

2

2

XEXXX

XEIXXCX

XFFXXFXFxx

dddd

dddd

dddddd T

+=

+==

=== XFx dd

Trong trng hp bin dng nh, ta c

T i t thi h bi d i t l h

( ) ( ) ( ) ( ) ( ) ( )212121

2 XXXXxx

dddddd +=


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114

Ti trng thi cha bin dng, cc vi vect vung gc ln nhau, suy ra

Gi l thay i gc

Bin dng gi thit lnh

=2

( ) ( ) ( ) ( ) ( ) ( )212121 2cos XXxx dddxdxdd ==

== sin)2

(coscos

( )

( )

( )

( ) ( ) ( )212

2

1

1

2

2cos NNdx

d

dx

d=

X

X

( ) ( )

( ) ( )2

1

2

1

dXdx

dXdx

Gi s rng, khi cha bin dng cc vect n v N vung gc vi nhau.Hy xt s thay i gc gia 2 vect

12

131211 0


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115

Tng t, c th chng

minh c bin i gc gia2 hng i vj xc nh quacc thnh phn khng nmtrn ng cho ca ten x

bin dng nh c gi l bin dng trt

ij

)jiijij = 2

[ ] 12

333231

23222112 2

0

10,0,12

=

=

-Xt ngha vt l ca tng cc thnh phn ten x bin dng nh nm trnng cho.

Xt bin i ca 1 phn t th tch sau khi chu bin dng nh. Gn vi htrc chnh ta c hnh v sau


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116

Vi gi thit bin dng nh, nu trng thi cha bin dng

cc vi vect vung gc vi nhau,ta c th coi trng thi bin dng, cc vi vect cng vung gc vi nhau Lc ny phn t th tch s c xc nh nh sau

idX

idx


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117

vi nhau. Lc ny, phn t th tch s c xc nh nhsau

Thay

Ch ti iu kin bin dng nh, ta thu c

Tng ca cc th

nh phn nm trn

ng cho xc nh bin itng i ca phn t th tch

( ) ( ) ( )

( ) ( ) ( )321

321

dXdXdXdV

dxdxdxdv

=

=

( ) ( ) ( )iii dXedx +=1

( ) ( ) ( ) iieeedVdVdv =++=

321

6.3.Tenx v vect quay tuyn tnh (nh)Xt vect vi chuyn v du ti imP.

j

i

j

j

i

i

j

j

ij

Pj

ii dX

Xu

Xu

Xu

XudX

Xudu

+

+=

= 21

21


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118

C th vit li nhsauTrong , tenx quay nh c nh ngha:

t - vect quay; c th chng minh

Nu ten x bin dng nh =0

Ta c

Khi ny, chuyn v ch l1 php quay c th

Vy, chuyn v nh = bin dng nh + quay c th

( )ijjijiij

uu ,,2

1==

ijijPj

jPijiji dXdu +=

kjijki 21= kkjiij =

0=ij

dXdudXdXdu jkikjjkkjii ===

6.4. iu kin tng thch ca bin dngTrong trng hp tng qut, ta t vn lnu bit 6 thnh phn ca tenx bin dng hu hn Euler t 6 phng trnh sau

jAiAijjiij uuuue ,,,,2 +=


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119

Chng ta c th tm c vect chuyn vu c 3 thnh phn hay khng(bi ton ngc, 6 phng trnh c 3 n s)?

iu kin tng thch bin dng liu kin rng buc ln cc thnh phn

ca tenx bin dng sao cho c th tm c trng chuyn v duy nht vlin tc tng ng. Khng d dng chng minh trong trng hp tng qutca bin dng hu hn. S ch tm iu kin khi bin dng lnh (tuyntnh). Lc ny c th chng minh c iu kin l

jijiji dxdu )( +=

02222

=

+

=

kj

il

li

jk

ki

jl

lj

ikijkl

xxxxxxxxR

0,,,, =+ ikjmjmikijkmkmij

Chng minh tham kho)

Bi ton:

Cho vch cho trng ten x bin dng nhe, hy tm iu kin tng

thch ca bin dng e, sao cho c th tm c 3 thnh phn ca vectchuyn v duy nht v lin tc.


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120

y y

Li gii

Xt 2 im P1 vP2 vi vect chuyn v u1 vu2 tng ng, C - ngcong bt k ni 2 im trn,

- chuyn v t ng i ca

im P2 so vi P1

Ta c

Tm iu kin lun c tch phn trn khng ph thuc vo bt kng tch phn no


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121

ng tch phn no.

Thay

Ta c

D dng chng minh c

Thay vo vt

Ta thu c


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122

R rng rng trong biu thc trn, thnh phn th nht c tch phn vch ph thuc vo 2 im, iu kin tch phn c thnh phn th 2khng ph thuc vo ng ly tch phn trong mt vng lin thng l

Tht vy, nu ta ly tch phn trn t P1 -> P2 -> P1 (ng vng C bt k),

Th dch chuyn tng i phi bng 0.

T nh l Stokes ta c

Suy ra iu kin tch phn vng ca tch v hng mt vect vi phn

t ng lRotor ca vect phi bng khng.

p dng vo iu kin trn ta chng minh c


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123

p dng vo iu kin trn, ta chng minh c

Tip tc bin i

T cc phng trnh trn ta c ng thc sau

Vi x bt k ta c


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124

y chnh liu kin tng thch ca bin dng nh, gm 81 iu kin,

nhng nu ch ti cc tnh cht i xng vphn i xng

Ta ch cn

6 iu kin

tng thch

6.5 Gradient vn tc, tc bin dng, xoy.

Cho trng vn tc trong to Euler v(x,t)

nh ngha gradient vn tc nhsau j

i

ij x

v

L

=


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125

D dng c th phn tch gradient vn tc thnh:

Trong

Tenx tc bin dng

Tenx xoy (spin)

Thay ten x xoy = vect xoy

ijijij WDL +=

+

=

i

j

j

iij

x

v

x

vD

2

1

=

i

j

j

iij

x

v

x

vW

2

1

vwvwjkijki ==

2

1

2

1,

6.6 o hm vt cht ca phn t ng, phn t din tchv phn t th tch

1. o hm vt cht ca phn t ng

Ly o hm vt cht ca phn t ng

t dd


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126

ta c

tm o hm vt cht theo thi gian caF, ta ly

xvxd jji di ,)( =

( ) xL.XL.F.X.FXFxx dddddt

dd

dt

dd =====

.)(

( ) ( ) X.FXF dd

dt

ddx

dt

ddx

=== .

Suy ra

2. o hm vt cht ca phn t din tch (t nghin cu vchng minh)


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127

3. o hm vt cht theo thi gian ca phn t th tch

Ta chng minh rng vi bin dng nh th

( ) ( ) ( )dVdVdv eee ii321 =++=


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128

Ly o hm vt cht

theo thi gian

( ) dv

dvdvdt

ddV

dt

ddv

dt

d

dVdv

vdv

v

ii

iiiiii

ii

,

,)()()(

)1(

=

==

+=

( ) dVvdivdVvdV ii ==

,

Bi kim tra th1

Cho quy lut chuynng ca mi tr ng dng sau

33122211 )1(;;2 XtxXtXxXtXx +=+=+=


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129

Cho trng nhit

1.Tm vect dch chuyn trong ta vt cht v ta khng gian2.Tm ten x bin dng hu hn Lagrange

3.Tm ten x bin dng hu hn Euler4.Tm ten x bin dng tuyn tnh (nh), ten x quay v vect quay5.Tm ten x tc bin dng, ten x xoy v vect xoy6.Tm gi tr chnh v hng chnh ca ten x bin dng nh khi t=1

7.Xc nh dn n tngi ca cc phn t ng hng theo theocc trc ta, thayi gc gia cc phn t ng nu trn, bini ca 1 phn t th tch khi t=1.

8.Tm o hm vt cht ca trng nhit cho

2

3

2

2

2

1

22

3

; xxxxx

e t ++==

co hoc moi truong lien tuc

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