Einstein Summation Convention

Transcription

1 Ensten Suaton Conventon Ths s a ethod to wrte equaton nvovng severa suatons n a uncuttered for Exape:. δ where δ or 0 Suaton runs over to snce we are denson No ndces appear ore than two tes n the equaton Indces whch s sued over s caed duy ndces appear ony n one sde of equaton Indces whch appear on both sdes of the equaton s free ndces. PH6_L

2 PH6_L Vector or Cross or Outer Product ( ( ( ˆ sn ( ( ( ( ( ( ˆ sn ssoctve Not C C n Dstrbutve C C Coutatve Not n + + θ θ Product vares under change of bass,.e. coordnate syste Drecton of the product s gven by rght hand screw rue Product gves the area of the paraeogra consstng the two vectors as ts ars

3 Cross Product: Graphca Representaton nˆ snθ θ PH6_L

4 Exapes: Magnetc force on a ovng charge Torque on a body F qv ag. τ r f f θ r PH6_L 4

5 In the coponent for eˆ eˆ eˆ x y z x Y Z X Y Z e ( + e ( + e ( x y z z y y z x x z z x y y x In the Ensten suaton notaton ( ε k k Where ε k s a Lev Cvta Tensor PH6_L 5

6 The tensor operator ε and k The tensor s defned for,,k,..., as ε k k ε ε k 0 uness,,and kare dstnct +, f rst s an even perutaton of -, f rst s an odd perutaton of odd even PH6_L 6

7 What s a Tensor? Tensor s a ethod to represent the Physca Propertes n an ansotropc syste For exape: You appy a force n one drecton and ook for the affect n other drecton (Pezo eectrcty Eastcty Deectrc constant Conductvty : Eastc Tensor : Deectrc Tensor : Conductvty Tensor PH6_L 7

8 Ths generazed notaton aows an easy wrtng of equatons of the contnuu echancs, such as the generazed Hook's aw : nd rank tensors Stress, stran Conductvty ( susceptbty Kroneker Deta δ rd rank tensors Pezoeectrcty Lev Cvta 4 th rank tensors Eastc odu n th rank tensor has n coponents n densona space J σ E th σ Stress on pane th n drecton PH6_L 8

9 Moent of Inerta tensor: When axs of rotaton s not gven, then we can generaze oent of nerta nto a tensor of rank. nguar oentu L I ω For dscrete partces For contnuous ass dstrbuton For axs of rotaton about ˆn the scaar for can be cacuated as PH6_L 9

10 Rank of a Tensor Rank 0 : Scaar Ony One coponent Rank : Vector Three coponents Rank Nne Coponents Rank Twenty Seven Coponents Rank 4 Eghty One Coponents Syetry pays a very portant roe n evauatng these coponents PH6_L 0

11 PH6_L Tensor notaton,, ( p p p p P In tensor notaton a superscrpt stands for a coun vector a subscrpt for a row vector (usefu to specfy nes atrx s wrtten as You know about Matrx Methods,, ( L,, ( M M M M M M M

12 Tensor notaton Tensor suaton conventon: an ndex repeated as sub and superscrpt n a product represents suaton over the range of the ndex. Exape: L P p + p + p PH6_L

13 Tensor notaton Scaar product can be wrtten as L P p + p + p where the subscrpt has the sae ndex as the superscrpt. Ths pcty coputes the su. Ths s coutatve Mutpcaton of a atrx and a vector Ths eans a change of P fro the coordnate syste to the coordnate syste (transforaton. L P P M P L P PH6_L

14 Lne equaton In cassca ethods, a ne s defned by the equaton ax + by + c 0 In hoogenous coordnates we can wrte ths as L T P x ( a, b, c y 0 In tensor notaton we can wrte ths as L P 0 PH6_L 4

15 Deternant n tensor notaton (, M, det( M ε k k PH6_L 5

16 Cross product n tensor notaton c a b c ( a b ε a b k k PH6_L 6

17 PH6_L 7 Exape Intersecton of two nes L: x+ y+ 0, M: x+ y+ 0 Intersecton: Tensor: Resut:, y x k k M L P E p p p 7

18 PH6_L 8 Transaton Cassc Tensor notaton T s a transforaton fro the syste to Hoogenous coordnates y x t y y t x x y x ty tx y x,,, wth P T P 8

19 PH6_L 9 Rotaton Hoogenous coordnates cos( sn( sn( cos( y a x a y y a x a x cos( sn( 0 sn( cos( y x a a a a y x Cassc Tensor notaton,,, wth P R P 9

20 Scaar Trpe Product x y z C.(.( C.( C x y z Cx Cy C z Can we take these vectors n any other sequence? PH6_L 0

21 vector trpe product The cross product of a vector wth a cross product The expanson forua of the trpe cross product s Exercse: Prove t: Hnt: use ε k ε δ δ k δ δ k Ths vector s n the pane spanned by the vectors b and (when these are not parae. c Note that the use of parentheses n the trpe cross products s necessary, snce the cross product operaton s not assocatve,.e., generay we have PH6_L

22 Coordnate Transforatons: transaton In engneerng t s often necessary to express vectors n dfferent coordnate fraes. Ths requres the rotaton and transaton atrxes, whch reates coordnates,.e. bass (unt vectors n one frae to those n another frae. ê z eˆx' eˆz' T Transaton of Coordnate systes ê y eˆy' Poston coordnate ' T ê x PH6_L