VECTORS AND TENSORS IV.1.1. INTRODUCTION

Transcription

1 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, IV. VECTORS AND TENSORS IV... INTRODUCTION In mthemtcs nd mechncs, we he to operte wth qunttes whch requre dfferent mthemtcl ojects for ther descrpton. Some of them such s temperture, densty, mss, etc. requre only one numer for ther descrpton (n correspondng unts); others such s elocty, ccelerton, force, etc. re chrcterzed y mgntude nd drecton. There re lso more complcted stutons when for some physcl qunttes we need to descre ther dstrutons n dfferent drectons (for exmple, sher stress etc.). The common mthemtcl ojects used for ths purpose re sclrs, ectors, mtrces etc. But ther pplcton ecomes dffcult f chnge of coordnte system s requred. For more conenent nd unersl descrpton whch does not depend on the coordnte system, more generl mthemtcl ojects re used. They re clled tensors. The tensors cn e of dfferent order. A zero order tensor whch s chrcterzed y sngle rel numer corresponds to sclr. A tensor of the frst order s defned y trple of rel numers nd t corresponds to ector. A second order tensor defned y nne rel numers corresponds to mtrx. In generl, n n th order tensor s chrcterzed y n components. The mn purpose of tensor nottons s specl orgnzton of ts components whch oey the so clled trnsformton lw of ts components under the chnge of the coordnte system. Opertons wth these ojects re studed y tensor nlyss. We wll restrct our study mnly to -dmensonl tensors whch re used for descrpton of the physcl qunttes n Euclden spce E.

2 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, IV... EUCLIDIAN SPACE E We ssume tht Eucldn -dmensonl spce E conssts of geometrcl ponts; nd tht n ths spce we cn drw lnes nd cures, plnes nd surfces whch oey the requrements of elementry Eucldn geometry. Also we ssume tht we re le to perform wth the help of ruler nd compss the constructon of segments nd ngles, drwng of rys, prllel lnes etc., nd tht we cn mesure the dstnce etween ponts n terms of the defned unt length. Recll the sc defntons (whch re more ntute thn rgorous) of the geometrcl ojects nd ther symolc sulzton nd nottons: A Pont defnes the poston n spce ut hs no prt. l Lne s set of ponts whch cn e treted s trnslton of pont n unounded strght lne. The ntersecton of two lnes yelds pont. A lne cn e defned y two ponts n spce (there s only one lne whch psses through two fxed ponts). P Plne s set of ponts otned y trnslton of one lne long nother lne. The ntersecton of two plnes yelds lne. A plne cn e defned y two ntersectng lnes. A AB B Segment s lne ounded on oth sdes ( lne connectng two ponts). Any fxed segment cn e chosen s the unt for mesurement for the lengths etween ponts. s Ry s lne ounded on one sde. A ry defnes drecton. Drecton s defned y ry. If two lnes lyng n the sme plne re prllel we sy tht they he the sme drecton. Ech lne decomposed nto two rys defnes two opposte drectons. termnl drecton α > 0 ngle Angle s formed y two rys; one ry determnes the termnl drecton nd the other ry determnes the ntl drecton (mesurng of the ngle from ntl to termnl drecton ccw yelds poste ngles; cw negte ngles ) ntl drecton α β β α Two lnes wth common pont defne two prs of ngles. (conjugte?)

3 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, IV... GEOMETRIC VECTORS IN EUCLIDIAN SPACE E We defne geometrc ector (free ector or just ector) s drected segment n the Eucldn spce E. It cn e sulzed s segment wth rrows ndctng ts drecton. termnl pont Vectors usully re desgnted y the lower cse old letters uwxyz,,,,,,,,... or y letters wth rrows oe,,u,... (there s lso specl desgnton for the unt ss ectors,, j or ˆˆ, j, ˆ ). ntl pont The rrowed end of ector ndctes the drecton nd s clled termnl pont, the other end s clled n ntl pont. Vectors cn e plced n ny locton of Eucldn spce. There s no need of coordnte system for ther defnton (lthough the coordnte system my e helpful for operton wth ectors nd for other types of ectors whch wll e defned lter (poston ectors)). norm (mgntude) The dstnce etween the ntl nd the termnl ponts of ector (the length of the segment) s sd to e the norm (solute lue, mgntude or modulus). It s denoted n one of the followng wys norm of ector equlty We sy tht two ectors re equl f they he the sme drecton nd norm. It mens tht geometrc ectors re not ssocted wth prtculr poston n the spce, nd they cn e moed to ny locton wthout loosng ther dentty (tht s why they re lso clled free ectors). Any ector s representte of whole fmly of ll ectors wth the sme norm nd drecton. If ector cn e otned y prllel trnslton of nother ector then t s the sme ector. In engneerng, comprson of ectors cn e performed only f ther norms re mesured wth the sme unts. zero ector 0 A zero ector 0 s ector wth zero norm. The drecton of such ector looses ts sense, ecuse the termnl pont concdes wth the ntl pont. Any pont n spce s representte of unque zero ector.

4 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, OPERATIONS WITH GEOMETRIC VECTORS (VECTOR ALGEBRA) multplcton y sclr After multplyng the ector y poste sclr > 0 the resultng ector hs the drecton of the ector nd norm. After multplyng the ector y negte sclr < 0 the resultng ector hs the drecton opposte to the drecton of the ector (the termnl nd the ntl ponts re nterchnged) nd norm. Therefore, Vectors whch re sclr multples of ech other re clled collner. Multplcton y 0 turns ny ector to zero ector, 0 0. A zero ector s collner to ny ector. sum The sum of two ectors nd s the ector + determned y the followng + rule: plce the ntl pont of ector to the termnl pont of the ector ; then the ector + hs the ntl pont of ector nd the termnl pont of ector (t s clled the trngle rule). sutrcton of ectors Defne formlly sutrcton of two ectors y ddton of the negte ector: + ( ) + Aeln group Defned n ths wy geometrc ectors wth the operton ddton form n eln group wth the zero ector s neutrl element (t mens tht they relly re ectors). Indeed, usng elementry geometrc constructon, t cn e shown tht the ssocte rule s ld ( + ) + c + ( + c ) The neutrl element s zero ector The nerse to ector s ector wth the sme norm nd opposte drecton And fnlly, the operton ddton s commutte + +

5 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, to deote ll my lfe to the cultton of my reson, nd to progress s much s possle n the nowledge of truth

6 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 0 IV..4. VECTOR SPACES Consder set of ll geometrc (free) ectors V { } unquely represented y the poston ectors wth the operton of ddton of ectors + nd wth the operton of multplcton of ector y sclr. Let us erfy tht ( V, +,) stsfes the xoms of ector spce ( Secton.): The closure xoms ) For ny, V there s the ector + V V V ): For ny V nd there s the ector V These xoms re the corollres of the xomtc propertes of the geometrcl Eucldn spce: tht ny two ponts of the Eucldn spce cn e connected y segment, nd tht ny segment cn e elongted y ny fctor or reduced y ny fctor. The ector xoms ) u+ + u commutte lw u+ u + u u From geometrcl constructon, t s seen the result of summton s the sme dgonl of the prllelogrm. It lso yelds the other equlent defnton of the summton rule clled the prllelogrm rule. Ths rule s used for summton of the poston ectors. ) ( + ) + + ( + ) u w u w ssocte rule Verfcton of ths xom lso cn e performed y geometrcl constructons yeldng the sme resultng ector. w w u u+ ( u+ ) + w u+ ( + w) u + w ) The neutrl element s zero ector ) The nerse to ector s ector (the ector wth the sme norm nd the opposte drecton). The smple geometrcl consdertons yeld the remnng propertes: u u ( u ) + u+ 5) If V u nd,, then ( ) ( ) 6) If, V 7) If V u nd, then ( ) u nd,, then ( ) u u ssocte lw u+ u+ dstrute lw + u u+ u dstrute lw 8) If u V, then u u The propertes 5,7, nd 8 re the propertes of collner ectors (ectors lyng on the sme lne re clled collner). Therefore, we erfed tht The set of ll geometrc ectors V wth opertons of ddton of ectors V, +, nd multplcton of ectors y sclr, form ector spce ( )

7 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 The generl fcts nd propertes of ector spces (consdered n Chpter x) cn e ppled to the ector spce of geometrcl ectors. Now we wnt to fnd wy for representton of geometrcl ectors, nmely we need to determne the dmenson of the ector spce nd construct the ss of the ector spce. Recll some defntons concernng the ector spces from Secton. nd formulte them n terms of geometrc ectors. Lner comnton s fnte sum of the form α + α α n n n α α, V Lner ndependence The set of ectors,,..., n s lnerly ndependent f ther lner comnton s equl to zero ector f nd only f ll coeffcents re equl to zero. Therefore, α + α αnn 0 α α... αn 0 If fnte set of ectors s not lnerly ndependent then t s sd to e lnerly dependent. Therefore, t s possle to construct lner comnton of lnerly dependent ectors equl to zero ector wth the coeffcents not ll equl to zero. If n set of ectors, one of them cn e represented s lner comnton of other ectors, then they re lnerly dependent. Also, n set of lnerly dependent ectors, one of them cn e represented s lner comnton of other ectors. If set of ectors ncludes zero ector, then t s lnerly dependent. If two ectors re lnerly dependent, they re collner (le on the sme lne). Any three lnerly dependent ectors re coplnr (le n the sme plne). VECTORS ON THE LINE -dmensonl ector spce (collner ectors): If two ectors u nd le on the sme lne or on prllel lnes, then one of the ectors cn e represented s the sclr multple of the other u α () Concluson: ny two collner ectors re lnerly dependent. u αu

8 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 VECTORS ON THE PLANE -dmensonl ector spce Let u nd e two lnerly ndependent ectors. If ector s coplnr wth u nd, then t cn e unquely represented s lner comnton αu+ β () Geometrcly, ths fct cn e esly confrmed. Set ll three ectors to the sme ntl pont. Buld prllelogrm wth ector s dgonl nd wth two sdes on the lnes long ectors u nd. Then scle ectors u nd to ectors αu nd β whch concde wth the sdes of the prllelogrm. Then, oously, αu+ β. To see tht ths expnson s unque, ssume tht there exsts the other expnson α u+ β Sutrct ths equton from the preous one, then 0 α α u+ β β ( ) ( ) Hence ectors u nd re lnerly ndependent, coeffcents n ths expnson should e equl to zero, nd therefore α α β β Concluson: ny three coplner ectors re lnerly dependent. VECTORS IN -D SPACE -dmensonl ector spce Let u, nd w e three lnerly ndependent ectors. Then ny ector cn e unquely represented s lner comnton αu+ β+ γw Agn, s proof, consder the followng geometrc constructon. Plce ll four ectors t the sme ntl pont. Prs of ectors u, w, nd wu defne three plnes n the spce. Through the termnl pont of ector drw three more plnes whch re prllel to them. Then ntersectons of the sx plnes form prllelepped wth the ector s dgonl. Scle ectors u, nd w to ectors αu, β nd γ w whch concde wth the edges of the prllelepped. Then from geometrc consderton t s oous tht αu+ β+ γw. Unqueness of ths expnson cn e checed smlrly to the preous cse. Concluson: ny four ectors re lnerly dependent. () BASIS Becuse ny ector n the set of ll geometrc ectors V cn e represented y lner comnton of ny three lner ndependent ectors e, e, e { } αe { } α e + α e + α e (4) the spn of the set e, e, e genertes the Eucldn ector spce V. { } A set of ny three lner ndependent ectors e, e, e s ss of V. Therefore, ector spce of geometrc ectors V s -dmensonl.

9 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 COORDINATE SYSTEM Plce the ss ectors e, e, e t the sme ntl pont O clled the orgn, nd drw lnes long the ectors clled Ox,Ox,Ox. Assocte ech of these lnes wth the rel xs whch drectons concde wth the drecton of ectors e Then they wll form n olque coordnte system, nd the coeffcents n the expnson (4) re clled the coordntes of ector. Denote them y wth the upper ndces. If ectors n the ss u u j 0 for j x e + x e + x e (5) re mutully orthogonl (see defnton elow) then they form n orthogonl coordnte system. If n ddton, the orthogonl,, re of unt length nd ss ectors { } x e { } e { u, u, u }, j,, j j δj 0 j, j,, then the ss s clled orthonorml (where δ s clled the Kronecer delt). j The coordnte system formed y the orthogonl (or orthonorml) ss s clled rectngulr coordnte system (or the Crthesn coordnte system). We wll use two nottons for the Crtesn coordnte system: 0xyz nd 0xxx. Expnson n the rectngulr coordnte system uses coeffcents wth the lower ndces: 0 0 x x+ x+ x x 0 x x (6) 0 0 Rght rectngulr coordnte system Left rectngulr coordnte system The rght rectngulr coordnte system s preferred n mthemtcl modelng n engneerng. The other stndrd notton for the ector components s xˆ+ yˆj+ z ˆ x+ yj+ z (6) Therefore, ny ector x unquely defnes pont n the Euclden spce wth the coordntes ( ) x,x,x or ( ) x,y,z. Therefore, lterntely to coordntes, ectors cn e used for specfctons of ponts, nd nsted of f r cn e used. functons of three rles f ( x,y,z ) the ector functons ( )

10 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 4 FREE VECTORS, BOUND VECTORS, POSITION VECTORS set of ectors equl to free ectors the sme ector ound ectors The equlty of the geometrcl ectors (free ectors) ws defned through the equlty of ther drecton nd length. Accordng to ths defnton, ny ector hs nfntely mny ectors equl to t otned y prllel trnslton of n Eucldn spce. We wll tret them s clss of ectors represented y ny one of them t mens tht ll of them re just the sme ector. It wll prode us unqueness of the result of opertons wth ectors; ut we stll he the flexlty wth hndlng the ectors we cn ssocte t wth ny conenent locton for nlyss. In three dmensons, t s unquely represented y three numers. There re lso stutons n mechncs when ectors re referred to specfed pont n spce, for exmple, they cn e ssocted wth the elocty nd the ccelerton of mong prtcles or wth the elocty feld of flud flow, or wth the forces ctng on odes, or grdents nd fluxes etc. These ectors re clled the ound ectors. For ther defnton we lso need specfcton of poston n spce; n dmensons, ound ectors re gen y sx numers. Lter n the secton Vector Clculus, the poston ectors wll e defned wth the help of ector functons. Free ectors re the most generl nd of ectors. The hndlng ound ectors lwys cn e reduced to opertons wth free ectors. Poston ectors 0 c There s lso specl cse of ound ectors poston ectors whch ll refer to fxed pont unquely defnng the zero poston ector 0. Therefore, they lso need only three numers for ther defnton, ut the opertons wth the ound nd poston ectors should e modfed n such wy tht the result s lso ound or poston ector. The comprson of the free ectors nd poston ectors s demonstrted n the Tle Vectors n Euclden Spce. Ths tle lso ncludes defnton of ectors s the st order tensors whch wll e studed n Secton IV..7. x x z z OP x y z OP OP P (x,y,z ) OP + OP P(x,y,z ) y P(x +x,y +y,z +z ) y P (x,y,z ) Poston ectors re the suset of ll geometrc ectors. Poston ectors re ll ectors wth the ntl pont t the sme fxed pont 0 clled the orgn nd some termnl pont P. A poston ector s denoted y 0P. The defnton of poston ector does not requre the ntroducton of coordnte system, howeer, descrpton of poston ectors s more conenent f coordnte system s ntroduced. Opertons wth poston ectors re smlr to opertons wth free ectors wth some modfctons: Two poston ectors re equl f ther termnl ponts re the sme. A zero ector s represented only y the orgn. Sclr multplcton s equlent to sclr multplcton of free ectors. The sum of two poston ectors s determned y the prllelogrm rule. The set of ll poston ectors wth the operton ector summton nd operton of sclr multplcton form ector spce.

11 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 5 Coordnte ectors Becuse ll poston ectors he the sme ntl pont, they re completely determned only y ther termnl pont. It mens tht dfferent dentfcton of the termnl pont cn lso form ector spce. If the rectngulr coordnte system Oxyz s set to the orgn, then such dentfcton of termnl ponts cn e performed y t s coordntes nd opertons wth poston ectors cn e expressed n terms of the coordntes of the termnl ponts. It s not exctly set of the poston ectors ecuse they re not geometrc ojects (drected segments) ut rther the ordered trple of rel numers, ut they wll e completely dentcl ector spces, nd llows them to e used nterchngely. Denote poston ectors y the coordntes of the termnl pont ( x,x,x ). Two coordnte ectors (,, ) nd (,, ) re equl, f,,. Sclr multplcton (,, ), Addton + ( +, +, + ) Other notton for coordnte ectors x,x,x. Row ectors (,, ) s the other nme for coordnte ectors. Column ectors Column ectors re dentcl to coordnte ectors, the dfference s only on the wy they re wrtten:,, Therefore, free ectors cn e defned n the form of coordnte ectors or n the form of column ectors. It mens tht f free ector s gen, then ts,, n the Crtesn coordnte system re gen. coordntes ( ) The defntons nd opertons wth these types of ectors re summrzed n the tle. The generlzton of the descrpton of ector spce nduced y the geometrcl Eucldn spce s performed wth the help of tensors whch we wll consder elow.

12 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 6 IV..5. DOT PRODUCT The geometrc constructons whch re used n trgonometry, nlytcl geometry or computer grphcs cn e formlzed n terms of opertons wth geometrc ectors. Angle etween two ectors Drw the ectors nd from the sme ntl pont. Then drw the rys n the drecton of ectors nd. These rys defne two poste ngles the sum of whch s equl to the full ngle π. For chrcterzton of the ngle etween two ectors choose those whch re etween 0 nd π. Use the followng notton for the ngle etween the ectors φ (, ) (, ) Dot product The dot product (nner product, sclr product) of ectors, V s defned s mp :V V clculted ccordng to ( ) cos, cosφ (7) nd n the form of the column ectors + + (7) We wll show tht the second defnton follows from the frst one (Property 6). The result of the dot product of two ectors s sclr (rel numer). It s poste f the ngle φ etween ectors nd s cute (less thn π ) nd negte f the ngle φ s otuse (greter thn π ). Orthogonl ectors We sy tht ectors nd re orthogonl nd denote t f the ngle π etween them s the rght ngle φ (, ). It s oous tht non-zero ectors nd re orthogonl f nd only f 0 (8) The condton on the coordntes of ectors to e orthogonl s It cn e shown tht the dot product stsfes propertes of the nner product n ector spce. Propertes of the dot product: ) The dot product s commutte: (commutte lw) ) The dot product of ector wth tself: ( ) ( ) cos, cos s squre of the norm (length) of ector. Therefore, + + Also 0, 0 only f 0

13 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 7 ) If ectors nd re collner (prllel) wth the sme drecton, then ( ) ( ) cos, cos 0 If ectors nd re collner (prllel) wth the opposte drecton, then ( ) ( π ) cos, cos 4) The dot products of the orthonorml ss ectors {,, j } : j j j j 0 or n more compct form for ss ectors n form of {,, } Kronecer delt j δ j j j δ 0 j, j,, δj where δ s clled the Kronecer delt. j 5) Dstrute propertes: ( + c) + c ( + ) c c + c ( α ) ( β ) αβ ( ) αβ, 6) Derton of the equton (7) usng propertes (4) nd (5): ( j ) ( j ) j+ + j + j j+ j + + j+ ( ) ( j) ( ) ( j) ( jj) ( j) ( ) ( j) ( )

14 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 8 PROJECTIONS Projecton of ector on ector s ector computed n the followng wy cos (, ) cos, (9) ( ) where s the length of the projecton clled the component of on Correspondngly, the projecton of ector on ector s wth the component. Then the dot product cn e wrtten n terms of the components n two forms: Projectons on the ss ectors drecton ngles: Ths mens tht the dot product of two ectors s equl to the product of the norm of one ector nd the component of the other ector on the frst one. Wth the help of the dot product, the projectons of ector on the ss ectors cn e determned s: cos, cos, cosα ( ) ( ) x cos, cos, cos β ( ) ( ) y cos, cos, cosγ ( ) ( ) z From these equtons, the drecton cosnes of the ngles etween ector nd the coordnte xs cn e defned s: cos x α + + x x y z y y cos β + + cos z γ x y z z + + x y z (0)

15 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 9 Vector s expnson n Crtesn coordntes Consder how expnson of ector V n the rectngulr coordnte system gen preously y Equton (6) cn e wrtten wth the help of the dot product: x x+ x+ x + + (sum of projectons on xs) + + ( ) ( ) ( ) cos, + cos, + cos, ( cosγ) ( cos β) ( cosγ) + + x y z + + () 0 0 x 0 y z If the coordntes of ector n the coordnte system Oxxx re nown then the projecton of ector on the drecton of the unt ector u cn e determned s (dered from equton (9)) u u u u ( uu ) ( x ) x x + + u u ( x ) x x u+ u+ uu ( ) ( ) ( ) x cos, u + y cos, u + z cos, u u () x cosα + y cos β + z cosγ u

16 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 0 Trnsformton of Coordntes Consder two orthogonl coordnte systems Oxyz nd Oxyz defned y the orthonorml ss {,, } nd {,, }. Consder frst how one ss cn e wrtten n terms of nother ss. Usng expnson (), wrte ectors {,, } n terms of {,, } : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) cos, + cos, + cos, α+ β+ γ cos, + cos, + cos, α + β + γ () cos, + cos, + cos, α+ β+ γ nd wrte ectors {,, } n terms of{,, } : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) cos, + cos, + cos, α + α + α cos, cos, cos, + + β + β + β (4) cos, + cos, + cos, γ + γ + γ 0 γ β α Here we use the followng notton for cosnes of the ngles etween nd : cos (, ) cos (, ) cos (, ) cos (, ) cos (, ) cos (, ) α β γ,,,, (5),, These equtons nclude nne coeffcents whch re cosnes of ngles etween the xes of the dfferent ss. We cn fnd the reltonshps for these coeffcents. Multply correspondngly ech of the equtons y ector whch s n the left hnd sde of the equton: α + β + γ α + β + γ α + β + γ α + β + γ α + β + γ α + β + γ α + α + α β + β + β γ + γ + γ α + α + α β + β + β γ + γ + γ Ths procedure yelds sx equtons for coeffcents, ut the frst three equtons re equlent to the lst three equtons. Now form cross products wth the ectors from the sme ss nd use the condton of orthogonlty α + β + γ αα + ββ + γγ 0 α + β + γ αα + ββ + γγ 0 α + β + γ αα + ββ + γγ 0 α + α + α αβ + αβ + αβ 0 α + α + α αγ + αγ + αγ 0 γ + γ + γ γβ + γβ + γβ 0

17 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 Agn, only three of these equtons for coeffcents re ndependent. Therefore, the nne coeffcents n the ss expnsons re connected only y sx equtons. Three coeffcents remn free t prodes three condtons for the rotton of the rectngulr coordntes system whch cn e defned y three prmeters (for exmple, y three Euler ngles). Consder now, wht wll hppen to the coordntes of some ector under the chnge of the coordnte system from Oxyz to Oxyz. Wrte the expnson of ector n Oxyz (Equton ): x cos (, ) + cos (, ) + cos (, ) ( ) ( ) ( ) x y z Use equton (): cos (, ) + cos (, ) + cos (, ) u u u u u, x y z then projectons of ector on the ectors {,, } re ( ) ( ) ( ) x cos, + y cos, + z cos, ( + + ) ( ) ( ) x cos (, ) + y cos (, ) + z cos (, ) ( ) ( ) ( ) x cos, + y cos, + z cos, α β γ x y z α + β + γ x y z α + β + γ x y z It mens tht coordntes of ector n the new coordnte system re: α + β + γ x x y z α + β + γ (6) y x y z α + β + γ z x y z They prode the drect trnsformton of the ector s coordntes under the chnge of coordnte system from Oxyz to Oxyz. Smlrly, t cn e shown tht under the chnge of coordnte system from Oxyz to Oxyz the coordntes of ector re trnsformed ccordng to α + α + α x x y z β + β + β (7) y x y z γ + γ + γ y x y z In prtculr, f we consder the trnsformton of coordntes of the pont x,y,z under the chnge of coordnte system from Oxyz to ( x,y,z ) to ( ) Oxyz wth the sme orgn, we he: the drect trnsformton of coordntes nd the nerse trnsformton x α x+ β y+ γ z y α x+ β y+ γ z (8) z α x+ β y+ γ z x α x + α y + α z y β x + β y + β z (9) z γ x + γ y + γ z

18 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 Alternte representton of trnsformton of coordntes (trnsotonl stge to tensor nottons) Consder two orthogonl coordnte systems Oxyz nd Oxyz defned y the orthonorml ss {,, } nd {,, }. j α j ( ) α cos, j j Defne the cosnes of the ngles etween coordnte ectors α cos (, ) α β cos (, ) α γ cos (, ) α α j Drectonl cosnes cn e wrtten n mtx form nd j α α α β β β γ γ γ α α α α α α α α α α j j α j ( ) α cos, j j α β γ α β γ α β γ α α α α α α α α α α j j Representton of coordnte ectors n coordnte system Oxyz ( ) + ( ) + ( ) ( ) + ( ) + ( ) ( ) + ( ) + ( ) α + β + γ α + β + γ α + β + γ α + α + α α + α + α α + α + α α β γ α β γ α β γ α α α α α α α α α Representton of coordnte ectors n coordnte system Oxyz ( ) + ( ) + ( ) ( ) + ( ) + ( ) ( ) + ( ) + ( ) α + α + α β + β + β γ + γ + γ α + α + α α + α + α α + α + α α α α β β β γ γ γ α α α α α α α α α

19 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 Representton of ector n Oxyz ( ) + ( ) + ( ) x x y x z x x + y + z x + x + x ( α+ β+ γ ) α ( ) + β ( ) + γ ( ) α x+ β y+ γ z α x + α x + α x ( α+ β+ γ ) α ( ) + β ( ) + γ ( ) α x+ β y+ γ z α x + α x + α x ( α+ β+ γ ) α ( ) + β ( ) + γ ( ) α x+ β y+ γ z x Ax αx + αx + αx x α β γ x y α β γ y z α β γ z x α α α x x α α α x x α α α x (8) Representton of ector n Oxyz ( ) + ( ) + ( ) T A x x y x z x x + y + z x + x + x ( α + α + α ) α ( ) + α ( ) + α ( ) αx + αy + αz α x + α x + α x ( β + β + β ) β ( ) + β ( ) + β ( ) βx + βy + βz α x + α x + α x ( γ + γ + γ ) γ ( ) + γ ( ) + γ ( ) γ x + γ y + γ z x x αx + α x + αx x y z α α α x β β β y γ γ γ z x α α α x x α α α x x α α α x (9)

20 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 4 IV..6. CROSS PRODUCT The cross product (outer product, ector product) of ectors, V s defned s mp :V V V. The result of the cross-product s ector whch s orthogonl to the plne defned y the ectors nd drwn from the sme pont nd t s orented ccordng to the rght-hnd rght-hnd rule. The norm of the ector s defned s rule ( ) sn, (0) It s equl to the re of prllelogrm formed y ectors nd. Propertes: Cross products of ss ectors ) If ectors nd re collner (prllel), then 0 ) The cross product s ntcommutte: ) Dstrute propertes: ( + ) c c+ c ( + c) + c ( ) ( ) ( ), It follows from the defnton nd the propertes tht j j 0 j j j j () j j Component form of cross product If ectors nd re gen s the column ectors: j j+ 0 0 Then usng the dstrute property nd Equtons () one cn otn: ( + j+ ) ( + j+ ) + j+ + j + j j+ j + + j+ j + + j ( ) ( ) j+ ( ) j (0)

21 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 5 Trple sclr product ( c) () c c c The geometrc sense of the trple sclr product s the olume of prllelepped formed y the ectors, nd c : Volume ( c) If the ectors nd, c re coplnr (le on the sme plne), then: ( c ) 0 Prllel ectors The non-zero ectors nd re prllel f nd only f ther cross product s zero ector: 0 () Trple ector product ( c) ( c ) ( c ) () Ths ector ( ) c s perpendculr to ectors nd c, nd therefore, t s n the plne formed y ectors nd c : The other form of the trple ector product s gen y the smlr equton: ( ) c ( c ) ( c ) (4) Lgrnge Identty ( ) ( c d) ( c )( d ) ( d )( c ) (5)

22 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 6 IV..7. EXAMPLES:. Wht s free ector defned y the ntl pont P (,4,) nd the termnl pont Q (,0,6 )? Ponts P nd Q re defned y the poston ectors 0P (,4,) nd 0Q (,4,). Then the ector cn e defned s 0 0Q 0P (,0,6) (,4,) ( 0, 4,) 4j+ 4. Fnd the unt ector n the drecton of ector 4j+. The norm of ector s u u ( ) Then the unt ector u n the drecton of ector cn e defned s 0 4j+ 4 4 u j x,y,z x,y,z 0 re collner. Show tht f the ectors ( ) nd ( ) then x y z x y z If the ectors re collner, then they re multples of ech other x,y,z x,y,z ( ) ( ) ( x, y,z ) ( x,y,z ) x nd, therefore, x x x y y y y z z z z From whch follows the requred dentty. x 0 y 0 z 0 4. (Wor y constnt force) Determne the wor done y constnt force ( 5,4,0) F on the oject long the x-xs on the dstnce s. The wor done y the force s defned y the product of the mgntude of ts projecton on the drecton of moton nd the dstnce tht the ody moes: ( ) ( ) W F s F s F s Let s e ector of mgntude s n drecton : s (,0,0) Then s. W Fs (6) W ( 5,4,0) (,0,0) 55

23 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, For ectors (,,) nd (,,) ) the norm of the ectors: ( ) , fnd: ( ) ) the sum of the ectors: +, +, + 0,,4 ( ) ( ) c) the dot product of the ectors: ( ) ( ) d) the ngle etween the ectors: 4 φ (, ) cos cos c) the projecton of ector on the drecton of ector : (,, ),, f) the cross product of ectors: j ( 9) ( + ) j+ ( ) 4j+ (, 4,) 6. The center of mss of two ponts: mr+ mr R m + m The center of mss of n ponts: m r + m r + + m r R m + m + m n n n Let P,P,...,P n e fxed ponts wth msses m,m,...,m n respectely. Let the ttrcton force of the pont P y the pont P e proportonl to j the dstnce etween the ponts nd to the mss of the pont P : j F r m where s the coeffcent of proportonlty j j j Determne the ttrcton force ctng on the pont P nd determne the equlrum poston of the pont P. m r m r R M r n m F M ( R r) Soluton: Let r e the poston ector wth the termnl pont P. Then the ttrcton force ctng on the pont P y the pont P s ( ) F m r r Then the totl force ctng on the pont P F + F + + F F n ( r r) ( r r) n( rn r) ( ) ( ) m + m + + m m r + m r + + m r m + m + + m r n n n mr+ mr+ + mnr n ( m + m + + mn) r m + m + + mn M ( R r ), where M m + m + + mn r P Pont P s the equlrum pont f F 0, therefore, R r. It mens tht the equlrum pont s locted t the center of mss of the system of ponts.

24 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, (Vector s representton y lner comnton) cw u w Let ectors u,, w u w e lnerly ndependent. Then u w ny ector cn e represented s lner comnton w c u cu c u+ c + c w (ths representton s equlent for wrtng the ector n the olque coordnte system (5)). Let us fnd the coeffcents c,c,c (coordntes of the ector n the coordnte system {,, } uw. Wrte lner comnton n the component form: u w c u c c w + + u w whch cn e wrtten s lner system for coeffcents c,c,c : or n the ector form: u w c u w c u w c Ac Becuse the set of columns n the mtrx A s lnerly ndependent, the mtrx A s nertle, nd the lner system hs unque soluton (Chpter x, sttements,4,5 of the Inerse Mtrx Theorem). Therefore, coeffcents c,c,c cn e found s c u w c u w c u w or usng Crmer s rule (Chpter x, Theorem x). For exmple, fnd the coordntes of ector ( 0,5, 5) coordnte system wth the ss ectors 4,, u w The determnnt of the mtrx of coeffcents 4 det A 5 0 n the olque Ths mens tht the column ectors re lnerly ndependent. Then the lner system hs the soluton 6 c c c Therefore, 6u 0+ 7w

25 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 9 w w w w u u u u 8. (A ector s representton y lner comnton of orthogonl ectors) uw,, e mutully orthogonl: Let ectors { } u uw u 0 Fnd the coeffcents n the representton of y lner comnton c u+ c + c w Construct dot product of the equton wth the ectors { uw,, } consequently : ( ) u c u+ c + c w u c uu + c u + c wu c uu c u ( ) c u+ c + c w c u + c + c w c c ( ) w c u+ c + c w w c u w+ c w+ c w w c w w c w Then the coeffcents cn e determned s: c u u, c, c w w If n ddton, ectors {,, } (7) uw re normlzed ( u w ), then c u, c, c w For the orthogonl ss, t ws not needed to sole the lner system ech coeffcent ws found nddully. Ths s the dntge of the orthogonl ss. In the next exmple, t s shown how the lnerly ndependent set cn e used for constructon of the orthonorml ss. ( ) u u u u u ( ) u u u ( ) + ( ) u u 9. (Grm-Schmdt orthogonlzton process) Let the set of ectors {,, } orthonorml ectors {,, } u u u e lnerly ndependent. Then the set of cn e constructed wth the help of the so clled Grm-Schmdt process whch conssts of the followng steps: ) Normlze the frst ector u nd cll t : u u ) Fnd the component of ector u orthogonl to ector, normlze t nd cll t : ( ) ( ) u u u u ) Projecton of the ector u on the plne defned y the ectors, cn e found s the sum of projectons on the drectons of nd : ( ) ( ) + ( ) u u u Then the ector ( ) ( ) ( ) u u u u u s orthogonl to the plne,. Normlze t nd cll t : ( ) ( ) ( ) ( ) u u u u u u Grm-Schmdt orthogonlzton: u u ( ) ( ) u u u u ( ) ( ) ( ) ( ) u u u u u u (8)

26 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 0 The homomorphsm etween these ector spces cn e estlshed nd n ths sense ll of them re equlent. Therefore, we cn e ery flexle wth ther hndlng nd cn use the nterchngely pproprte to the stuton: physcl modelng s sul wth geometrc ectors, clculton s more conenent wth coordnte ectors, nd derton of equtons s smpler wth tensors.

27 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 IV..8. TENSORS The ndex nottons nd the summton conenton Tensor s n orgnzed multdmensonl rry of numercl lues (numers) whch re clled the components of tensor. Ech tensor comes equpped wth trnsformton lw tht detls how the components of the tensor respond to chnge of ss. Order of tensor s numer of dmensons needed for ts representtons (numer of ndces needed to lel the components). The followng conenton s unerslly ccepted n the modern mthemtcl nd physcl lterture:. Any ndex cn pper n lower or upper poston:, x, c j. Any ndex whch ppers once n the expresson cn te lues,, denotes qunttes:,, denotes 9 qunttes: j,,..., A denotes 7 qunttes: j A, A,..., A. Any ndex whch ppers twce n the expresson denotes summton wth respect to ths ndex from to (Ensten conenton): j j j j j Wth ths conenton the summton sgn cn e dropped nd expressons re smplfed. Note tht ndex of summton s dummy rle, tht mens tht ny other ndex n the sme poston produces the sme result: A A A + A + A 4. The coordntes of pont re usully denoted: n the olque coordnte system y n the rectngulr coordnte system y x x (upper ndex) (lower ndex) 5. The chnge of coordnte system s denoted y prme. The coordntes of the sme pont re denoted n the rectngulr coordnte system Oxyz y x n the rectngulr coordnte system Oxyz y x We wll consder tensors n the lner rectngulr coordnte systems. They re clled Crtesn tensors (or ffne orthogonl tensors).

28 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 Trnsformton of coordntes Consder how the equtons for trnsformton of coordntes cn e rewrtten usng the ndex conenton nd produce some ddtonl results. Coordnte xes: x wth tensor conenton: x x x y x y x z x z x x j x Drectonl cosnes α j j cos (, j ) () α j α α α α α α α α α 0 s fxed trnsformton s rotton α j α α α α α α α α α Trnsformton of the ss ectors: α j (repeted ndex mens summton!) j α j j Then trnsformton of coordntes (8) wth the Ensten conenton ecomes x α x () j j nd the nerse trnsformton (9) s gen y x α x () j j Some useful denttes for coeffcents cn e dered wth tensor nottons (whch correspond to reltonshp etween drectonl cosnes fter Eqn.(5)): α j j α j j αα δ (6) j j α j j α j j α α δ (7) j j

29 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 The zero order tensors (sclrs) Defnton of the 0 th order tensor The zero-order tensors re the elements of the feld of rel numers, whch re unquely specfed n ny coordnte system y sngle numer, nd re nrnt under the chnge of coordnte system: Exmple: The dstnce etween two ponts s the sme n ny coordnte system nd s represented y the zero order tensor (sclr). Indeed, consder two ponts: Pont A wth coordntes Pont B wth coordntes n Oxyz nd n Oxyz nd n Oxyz n Oxyz Let the coordntes of the orgn 0 n the system Oxyz e the coordntes of the orgn 0 n the system Oxyz e o x o x Then + x o α j j α + x o j j + α j ( j j ) o α j j + α j ( j j ) o α j j x x (9) By the Pythgoren Theorem: ( d ) ( ) ( ) α ( ) α j j j ( )( ) α α j j j ( )( ) δ j j j from (6) α jα δ j ( ) δ j j j ( j j) j d Therefore, y tng the squre roots, we formlly estlshed ths smple geometrcl fct tht d d.e. dstnce etween ponts does not depend on the choce of coordnte system nd t s nrnt under trnsformton of coordntes.

30 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 4 The frst-order tensors (ectors) Consder the ector spce. Let Oxyz nd Oxyz e two rectngulr coordnte systems n ths spce. Consder two ponts Pont A wth coordntes n Oxyz nd Pont B wth coordntes n Oxyz nd n Oxyz n Oxyz Let the coordntes of the orgn 0 n the system Oxyz e the coordntes of the orgn 0 n the system Oxyz e o x o x Then + x o α j j α + x o j j + x o α j j α + x (0) o j j The ncrements of coordntes n two systems re connected through the relton x o o ( αj j x ) ( αj j x ) α j ( j j ) + + α j x j Ths equton determnes the trnsformton of the dfference etween the coordntes of two ponts under the chnge of coordnte system from Oxyz to Oxyz. Tht trnsformton s lso equlent to the trnsformton of the coordntes of the pont under the chnge of coordnte system from Oxyz to Oxyz when the orgn of the coordnte system s fxed (just the rotton): x α x j j Ths consderton s foundton for the followng defnton: Defnton of the st order tensor The frst-order tensor (ffne ector or just ector) s gen n ny coordnte system Oxyz y trple x whch s trnsformed under the chnge of coordnte system to Oxyz ccordng to the lw: x α x () j j x α x () j j. Note tht zero ector s zero ector n ll coordnte systems. The st order tensors re equlent to coordnte ectors; the comprson of them n the Tle of Vectors n Euclden Spce (p.0) shows only some smplfcton n the nottons. But the dntge s n the posslty of generlzng them to rtrry order tensors.

31 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 5 Exmple Suppose tht the functon x ( ) t determnes the poston (trjectory) of prtcle of mss m n spce wth the coordnte system Oxyz. Show tht the force ctng on ths prtcle s ector. For the tme nterl from t to t, the dsplcement of the prtcle coordnte system Oxyz s gen y x ( t+ t) x ( t) x whch n the other coordnte system Oxyz s wrtten s: x ( t+ t) x ( t) x If we ssume tht tme does not depend on coordnte system, t t, then ccordng to () ( + ) ( ) α ( + ) ( ) x t t x j t j x t t x t Therefore, dsplcement x s ector, nd proded tht the lmt exsts, x t s lso ector. Moreoer, x lm t 0 t s lso ector, whch defnes nstntneous elocty of the prtcle t the moment of tme t. By smlr rguments, the ccelerton s lso ector wth components. Then, ccordng to Newton s Second Lw F m holds n ny coordnte system, nd force s ector F m.e. force s defned oth y mpltude nd drecton nd cnnot ecome sclr y choce of coordnte system. Dot product + + Norm or mgntude + +

32 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 6 The second-order tensors (mtrces) Consder the ordered trple of ectors whch n the coordnte system Oxyz re wrtten s,, They re descred y the 9 components,, whch cn e orgnzed n one unt s Aj () Accordng to (), under the chnge of coordntes from Oxyz to Oxyz, ectors,, re trnsformed to α α α The smultneous trnsformton of the components of ll three ectors under the chnge of coordnte system cn e performed n the followng wy A α α A j jm m Defnton of the nd order tensor The quntty defned y nne components Aj whch re trnsformed under the chnge of coordnte system ccordng to the lw A α α A () j jm m s clled second-order tensor. A second-order tensor cn e wrtten s A or n the mtrx form (). j A second order tensor defned n one coordnte system cn e determned n ny other coordnte system ccordng to trnsformton (). If ll components of tensor re equl to zero, then the tensor s clled zero tensor. It s oous tht zero tensor s zero tensor n ll coordnte systems.

33 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 7 Exmple A second-order tensor cn lso e otned s lstng of ll cross products of the components of two ectors nd j dydc j dydc (outer product) (4) The trnsformton of the ectors s gen y α α j jm m Then trnsformton of (4) s defned y α α j jm m Therefore (4) s second-order tensor. It s clled dydc nd s denoted y j Note tht. Exmple (stress tensor) edt! Consder pont M n spce. A force ctng on some element of re ds contnng pont M s f p ds where p s stress. Vector p cn e expnded nto pds p ds + p ds + p ds where ectors p, p, p re the components of the stress tensor. Exmple (the deformton tensor n the lner theory of elstcty) u u u x x + Exmple 4 (the rte of the deformton tensor) x x +

34 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 8 IV..9. TENSOR ALGEBRA Consder some opertons wth tensors of dfferent order. Trnspose A T j A j Addton Let A, j Cj Aj + Bj B e the nd order tensors, then defne j Chec f the result s nd order tensor: A j αjm A, m B j αjmbm ( ) C α A + α B α A + B α C j jm m jm m jm m m jm m Therefore, the defned sum s tensor nd trnsforms the sme wy. Multplcton y sclr c, cx, ca j Multplcton of tensors Cjm AB fourth-order tensor (outer product) j m Contrcton of tensors A zero order tensor A multplcton of mtrx y ector, the result s ector ( st order tensor) Mtrx multplcton AB j Dot product mtrx multplcton ( nd order tensor) nner product, dot or sclr product (0 th order tensor) Note tht contrcton reduces the order of tensor. Symmetry Aj A symmetrc tensor j A j A ntsymmetrc tensor j Symmetry propertes of tensors re not chnged under the chnge coordntes: tensor whch s symmetrc (ntsymmetrc) n one coordnte system s symmetrc (ntsymmetrc) n ny other coordnte system. Generl forms symmetrc Aj ntsymmetrc 0 Aj 0 0 Any nd order tensor T cn e represented s sum of symmetrc nd j ntsymmetrc tensors. In fct, such n expnson cn e wrtten s +, where Sj ( Tj Tj ) Tj Sj Aj + symmetrc Aj ( Tj Tj ) ntsymmetrc

35 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 9 Kronecer delt δ j The Kronecer delt s symmetrc tensor (unt tensor) defned s 0 0 j δj j 0 0 (5) 0 j 0 0 Opertons wth δ j chnge of ndex δ Ths expresson ccordng to conenton yelds: δ δ + δ + δ δ δ + δ + δ δ δ + δ + δ chnge of ndex δ Am Am A δ A δ A j j j j fctorng A c ( δ ) A cδ A c tensor nerse j δj A A contrctons δ δ δ δ j j δmδmδj δj, BYU 6 th order tensor

36 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, Alterntng unt tensor ε (Le-Ct tensor, pseudotensor) s specl rd order tensor defned s j ( ) ε j j ε j f j,, or 0 f ny two ndces re le f j,, or (6) rules εj ε j ε ε j j ε ε j j sgn s chnged under nterchnge of ny pr of suscrpts ( totlly ntsymmetrc tensor) εj ε j εj repeted nterchnge of suscrpts contrctons th εjεlm δ jlδm δ jmδ (7) l 4 order tensor ε ε nd δ order tensor (7) j mj m cross-product ( ) ( ) ( ) + ( ) εj (8) j ( ) ε j j + ε j ε j + ε ( ) ε j j + ε j ε j + ε ( ) ε j j + ε j ε j + ε Exercse Trnsformton of δ j oeys the tensor rule (). Trnsformton of ε j oeys the tensor rule (44). Therefore, ndeed, they re the nd oder nd the rd order tensors.

37 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 4 Exmples of pplcton of tensors δ nd j ε to proe the ector denttes j Exmple Proe tht the cross product s ntcommutte: Proof: pply Eqn.(8), nd snce εj ε j ( ) ε ε ε ( ) j j j j j j Exmple Proe the Lgrnge dentty, Eqn.(5): ( ) ( c d) ( c )( d ) ( d )( c ) Proof: pply Eqn. (8) for tensor representton of cross-product ( ) ( c d ) ( ) ( c d ) ε ε cd j j lm l m ε ε cd contrcton (7) j lm j l m ( δ δ δ δ ) cd jl m jm l j l m δ δ cd δ δ cd jl m j l m jm l j l m ( δ )( δ ) ( δ )( δ ) c d d c chnge of ndex jl l m m j jm m l l j cd dc j j j j d d d c j j m m l l ( c )( d ) ( d )( c )

38 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 4 Reducton to prncple xes Contrcton operton A results n ector. It cn e treted s rotton of ector nd chngng of ts length (lner trnsformton). For gen nd order tensor A t s mportnt to determne f there re some ectors whch re not rotted fter contrcton (trnsformton). Ths queston s formulted n the fmlr form of n egenlue prolem: Egenlue prolem Fnd lues of prmeter λ for whch equton A x λx (9) hs non-trl soluton x. They re clled: λ egenlue x egenector Egenectors f they exst determne the prncple xes (coordnte system) of the tensor A. The prolem s to fnd ths coordnte system nd to trnsform tensor j to t. Rewrte equton (5) n the form: ( λδ ) A x 0 The necessry condton for ths equton to he non-trl soluton s: ( λδ ) det A 0 chrcterstc equton (40) The tensor wrtten n the prncple coordnte system hs the smplest form. Formulte the egenlue prolem n the trdtonl ector-mtrx form.

39 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 4 IV..0. SUMMARY OF TENSORS The Crtesn tensors re defned n the rectngulr coordnte system s the qunttes whch under the chnge of the coordnte system oey the followng lws of trnsformton of ts components: Zero-order tensors (sclrs) (4) x α x Frst-order tensors (ectors) (4) j j A α α A Second-order tensors (mtrces) (4) j jm m A α α α A n th -order tensors (44) n n n n The Ensten conenton on ndex notton nd summton (IV..8) s used n these defntons. The coeffcents ( ) α cos, j j j re the cosnes of ngles etween the ss ectors of the coordnte systems (). The tensors wth the hgher order n consst of n rel numers. If the components of tensor n one coordnte system 0xyz re nown, then usng equtons (4-44), we cn determne the components of tensor n the rotted coordnte system 0xyz. Vector spce of tensors V n The set of ll tensors of order n together wth opertons multplcton y sclr nd ddton, form ector spce V n A Vn (45) n Tensors cn lso e defned n the generlzed curlner coordnte systems. Defntons n the m-dmensonl geometrcl spce yeld n th order tensors whch he m n components.

40 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, Orthonorml rght coordnte system {,, }, δ j j ( ) ε j j f j,, or 0 f ny two ndces re le f j,, or 0 Kronecer delt nd Le-Ct tensor δ chnge of ndex δ Am δ Am Am A c ( A cδ ) fctorng j j A A δ tensor nerse ε ε ε j j j ε ε ε j j j εj ε j εj εjεlm δ jlδm δ jmδl Trnsformton of coordntes {,, } nd {,, } 0xyz 0xyz (rotton) α j j α j j α j j α j j αα δ j j α α δ j j Crtesn tensors zero-order tensors (sclrs) x α x frst-order tensors (ectors) j j A α α A second-order tensors (mtrces) j jm m

41 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, Vector opertons + + ε j j Propertes ( ) ( ) ( ) ( + c ) + c ( + c ) + c 0 Trple Sclr product ( ) c ε jc j Trple Vector product ( ) c εjε jmnc j m n Identtes ( c ) ( c ) ( ) c ( ) c ( c ) ( ) c ( ) ( c d ) ( )( ) ( )( ) c d d c Lgrnge denttty ( ) ( c d ) (( ) ) (( ) ) (( ) ) (( ) ) (( ) ) (( ) ) ( ) ( c ) (( ) ) d c c d c d c d c d c d ( c) + ( c ) + c ( ) 0 c? ( ) ( c d) + ( c) ( d) + ( c ) ( d ) 0

42 Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, Ensten s mnuscrpts