Tensors

1. Tensors

2. A general transformation can be expressed as a matrix. Partial derivatives between two systems JacobianNN real matrix Element of the general linear group Gl(N, r) Cartesian coordinate transformations have an additional symmetry. Not generally true for other transformations Jacobian Matrix

3. Covariant Transformation • The components of a gradient of a scalar do not transform like a position vector. • Inverse transformation • This is a covariant vector. • Designate with subscripts • Position is a contravariant vector. • Designate with superscripts

4. Volume Element • An infinitessimal volume element is defined by coordinates. • dV = dx1dx2dx3 • Transform a volume element from other coordinates. • components from the transformation • The Jacobian determinant is the ratio of the volume elements. x3 x2 x1

5. Two vectors can be combined into a matrix. Vector direct product Covariant or contravariant Indices transform as before This is a tensor of rank 2 Vector is tensor rank 1 Scalar is tensor rank 0 Continued direct products produce higher rank tensors. Direct Product Transformation defines the tensor

6. Tensor algebra many of the same properties as vector algebra. Scalar multiplication Addition, but only if both match in number of covariant and contravariant indices Kronecker delta is a tensor. dij or dij or dij Jacobian matrix is a tensor. Permutation epsilon eijk is a rank-3 tensor. Including permutations of covariant and contravariant subscripts Tensor Algebra

7. The summation rule requires that one index be contravariant and one be covariant. A tensor can be contracted by summing over a pair of indices. Reduces rank by 2 Example Permitted Not permitted Note: the usual dot product is not permitted. Contraction

8. The wedge product was defined on two vectors. Magnitude gives area in the plane It can be generalized to a set of basis vectors. Associative Anticommutative Forms a tensor It can create a generalized volume element. Wedge Product

9. Volume Preservation • The group of Jacobian transformations of real vectors Gl(N,r) does not generally preserve the volume element. • Some subsets of transformations do preserve volume. • Special Linear Group Sl(N,r) next