3. Tensor (continued) Definitions
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1 atheatcs Revew. ensor (contnued) Defntons Scalar roduct of two tensors : : : carry out the dot roducts ndcated ( )( ) δ δ becoes becoes atheatcs Revew But, what s a tensor really? tensor s a handy reresentaton of a Lnear Vector Functon scalar functon: y f ( x) x + x + a ang of values of x onto values of y vector functon: w f (v) a ang of vectors of v nto vectors w How do we exress a vector functon?
2 atheatcs Revew What s a lnear functon? Lnear, n ths usage, has a recse, atheatcal defnton. Lnear functons (scalar and vector) have the followng two roertes: f ( λx) λf ( x) f ( x + w) f ( x) + f ( w) It turns out... ultlyng vectors and tensors s a convenent way of reresentng the actons of a lnear vector functon (as we wll now show). atheatcs Revew ensors are Lnear Vector Functons Let f(a) b be a lnear vector functon. We can wrte a n Cartesan coordnates. a a + a + a f ( a + a + a ) b Usng the lnear roertes of f, we can dstrbute the functon acton: f a) a f ( ) + a f ( ) + a f ( ) b ( hese results are just vectors, we wll nae the v, w, and.
3 atheatcs Revew ensors are Lnear Vector Functons (contnued) f a) a f ( ) + a f ( ) + a f ( ) b ( v w a v + a w + a b Now we note that the coeffcents a ay be wrtten as, a a ˆ ˆ a a e a a e Substtutng, a v + a w + a e b ˆ he ndeternate vector roduct has aeared! atheatcs Revew Usng the dstrbutve law, we can factor out the dot roduct wth a: ( e v + e w + e ) b a ˆ ˆ ˆ hs s just a tensor (the su of dyadc roducts of vectors) ( e v e w + e ) + ˆ ˆ ˆ a b CONCLUSION: ensor oeratons are convenent to use to exress lnear vector functons.
4 atheatcs Revew. ensor (contnued) ore Defntons Identty ensor I ee ˆ ˆ ee ˆ ˆ + + I ee ˆ ˆ δ ee ˆ ˆ atheatcs Revew. ensor (contnued) ore Defntons Zero ensor agntude of a ensor : + : ee ˆ ˆ : ( )( ) roducts across the dagonal 4
5 atheatcs Revew. ensor (contnued) ore Defntons ensor ransose ( ee ˆ ˆ) Exchange the coeffcents across the dagonal CUION: ( C) ( ee ˆ ˆ Cj j) ( Cj ee ˆ ˆjδ) ( C ee ˆ ˆ ) C j j j j It s not equal to: ( C) ( C ee ˆ ˆ ) C j j j ee ˆ ˆ j I recoend you always nterchange the ndces on the bass vectors rather than on the coeffcents. atheatcs Revew. ensor (contnued) ore Defntons Syetrc ensor e.g. 4 6 ntsyetrc ensor e.g.
6 atheatcs Revew. ensor (contnued) ore Defntons ensor order Scalars, vectors, and tensors ay all be consdered to be tensors (enttes that exst ndeendent of coordnate syste). hey are tensors of dfferent orders, however. order degree of colexty scalars vectors tensors hgherorder tensors th -order tensors st -order tensors nd -order tensors rd -order tensors Nuber of coeffcents needed to exress the tensor n D sace atheatcs Revew. ensor (contnued) ore Defntons ensor Invarants Scalars that are assocated wth tensors; these are nubers that are ndeendent of coordnate syste. vectors: tensors: v v he agntude of a vector s a scalar assocated wth the vector It s ndeendent of coordnate syste,.e. t s an nvarant. here are three nvarants assocated wth a second-order tensor. 6
7 atheatcs Revew ensor Invarants I trace tr For the tensor wrtten n Cartesan coordnates: trace + + II ( ) trace : III trace ( ) jjhh Note: the defntons of nvarants wrtten n ters of coeffcents are only vald when the tensor s wrtten n Cartesan coordnates. 7
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