02---tensor-calculus---tensor-algebra

102-tensorcalculus02-tensorcalculus-tensoralgebra2tensorcalculustensorthewordtensorwasintroducedin1846bywilliamrowanhamilton.itwasusedinitscurrentmeaningbywoldemarvoigtin1899.tensorcalculuswasdeve-lopedaround1890bygregorioricci-curba-strounderthetitleabsolutedifferentialcalculus.inthe20thcentury,thesubjectcametobeknownastensoranalysis,andachievedbroaderacceptancewiththeintro-ductionofeinsteins'stheoryofgeneralrelativityaround1915.tensorsareusedalsoinotherfieldssuchascontinuummechanics.tensorcalculus3tensorcalculustensorcalculus-repetition•tensoranalysis•vectoralgebranotation,euklidianvectorspace,scalarproduct,vectorproduct,scalartripleproductnotation,scalarproducts,dyadicproduct,invariants,trace,determinant,inverse,spectraldecomposition,sym-skewdecomposition,vol-devdecomposition,orthogonaltensorderivatives,gradient,divergence,laplaceoperator,integraltransformations•tensoralgebra4tensorcalculusvectoralgebra-notation•summationoveranyindicesthatappeartwiceinaterm•einstein‘ssummationconvention5tensorcalculusvectoralgebra-notation•permutationsymbol•kroneckersymbol6tensorcalculusvectoralgebra-euklidianvectorspace•zeroelementandidentity•euklidianvectorspace•isdefinedthroughthefollowingaxioms•linearindependenceofifistheonly(trivial)solutionto7tensorcalculusvectoralgebra-euklidianvectorspace•euklidianvectorspaceequippedwithnorm•normdefinedthroughthefollowingaxioms8tensorcalculusvectoralgebra-euklidianvectorspace•euklidianvectorspaceequippedwith•representationof3dvectoreuklidiannormwithcoordinates(components)ofrelativetothebasis9tensorcalculusvectoralgebra-scalarproduct•euklidiannormenablesdefinitionofscalar(inner)product•propertiesofscalarproduct•positivedefiniteness•orthogonality10tensorcalculusvectoralgebra-vectorproduct•vectorproduct•propertiesofvectorproduct11tensorcalculusvectoralgebra-scalartripleproduct•scalartripleproduct•propertiesofscalartripleproductareavolume•linearindependency12tensorcalculustensoralgebra-secondordertensors•secondordertensor•transposeofsecondordertensorwithcoordinates(components)ofrelativetothebasis13tensorcalculustensoralgebra-secondordertensors•secondorderunittensorintermsofkroneckersymbol•matrixrepresentationofcoordinateswithcoordinates(components)ofrelativetothebasis•identity14tensorcalculustensoralgebra-thirdordertensors•thirdordertensor•thirdorderpermutationtensorintermsofpermutationwithcoordinates(components)ofrelativetothebasissymbol15tensorcalculustensoralgebra-fourthordertensors•fourthordertensor•fourthorderunittensorwithcoordinates(components)ofrelativetothebasis•transposeoffourthorderunittensor16tensorcalculustensoralgebra-fourthordertensors•symmetricfourthorderunittensor•screw-symmetricfourthorderunittensor•volumetricfourthorderunittensor•deviatoricfourthorderunittensor17tensorcalculustensoralgebra-scalarproduct•scalar(inner)product•propertiesofscalarproductofsecondordertensorandvector•zeroandidentity•positivedefiniteness18tensorcalculustensoralgebra-scalarproduct•scalar(inner)product•propertiesofscalarproductoftwosecondordertensorsand•zeroandidentity19tensorcalculustensoralgebra-scalarproduct•scalar(inner)productoffourthordertensorsandsecondordertensor•zeroandidentity•scalar(inner)productoftwosecondordertensors20tensorcalculustensoralgebra-dyadicproduct•dyadic(outer)product•propertiesofdyadicproduct(tensornotation)oftwovectorsintroducessecondordertensor21tensorcalculustensoralgebra-dyadicproduct•dyadic(outer)product•propertiesofdyadicproduct(indexnotation)oftwovectorsintroducessecondordertensor