Tensors

1. Tensors

2. Transformation Rule • A Cartesian vector can be defined by its transformation rule. • Another transformation matrix T transforms similarly. x3 x2 x1

3. 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. Order and Rank Scalars are independent of coordinate system.

4. 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 Direct Product

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

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

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

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