Fundamentals of Tensor Analysis

MCEN 503/ASEN 50 Chptr Fundmntls of Tnsor Anlysis Fll, 006

Fundmntls of Tnsor Anlysis Concpts of Sclr, Vctor, nd Tnsor Sclr α Vctor A physicl quntity tht cn compltly dscrid y rl numr. Exmpl: Tmprtur; Mss; Dnsity; Potntil. Th xprssion of its componnt is indpndnt of th choic of th coordint systm. A physicl quntity tht hs oth dirction nd lngth. Exmpl: Displcmnt; Vlocity; Forc; Ht flow;. Tnsor A Th xprssion of its componnts is dpndnt of th choic of th coordint systm. A nd ordr tnsor dfins n oprtion tht trnsforms vctor to nothr vctor A tnsor contins th informtion out th dirctions nd th mgnituds in thos dirctions. In gnrl, Sclr is 0 th ordr tnsor; Vctor is A st ordr tnsor; nd ordr tnsor; 3 rd ordr tnsor

Fundmntls of Tnsor Anlysis Vctors nd Vctor Algr A vctor is physicl quntity tht hs oth dirction nd lngth Wht do w mn th two vctors r qul? Th two vctors hv th sm lngth nd dirction X3 Wht is unit vctor? X Th lngth of unit vctor is on X 3

Fundmntls of Tnsor Anlysis Vctors nd Vctor Algr X3 3 In Crtsin coordint, vctor cn xprssd y thr ordrd sclrs = + Summtion convntion + 3 3 = i = i 3 i= i i X X Dummy indx nd fr indx 4

Fundmntls of Tnsor Anlysis Vctor Algr Sum: ( i i ) i + = + Sclr Multipliction α = α i i Dot Product ( ) = cosθ, Cross Product = 3 3 3 5

Fundmntls of Tnsor Anlysis Vctor Algr Sum: ( i i ) i + = + Proprtis of Sum + = + ( + ) + c = + ( + c) ( ) o + = X + X Prlllogrm lw of ddition + o = 6

Fundmntls of Tnsor Anlysis Vctor Algr Sclr Multipliction α = α i i X Proprtis of Sclr Multipliction ( αβ ) = α( β ) ( α + β ) = α + β ( + ) = α α α + α (α<0) α (α>0) X 7

8 Vctor Algr Fundmntls of Tnsor Anlysis Proprtis of Dot Product Dot Product ( ), cosθ = = o o = ( ) ( ) ( ) c c + = + β α β α o > 0 nd o = = 0 = norm of = o = is orthogonl to Dirction of vctor: n = =

Fundmntls of Tnsor Anlysis Vctor Algr Componnts of vctor θ Symolic xprssion i i Componnt xprssion X X X X 9

Fundmntls of Tnsor Anlysis Vctor Algr Dot Product = cosθ (, ) Thr sis vctors of Crtsin coordint i i=,,3 i j = 0 i = i j j = δ ij δ ij = 0 i = i j j Kronckr dlt Proprtis of δ ij 0

Fundmntls of Tnsor Anlysis Vctor Algr Cross Product = 3 3 3 Prmuttion symol ε ijk = 0 vn prmuttion odd prmuttion rptd indx

Fundmntls of Tnsor Anlysis Vctor Algr Physicl Mning of Cross Product

Fundmntls of Tnsor Anlysis Vctor Algr Proprtis of Cross Product = 3 3 3 = 3

Fundmntls of Tnsor Anlysis Vctor Algr Tripl sclr product ( ) c c ( ) c = ( c) = ( c ) 4

Fundmntls of Tnsor Anlysis Concpt of Tnsor A nd ordr tnsor is linr oprtor tht trnsforms vctor into nothr vctor through dot product. A: or =A X =A X Proprtis du to linr oprtion ( α + ) = αa A A + ( A ± B) = A ± B 5

Fundmntls of Tnsor Anlysis Antomy of Tnsor: Concpts of Dyd nd Dydic Dyd ( ) A dyd is tnsor. It trnsforms vctor y A dydic is lso tnsor. It is linr comintion of dyds with sclr cofficints. B= + c d 6

Fundmntls of Tnsor Anlysis Concpts of Dyd nd Dydic Now, considr spcil dyd i j 7

Fundmntls of Tnsor Anlysis Concpts of Dyd nd Dydic Now, considr spcil dyd i j 8

9 Fundmntls of Tnsor Anlysis Proprtis of Dyd nd Dydic ( )( ) ( ) ( )d c d c + = + α α ( ) ( ) ( ) c c c + = + β α β α ( ) ( ) A A =

Fundmntls of Tnsor Anlysis Spcil Tnsors Positiv smi-dfinit tnsor: Positiv dfinit tnsor: A 0 for ny 0 A > 0 for ny 0 Unit tnsor: I I = δ ij i j AI = A Trnspos of tnsor T A 0

Fundmntls of Tnsor Anlysis Dot Product AB Proprtis of dot product AB BA ( AB ) C = A( BC) = ABC A = AA n A = AA... A ( ) T T T AB = B A

Fundmntls of Tnsor Anlysis Trc nd Contrction tr A = A Trc ( ) ii tr ( ) ( T A = tr A ) ( AB) tr( BA) ( A + B) = tr( A) tr( B) ( α A) = α tr( A) tr = tr + tr Contrction (doul dot) A : B

Fundmntls of Tnsor Anlysis Tnsor Rviw Dyd: vctor is sitting in front vctor ( ) c = ( c) c ( ) = ( c) Cross Product = 3 3 3 3

Fundmntls of Tnsor Anlysis Rul of Thum: For lgr on vctors nd tnsors, n indx must show up twic nd only twic. If n indx shows up onc on th lft hnd sid (LHS) of = sign, it must show up onc nd only onc on th right hnd sid (RHS) of = sign. This indx is fr indx. If n indx shows up twic on ithr LHS or RHS of =, it dos not hv to show up on th othr sid of =. This indx is dummy indx. You r fr to chng th lttrs tht rprsnt fr indx or dummy indx. But you hv to chng it in pir. 4

Fundmntls of Tnsor Anlysis Dtrminnt nd invrs of tnsor Dtrminnt dt A A = dt A A 3 A A A 3 ( AB) AdtB dt = dt T dt A = dta A A A 3 3 33 Invrs A is singulr if nd only if dta=0 if dta is not zro AA = I = A A 5

Fundmntls of Tnsor Anlysis Proprtis of invrs tnsor ( ) AB = B A ( A ) = A ( ) α A = A α ( ) T ( ) A = A T ( ) T T A =A A = A A A n = A A... A dt ( ) ( dt ) A = A 6

Fundmntls of Tnsor Anlysis Orthogonl tnsor T Q Q = I Q Q = Q θ θ dt Q = ± Q dt Q = dt Q = Q is propr orthogonl tnsor nd corrsponds to rottion Q is n impropr orthogonl tnsor nd corrsponds to rflction 7

Fundmntls of Tnsor Anlysis Rottion Q = cosθ sinθ 0 sinθ cosθ 0 0 0 X θ X Rottion of th vctor X X θ X X Rottion of th coordint systm 8

Fundmntls of Tnsor Anlysis Symmtric nd skw tnsor A=S+W ( T S = A + A ) ( T W A A ) = 9

Fundmntls of Tnsor Anlysis Eignvctors nd ignvlus of tnsor A Th sclr λ is n ignvlu of tnsor A if thr is non-zro vctor unit ignvctor of A so tht nˆ Anˆ = λ nˆ X m=an X λˆ n = Anˆ n nˆ X Gnrl cs: m=an Eignvctor: X λˆ n = Anˆ 30

Fundmntls of Tnsor Anlysis Eignvctors nd ignvlus of tnsor A Anˆ = λnˆ ( A λ I) nˆ = 0 ( A λ I) 0 dt = 3

Fundmntls of Tnsor Anlysis Spctrl Dcomposition 3