1. Tensors

2. Transformation Rule  A Cartesian vector can be defined by its transformation rule.  Another transformation matrix T transforms similarly. x1x1 x2x2 x3x3

3. 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: scalarRank 0: scalar Rank 1: vectorRank 1: vector Rank 2 and up: TensorRank 2 and up: Tensor  The Kronecker delta is the unit rank-2 tensor. Scalars are independent of coordinate system.

4. Direct Product  A rank 2 tensor can be represented as a matrix.  Two vectors can be combined into a matrix. Vector direct productVector direct product Old name dyadOld name dyad Indices transform as separate vectorsIndices transform as separate vectors

5. Tensor Algebra  Tensors form a linear vector space. Tensors T, U Scalars f, g  Tensor algebra includes addition and scalar multiplication. Operations by component Usual rules of algebra

6. Contraction  The summation rule applies to tensors of different ranks. Dot productDot product Sum of ranks reduce by 2Sum of ranks reduce by 2  A tensor can be contracted by summing over a pair of indices. Reduces rank by 2Reduces rank by 2 Rank 2 tensor contracts to the traceRank 2 tensor contracts to the trace

7. Symmetric Tensor  The transpose of a rank-2 tensor reverses the indices. Transposed products and products transposedTransposed products and products transposed  A symmetric tensor is its own transpose. Antisymmetric is negative transposeAntisymmetric is negative transpose  All tensors are the sums of symmetric and antisymmetric parts.

8. Eigenvalues  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