TensorsJacobian MatrixCovariant TransformationVolume ElementDirect ProductTensor AlgebraContractionWedge ProductVolume PreservationTensorsTensorsJacobian MatrixJacobian MatrixA general transformation can A general transformation can be expressed as a matrix.be expressed as a matrix.•Partial derivatives between Partial derivatives between two systemstwo systems•JacobianJacobian NNNN real matrix real matrix•Element of the general Element of the general linear group linear group GlGl((NN, , rr))Cartesian coordinate Cartesian coordinate transformations have an transformations have an additional symmetry.additional symmetry.•Not generally true for other Not generally true for other transformationstransformationsmmiiqqxxmiqxJmimiqJxijjiijxxxxJijjijieexxcosCovariant TransformationCovariant TransformationThe components of a The components of a gradient of a scalar do not gradient of a scalar do not transform like a position transform like a position vector.vector.•Inverse transformationInverse transformationThis is a covariant vector.This is a covariant vector.•Designate with subscriptsDesignate with subscriptsPosition is a contravariant Position is a contravariant vector.vector.•Designate with superscriptsDesignate with superscriptsmmiivlv    mimiliixe immiixqqx mmqemmiiqqxxVolume ElementVolume ElementAn infinitessimal volume An infinitessimal volume element is defined by element is defined by coordinates.coordinates.•dVdV = = dxdx11dxdx22dxdx33Transform a volume element Transform a volume element from other coordinates.from other coordinates.•components from the components from the transformationtransformationThe Jacobian determinant is The Jacobian determinant is the ratio of the volume the ratio of the volume elements.elements.x1x2x3111qqxxdiVdJqqqJVqqxqqxqqxVxxxViiiijk321332211321)(321qqqV321xxxVDirect ProductDirect ProductTwo vectors can be Two vectors can be combined into a matrix.combined into a matrix.•Vector direct productVector direct product•Covariant or contravariantCovariant or contravariant•Indices transform as beforeIndices transform as beforeThis is aThis is a tensor tensor of rank 2of rank 2•Vector is tensor rank 1Vector is tensor rank 1•Scalar is tensor rank 0Scalar is tensor rank 0Continued direct products Continued direct products produce higher rank tensors.produce higher rank tensors.332313322212312111bababababababababaBABATCClkljikjiCxxxxCTransformation defines the tensorTensor AlgebraTensor AlgebraTensor algebra many of the Tensor algebra many of the same properties as vector same properties as vector algebra.algebra.•Scalar multiplicationScalar multiplication•Addition, but only if both Addition, but only if both match in number of match in number of covariant and contravariant covariant and contravariant indicesindicesKronecker delta is a tensor.Kronecker delta is a tensor.• ijij or or iijj or or ijij Jacobian matrix is a tensor.Jacobian matrix is a tensor.Permutation epsilon Permutation epsilon ijkijk is a is a rank-3 tensor.rank-3 tensor.•Including permutations of Including permutations of covariant and contravariant covariant and contravariant subscriptssubscriptsTxQJContractionContractionThe summation rule requires The summation rule requires that one index be that one index be contravariant and one be contravariant and one be covariant.covariant.A tensor can be contracted A tensor can be contracted by summing over a pair of by summing over a pair of indices.indices.•Reduces rank by 2Reduces rank by 2ExampleExamplePermittedPermittedNot permittedNot permitted•Note: the usual dot product Note: the usual dot product is not permitted.is not permitted.iijkijTT jjiibac iibas nmnmugv Wedge ProductWedge ProductThe wedge product was The wedge product was defined on two vectors.defined on two vectors.•Magnitude gives area in the Magnitude gives area in the planeplaneIt can be generalized to a set It can be generalized to a set of basis vectors.of basis vectors.•AssociativeAssociative•AnticommutativeAnticommutative•Forms a tensorForms a tensorIt can create a generalized It can create a generalized volume element.volume element.kjikijebabaijjijieeeeee kjieee 321xxxVVolume PreservationVolume PreservationThe group of Jacobian transformations of real vectors The group of Jacobian transformations of real vectors GlGl((NN,,rr) does not generally preserve the volume ) does not generally preserve the volume element.element.Some subsets of transformations do preserve Some subsets of transformations do preserve volume.volume.•Special Linear Group Special Linear Group