Tensors

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

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

Volume Element  An infinitessimal volume element is defined by coordinates. dV = dx 1 dx 2 dx 3dV = dx 1 dx 2 dx 3  Transform a volume element from other coordinates. components from the transformationcomponents from the transformation  The Jacobian determinant is the ratio of the volume elements. x1x1 x2x2 x3x3

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

Tensor Algebra  Tensor algebra many of the same properties as vector algebra. Scalar multiplicationScalar multiplication Addition, but only if both match in number of covariant and contravariant indicesAddition, but only if both match in number of covariant and contravariant indices  Kronecker delta is a tensor.  ij or  i j or  ij  Jacobian matrix is a tensor.  Permutation epsilon  ijk is a rank-3 tensor. Including permutations of covariant and contravariant subscripts

Contraction  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 2Reduces rank by 2 Example  Permitted  Not permitted Note: the usual dot product is not permitted.

Wedge Product  The wedge product was defined on two vectors. Magnitude gives area in the planeMagnitude gives area in the plane  It can be generalized to a set of basis vectors. AssociativeAssociative AnticommutativeAnticommutative Forms a tensorForms a tensor  It can create a generalized volume element.

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)Special Linear Group Sl(N,r) next