temperature T speed v time t density ρ scalars may be constant or may be variable yes distributive a(b+c) = ab+ac

Size: px

Start display at page:

Download "temperature T speed v time t density ρ scalars may be constant or may be variable yes distributive a(b+c) = ab+ac"

Esmond Lang
5 years ago
Views:

1 Mthmtics Riw. Sclr mthmticl ntity tht hs mgnitud only.g.: tmprtur T spd tim t dnsity ρ sclrs my constnt or my ril Lws of Algr for Sclrs: ys commutti ys ssociti (c) ()c ys distriuti (c) c Fith A. Morrison, Michign Tch U. Mthmtics Riw Polymr Rhology. Vctor mthmticl ntity tht hs mgnitud nd dirction.g.: forc on surfc f locity ctors my constnt or my ril Dfinitions mgnitud of ctor sclr ssocitd with ctor f unit ctor ctor of unit lngth f unit ctor in th dirction of Fith A. Morrison, Michign Tch U.

2 Mthmtics Riw Polymr Rhology Lws of Algr for Vctors:. Addition. Sutrction (-) Fith A. Morrison, Michign Tch U. Lws of Algr for Vctors (continud):. Multipliction y sclr α ys commutti ys ssociti ys distriuti α α ( β ) ( αβ ) αβ ( w) α α w α α 4. Multipliction of ctor y ctor 4. sclr (dot) (innr) product w w cosθ Not: w cn find mgnitud with dot product cos 0 w θ Fith A. Morrison, Michign Tch U.

3 Lws of Algr for Vctors (continud): 4. sclr (dot) (innr) product (con t) ys commutti NO ssociti ys distriuti 4. ctor (cross) (outr) product w w w ( w) w no such oprtion w wsinθ ê is unit ctor prpndiculr to oth nd w following th right-hnd rul w θ Fith A. Morrison, Michign Tch U. Lws of Algr for Vctors (continud): 4. ctor (cross) (outr) product (con t) NO commutti w w ys ssociti ys distriuti ( w) ( w ) ( w) ( ) ( w) w Fith A. Morrison, Michign Tch U.

4 Mthmtics Riw Polymr Rhology Coordint Systms Allow us to mk ctul clcultions with ctors Rul: ny thr ctors tht r non-ro nd linrly indpndnt (non-coplnr) my form coordint sis Thr ctors r linrly dpndnt if α, β, nd γ cn found such tht: α β γ c 0 for α, β, γ 0 If α, β, nd γ r found to ro, th ctors r linrly indpndnt. Fith A. Morrison, Michign Tch U. Mthmtics Riw Polymr Rhology How cn w do ctul clcultions with ctors? Rul: ny ctor my xprssd s th linr comintion of thr, non-ro, non-coplnr sis ctors cofficint of in th ê y dirction ny ctor x x y y x y xy Fith A. Morrison, Michign Tch U. 4

5 5 Mthmtics Riw Fith A. Morrison, Michign Tch U. Polymr Rhology ( ) ( ) ( ) ( ) ( ) Tril clcultion: dot product of two ctors If w choos th sis to orthonorml - mutully prpndiculr nd of unit lngth - thn w cn simplify. Mthmtics Riw Fith A. Morrison, Michign Tch U. Polymr Rhology 0 0 If w choos th sis to orthonorml - mutully prpndiculr nd of unit lngth, thn w cn simplify. W cn gnrli this oprtion with tchniqu clld Einstin nottion.

6 Mthmtics Riw Polymr Rhology Einstin Nottion systm of nottion for ctors nd tnsors tht llows for th clcultion of rsults in Crtsin coordint systms. m m th initil choic of suscript lttr is ritrry th prsnc of pir of lik suscripts implis missing summtion sign Fith A. Morrison, Michign Tch U. Mthmtics Riw Polymr Rhology Einstin Nottion (con t) Th rsult of th dot products of sis ctors cn summrid y th Kronckr dlt function 0 0 δ i p ip 0 i i Kronckr dlt p p Fith A. Morrison, Michign Tch U. 6

7 7 Mthmtics Riw Fith A. Morrison, Michign Tch U. Polymr Rhology Einstin Nottion (con t) To crry out dot product of two ritrry ctors... ( ) ( ) m m m m δ Einstin Nottion Dtild Nottion

This Week. Computer Graphics. Introduction. Introduction. Graphics Maths by Example. Graphics Maths by Example

This Week. Computer Graphics. Introduction. Introduction. Graphics Maths by Example. Graphics Maths by Example This Wk Computr Grphics Vctors nd Oprtions Vctor Arithmtic Gomtric Concpts Points, Lins nd Plns Eploiting Dot Products CSC 470 Computr Grphics 1 CSC 470 Computr Grphics 2 Introduction Introduction Wh do