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Tensors Transformation Rule x3 x3 x2 A Cartesian vector can be defined by its transformation rule. Another transformation matrix T transforms similarly. x2 x j l ij xi x1 x1 xi lij x j Tpq l ipl jqTij Tij l ip l jqTpq Order and Rank For a Cartesian coordinate system a tensor is defined by its transformation rule. The order or rank of a tensor determines the number of separate transformations. • Rank 0: scalar • Rank 1: vector • Rank 2 and up: Tensor The Kronecker delta is the unit rank-2 tensor. ss Scalars are independent of coordinate system. x j l ij xi Tpq l ipl jqTij Tp1pn l i1 p1 l in pnTi1in pq l ip l jq ij Direct Product A rank 2 tensor can be represented as a matrix. Two vectors can be combined into a matrix. • Vector direct product • Old name dyad • Indices transform as separate vectors C11 C12 C13 C C21 C22 C23 C31 C32 C33 T C AB A B a1b1 C a2b1 a3b1 a1b2 a2b2 a3b2 a1b3 a2b3 a3b3 Tensor Algebra 1T T Tensors form a linear vector space. • Tensors T, U • Scalars f, g f ( gT) ( fg )T ( f g )T fT gT f ( T U ) fT fU T U UT T U Tij U ij Tensor algebra includes addition and scalar multiplication. • Operations by component • Usual rules of algebra Contraction The summation rule applies to tensors of different ranks. ci Aijb j • Dot product • Sum of ranks reduce by 2 Tik ijk v j A tensor can be contracted by summing over a pair of indices. • Reduces rank by 2 • Rank 2 tensor contracts to the trace s ai bi Tij Tii 3 tr T ijTij Tii i 1 Symmetric Tensor The transpose of a rank-2 tensor reverses the indices. ~ T T ij Tij T ji • Transposed products and products transposed (TU ) T U T TT A symmetric tensor is its own transpose. • Antisymmetric is negative transpose All tensors are the sums of symmetric and antisymmetric parts. Sij S ji Aij A ji ~ ~ T 12 T T 12 T T TSA Eigenvalues viTij v j viTij vi ij vi Tij ij 0 T11 T12 T13 det Tij ij T21 T22 T23 0 T31 T32 T33 A tensor expression equivalent to scalar multiplication is an eigenvalue equation. • Equivalent to determinant problem The scalars are eigenvalues. • Corresponding eigenvectors • Left and right eigenvectors next