object exerting a force F with velocity v along a straight-line path: tangential velocity v at a point x of a rotating rigid body with angular velocity ω: potential energy U of a magnetic dipole of magnetic moment m in a uniform external magnetic field B: angular momentum J for a particle with position vector r and momentum p: torque τ acting on an electric dipole of electric dipole moment p in a uniform external electric field E: induced surface current density jS in a magnetic material of magnetization M on a surface with unit normal n: How these products transform under orthogonal transformations is illustrated below. Dot product, Kronecker delta, and metric tensor The dot product ⋅ of each possible pairing of the basis vectors follows from the basis being orthonormal. For perpendicular pairs we have while for parallel pairs we have Replacing Cartesian labels by index notation as shown above, these results can be summarized by where δij are the components of the Kronecker delta. The Cartesian basis can be used to represent δ in this way. In addition, each metric tensor component gij with respect to any basis is the dot product of a pairing of basis vectors: For the Cartesian basis the components arranged into a matrix are: so are the simplest possible for the metric tensor, namely the δ: This is not true for general bases: orthogonal coordinates have diagonal metrics containing various scale factors (i.e. not necessarily 1), while general curvilinear coordinates could also have nonzero entries for off-diagonal components. The dot product of two vectors a and b transforms according to which is intuitive, since the dot product of two vectors is a single scalar independent of any coordinates. This also applies more generally to any coordinate systems, not just rectangular ones; the dot product in one coordinate system is the same in any other. Cross product, Levi-Civita symbol, and pseudovectors For the cross product × of two vectors, the results are (almost) the other way round. Again, assuming a right-handed 3d Cartesian coordinate system, cyclic permutations in perpendicular directions yield the next vector in the cyclic collection of vectors: while parallel vectors clearly vanish: and replacing Cartesian labels by index notation as above, these can be summarized by: where i, j, k are indices which take values 1, 2, 3. It follows that: These permutation relations and their corresponding values are important,