PART 1: VECTOR & TENSOR ANALYSIS

Transcription

1 PART : VECTOR & TENSOR ANALYSIS wth LINEAR ALGEBRA Obectves Introduce the concepts, theores, nd opertonl mplementton of vectors, nd more generlly tensors, n dvnced engneerng nlyss. The emphss s on geometrc nd physcl nterprettons for engneerng pplctons. Study some of the fundmentl rules of lner lgebr nd show nloges wth tensor nlyss. We wll study elementry topcs of lner lgebr: Mtrces, determnnts, systems of lner equtons, nd egenvlues nd egenvectors.

2 Vector Defntons Descrpton of Physcl Qunttes Sclr: A quntty descrbed only by mgntude; descrbed sngle number, e.g., temperture, pressure, Vector: A quntty descrbed by both mgntude nd drecton, e.g., velocty, dsplcement, Tensor: A hgher-order vector, gves nformton n ddton to mgntude nd drecton, e.g., the stte of stress nd strn n contnuous medum re secondorder tensors. 2

3 Vector Defntons A free vector cn be dsplced prllel to tself nd ct t ny pont; requres three numbers to specfy free vector, e.g., velocty. A sldng vector cn only be dsplced long lne through fxed pont contnng the vector; requres fve numbers to specfy sldng vector, ntersecton of lne nd coordnte plne (2) nd the vector (3), e.g., force. α β A bound vector requres sx numbers (coordntes ponts x, x 2 ), e.g., dsplcement. x x 2 3

4 Vector Defntons b Vectors hve mgntude nd drecton nd stsfy the prllelogrm lw of ddton. + b Exmple: A fnte rotton hs mgntude nd drecton, but s t vector but the lne segment (rc) tht connects nd b. Therefore, fnte rotton s not vector snce t does not stsfy the geometrc defnton. (Wht bout dfferentl rotton?) b + b 4

5 Vector Defntons Invrnce Vectors re nvrnt under coordnte trnsformton. Exmple: The poston vectors r nd r 2 ndcte the poston of the fxed pont s the coordnte system trnsltes. After the coordnte trnslton, r r 2 therefore, poston vector s relly not vector snce t s not nvrnt under coordnte trnslton! (Wht bout pure coordnte rotton?) r r 2 5

6 Vector Algebr Elementry Opertons Addton + b = b + commuttvty + b + c = ( + b) + c = + (b + c) ssoctvty + 0 = ddtve dentty b + b b + c + b + c c Subtrcton b = + ( b) b b + b b 6

7 Vector Algebr Sclr Multplcton m = m m = m, m > 0 m = m, m < 0 0 = 0 Dvson s not defned vector operton Unt Vector (/2) ( ) 2 eˆ = = ˆ e = A vector cn lwys be wrtten n terms of pure mgntude nd drecton usng unt vector = ˆ e drecton mgntude 7

8 Vector Algebr Lner Dependence Gven vectors {, 2,, n } nd sclrs {β,β 2,,β n }, not ll zero. If one cn wrte, β + β + + β = n n () then the vectors re lnerly dependent,.e., one s lner combnton of the others. Exmple: n n β = 2, = colner 2 β2 = 3, = ( β + β ) β3 coplnr If () cnnot be stsfed, the vectors re lnerly ndependent. 8

9 Vector Algebr Expnson of Vector wth Respect to Other Vectors Gven nd b, lnerly ndependent m (non-colner) then, vectors c nd d cn lwys be constructed: 2D: c = m + nb 3D: d = m + nb + pc Sclr (Dot, Inner) Product b= b cos(, b) = bcos θ, 0 θ π Rules b = b ( b+ c) = ( b) + ( c) If b (orthogonl) b = bcos( π / 2) = 4. If b = 0 b or = 0 or b = 0 0 b nb pc θ π θ = 2 nb + pc b b c d 9

10 Vector Algebr 4. = 2 = = 2 5. e ˆ = cos θ = e proecton of n drecton of eˆ ê θ Exmple: Vector representton of work: e ˆ proecton of force mgntude of work = n drecton dsplcement of dsplcement dw = ( f cos θ ) ( ds) = f ds Vector (Cross, Skew, Outer) Product b= c= bsnθ eˆ b The vector product obeys the rght-hnd rule: Brngng nto b dvnces eˆ b n the drecton of rght-hnded screw. c eˆ b b θ 0

11 Vector Algebr Other rules:. b= b ntcommuttvty 2. If b θ = 0 or θ = π snθ = 0 b= 0. If b= 0 then ether b or = 0 or b= ( + b) c= ( c) + ( b c) dstrbutve but order must be preserved. Exmple: Moment (torque) bout some pont O from force ctng t pont P. r poston of pont P wth respect to O f m l force moment (torque) perpendculr dstnce from O to lne through f m= r f = rf snθeˆ = fleˆ m m eˆm O O l m r f θ r P P f θ

12 Vector Algebr Ths defnton of plne re cn be generlzed to descrbe generl plne re s vector quntty. By conventon, the re s enclosed on the left sde when trversng the closed contour n counterclockwse drecton. ˆn S s = Snˆ C Exmple: Determne the proected re from the oblque cut through crculr cylnder S = mgntude of slnt re S = mgntude of proected re nˆ = unt norml to re S s = S nˆ, s= Snˆ S s the proecton of s n drecton of S = sn ˆ = Snn ˆ ˆ ˆ n ˆn S ˆ n S 2

13 Vector Algebr Rgd-Body Rotton Determne the velocty t ny pont n n rbtrrly shped, 3-D body rottng bout some rbtrry xs. eˆω ω r = poston vector v = lner velocty ω = ngulr velocty t pont P: v= ωˆ e from geometry: = rsnθ v= ωr snθeˆ = ω r O eˆr θ r ê v v = ω r 3

14 Vector Algebr Multple Products sclr trple product: ( b c). ( b c) = b c [ bc] 2. ( b c) = c b= b c (cyclc permutton) 3. ( b c) = c b= c b = b c (noncyclc permutton) 4. If three vectors, b, nd c re coplnr, then [bc] = 0 ( necessry nd suffcent condton. 5. [bc] represents the volume of prllelpped b c volume = b c c b 4

15 Vector Algebr Multple Products vector trple product: ( b c). Prentheses preserve the order of the operton nd must be retned,.e., ( b c) ( b) c 2. ( b c) s n the plne of b nd c. 3. ( b c) = b( c) c( b) 3 e 3 Vector Components nd Bss A bss n n-spce contns n lnerly ndependent bss vectors. {e, e 2, e 3 } represents bss. = e + e + e sclr component e e 3 e e 2 2 e2 vector component 5

16 Vector Components nd Bss Dul (Recprocl) Bss We cn construct nother bss {e, e 2, e 3 } from {e, e 2, e 3 } tht enbles us to obtn the sclr component of vector. Snce e e 2 s perpendculr to both e nd e 2, 3 e 3 s the only nonzero component from the dot product,.e., ( e e ) = e ( e e ) + e ( e e ) + e ( e e ) or = e3 ( e e2) ( e e ) = = e e ( e e ) , where, 3 e e2 e2 e3 2 e3 e e =, nd smlrly, e =, nd e =. [ eee ] [ eee ] [ eee ]

17 Vector Components nd Bss Now we sy {e, e 2, e 3 } s the dul or recprocl bss of {e, e 2, e 3 } (nd vce vers) snce, e e e e e e 2 3 = 2 = 3 =. Summton Conventon (Ensten or Index Notton) n = = e e For exmple, n 3-spce: = e + e + e sum over repeted (dummy) ndex 7

18 Vector Components nd Bss Kronecker Delt Wth the dul bss we cn now ntroduce symbol clled the Kronecker delt δ defned by, = e e = δ = 0 Snce vector s nvrnt to coordnte trnsformton, t cn be wrtten n terms of ny bss. In prtculr, we cn represent n rbtrry vector usng the dul bss, 8

19 Vector Components nd Bss = ( e ) e + ( e ) e + ( e ) e e e e e e2 e e e3 e 2 3 δ e δ2 e δ3 e 2 3 e 2e 3e = ( ) + ( ) + ( ) = ( ) + ( ) + ( ) = + + Note tht the sclr components for the dul bss re wrtten wth subscrpts. In generl, we defne, = e cogredent sclr components = e contrgredent sclr components Note tht trnsform lke e nd trnsform lke e snce n ddton to (2), we cn wrte, = ( e ) e + ( e ) e + ( e ) e e e2 e3 = + + (2) (3) 9

20 Vector Components nd Bss For n rbtrry vector wrtten n terms of n rbtrry bss, (2) nd (3) cn be wrtten s = ( e ) e nd ( e ) e (4) Exmples: Let = e nd b= b e, then b = b( e e) = bδ = b = b + b + b. A second-order tensor mght be wrtten s, σ = ee = ee + ee + 2 σ σ σ

21 Vector Components nd Bss Orthonorml Bss In generl, ech sclr component nd bss vector hs dfferent unts. For n orthonorml bss, the bss vectors re unt vectors (dmensonless) tht re mutully perpendculr. The sclr components then hve the unts of the vector,.e., unt + orthogonl = orthonorml In ths cse the cogredent nd contrgredent components re the sme, so [ eee 2 3] = e ( e2 e3) = e e= nd = ˆeˆ + ˆ2eˆ2 + ˆ 3eˆ3. Here, the ˆ re physcl components tht hve the unts of the vector. 2

22 Vector Components nd Bss Most engneerng pplctons requrng reference to specfc coordnte system employ n orthonorml system. The most commonly used re the rectngulr Crtesn, cylndrcl, nd sphercl coordnte systems. We wll lter exmne ech of these systems n consderble detl. Grm-Schmdt Orthonormlzton Purpose: Construct n orthonorml bss from n rbtrry set of lnerly ndependent vectors,.e., strtng wth the generl bss { e, e2,, e n } we wll construct the orthonorml bss { eˆ ˆ ˆ, e2,, e n }. 22

23 Vector Components nd Bss Why go through the trouble of cretng n orthonorml bss? Becuse, t s generlly eser to work wth n orthonorml bss. Procedure:. Gven {e, e 2, e 3 }, normlze e eˆ. e = e 2. Choose e 2 nd set e 2 = e2 αeˆ. e 2 e 2 αeˆ 23

24 Vector Components nd Bss 3. Requre, e e 2 e 2 e 2 αeˆ eˆ ( e αeˆ ) = eˆ e α eˆ = 0 α = eˆ e Normlze e e ( eˆ e ) eˆ e ˆ 2 e2 = = e e For the remnng vectors, employ the recurson relton, e = e ( eˆ e ) eˆ ( eˆ e ) eˆ ( eˆ e ) eˆ r+ r+ r+ 2 r+ 2 r r+ r 6. Fnlly, normlze e e ˆ 2 er+ = e r+ r+ 24

25 Vector Components nd Bss Note: Grhm-Schmdt orthonormlzton does not necessrly yeld rght-hnded system. For left-hnded system, n pproprte renumberng of the orthonorml bse vectors wll crete rght-hnded system. Permutton nd Kronecker Delt Symbols The Kronecker delt ws ntroduced erler where t s used n the dot product of bss nd ts dul bss,.e., = e e = δ =. 0 For n orthonorml system, the bss nd the dul bss re dentcl, eˆ ˆ e. The conventon s to choose the subscrpt (cogredent bss) so, 25

26 Vector Components nd Bss eˆ eˆ = δ = 0 = The cross product operton cn be represented n ndex notton by ntroducng the permutton symbol (ctully thrd-order tensor, often clled the permutton tensor or lterntng tensor): eˆ eˆ =ε eˆ k k for rght-hnded orthonorml system, ε k cyclc permutton of k = noncyclc permutton of 0 repeted ndex k 26

27 Vector Components nd Bss Index Notton Exmples Applctons of δ nd ε k : b = ( ˆ eˆ ) ( bˆ eˆ ) = b ˆ ˆ δ = b ˆ ˆδ + b ˆ ˆδ + b ˆ ˆδ b ˆ ˆδ + b ˆ ˆδ + b ˆ ˆδ b ˆ ˆδ + b ˆ ˆδ + b ˆ ˆδ = b ˆ ˆ + b ˆ ˆ + b ˆ ˆ = b ˆ ˆ 27

28 Vector Components nd Bss eˆ eˆ2 eˆ3 b= ˆ ˆ ˆ 2 3 bˆ ˆ ˆ b2 b3 = ( b ˆ ˆ b ˆ ˆ ) eˆ + ( b ˆ ˆ b ˆ ˆ ) eˆ + ( b ˆ ˆ b ˆ ˆ) eˆ = ˆ eˆ bˆ eˆ = ˆ bε ˆ eˆ k k The ε δ dentty: εkεmn = δ mδkn δ nδkm Note: Vector reltons re nvrnt. It s convenent to develop reltons nd do proofs n n orthonorml (usully rectngulr Crtesn) coordnte system becuse we cn employ δ nd ε k. 28

29 Vector Components nd Bss Exmple: Show tht ( b) ( c d) = ( c )( bd ) ( d )( bc ) ( b) ( c d) = ( c )( bd ) ( d )( bc ) = b ˆ ˆε eˆ cˆ dˆ ε eˆ k k m n mnp p = bc ˆ ˆ ˆ dˆ ε ε m n k mnp kp = bc ˆ ˆ ˆ dˆ ε ε m n k mnk δ = bc ˆ ˆ ˆ dˆ ( δ δ δ δ ) m n m n n m = cbd ˆˆ ˆˆ dbc ˆˆˆˆ = ( c )( bd ) ( d )( bc ) Q.E.D. Note: An orthonorml system mples tht the sclr components re the physcl components, we wll no longer employ the cret ^ bove the physcl components for n orthonorml system. 29

30 Vector Components nd Bss Bss, Dul, nd Components: A Grphcl Illustrton Recll from Eq. (4), = ( e ) e nd = ( e ) e. Then e = e cos( e, ) nd ( e ) e= e cos( e, ) orthogonl proecton of = cos(, e ) e n the drecton of e orthogonl proecton of = cos(, e ) e n the drecton of e 30

31 Vector Components nd Bss 2-D llustrton for bss {e, e 2 } nd dul bss {e, e 2 }: e e 2 e 2 e e 2 e e 2 e e = δ e e 2 = e + e 2 2 = e + e 2 2 e e e e e e 3

32 Vector Components nd Bss An mportnt thng to note n the fgure s tht vector does not chnge n orentton or mgntude when represented n ether coordnte system t s nvrnt. Note, however, tht n generl, both the sclr nd vector components re dfferent for dfferent coordnte systems. Specfcton of Vector Gven vector n terms of generl bss {e,e 2, e 3 } 3 e 3 γ e e 2 cos α =, cos β = e e e 3 cosγ = e 2 3 e α e 2 β 2 32

33 Vector Components nd Bss Assocted wth the bss {e,e 2,e 3 } one my wrte n ordered trple, e.g., (, 2, 3 ), (,β,γ), etc. The numbers re ordered n tht they re plced n the order of the bse vector to whch they re ssocted nd they specfy the mgntude nd drecton of. Ths leds to n nlytcl defnton of vector s: An ordered set of numbers tht obey certn specfc vector rules. Our tsk now s to develop these rules. 33

34 Vector Components nd Bss Invrnce & Trnsformton Lws We stted tht vector s nvrnt under coordnte trnsformton. Ths mens tht we my represent ny vector n terms of the bss of ny coordnte system. We strted wth gven generl covrnt bss denoted s { e, e2, e3}. We then ntroduced method of constructng the dul (recprocl) bss. Agn, the purpose of the dul bss thus fr s to enble smple recprocl sclr (dot) product operton. In engneerng pplctons, we often hve need to trnsform from one coordnte system to nother; for exmple n n strodynmcs pplcton, we mght trnsform from coordnte system fxed to n orbtng stellte to geocentrc (Erth-centered) system. In flud mechncs we mght trnsform from body-ft coordnte system to rectngulr Crtesn computtonl system 34

35 Vector Components nd Bss We now develop the generl rules for such coordnte trnsformton. Introduce new cogredent bss ssocted wth some new coordnte system, nd ts contrgredent dul { e, e, e } { e, e, e } As explned erler, we know gven vector cn be represented n terms of ech bss nd ts dul, e.g., = e = e = e = e Now usng ths relton for the contrvrnt components nd the smlr relton for the covrnt components, we hve for the brred system, 35

36 Vector Components nd Bss = ( e ) e = ( e e ) s s s s s = ( e ) e = ( e e ) = ( e ) e = ( e e ) s s s = ( e ) e = ( e e ) s s Once gn use Eqs. (2) nd (3), = ( e ) e nd = ( e) e to wrte the trnsformton reltons for the bss vectors nd duls. Replcng the rbtrry vector wth the specfc bss vectors from the brred system gves the brred bss vectors n terms of the unbrred bss vectors, e = ( e e ) e s s s = ( e e ) e e = ( e e ) e s s = ( e e ) e s 36

37 Vector Components nd Bss Now defne the dot products wth the ssocted trnsformton lws: Cogredent lw: Contrgredent lw: Mxed lws: e e = e nd = s s s s = b e nd = b s s s s es = cse nd s = cs s s s s e = d e nd = d where b e e c e e s s s s e e d e e s s s s 37

38 Vector Components nd Bss Be sure to recognze tht the choce of letter for the dummy vrbles s rbtrry nd were mde for convenence. If both systems re orthonorml, then eˆ = ( eˆ eˆ ) eˆ =γ eˆ where γ = cos( eˆ, eˆ ) The, etc. re not drecton cosnes snce s s s e es nd cos( e, es) = e e s Before contnung, we now vst the elements of lner lgebr 38