material or object, given by: |- | unit vector n || Cauchy stress tensor σ || a traction force t || |- | scope="row" rowspan="2"| angular velocity ω | rowspan="2" | moment of inertia I | an angular momentum J || |- | a rotational kinetic energy T || |- | scope="row" rowspan="2"| electric field E | electrical conductivity σ || a current density flow J || |- | polarizability α (related to the permittivity ε and electric susceptibility χE) || an induced polarization field P || |- | magnetic H field || magnetic permeability μ || a magnetic B field || |} For the electrical conduction example, the index and matrix notations would be: while for the rotational kinetic energy T: See also constitutive equation for more specialized examples. Vectors and tensors in n dimensions In n-dimensional Euclidean space over the real numbers, , the standard basis is denoted e1, e2, e3, ... en. Each basis vector ei points along the positive xi axis, with the basis being orthonormal. Component j of ei is given by the Kronecker delta: A vector in takes the form: Similarly for the order-2 tensor above, for each vector a and b in : or more generally: Transformations of Cartesian vectors (any number of dimensions) Meaning of "invariance" under coordinate transformations The position vector x in is a simple and common example of a vector, and can be represented in any coordinate system. Consider the case of rectangular coordinate systems with orthonormal bases only. It is possible to have a coordinate system with rectangular geometry if the basis vectors are all mutually perpendicular and not normalized, in which case the basis is orthogonal but not orthonormal. However, orthonormal bases are easier to manipulate and are often used in practice. The following results are true for orthonormal bases, not orthogonal ones. In one rectangular coordinate system, x as a contravector has coordinates xi and basis vectors ei, while as a covector it has coordinates xi and basis covectors ei, and we have: In another rectangular coordinate system, x as a contravector has coordinates i and bases i, while as a covector it has coordinates i and bases i, and we have: Each new coordinate is a function of all the old ones, and vice versa for the inverse function: and similarly each new basis vector is a function of all the old ones, and vice