Effective mass (solid-state physics) from both sides gives where is the th element of , is the th element of , and are the th and th elements of , respectively, and is the total energy of the particle according to the Planck–Einstein relation. The index is contracted by the use of Einstein notation (there is an implicit summation over ). Since Newton's second law uses the inertial mass (not the gravitational mass), we can identify the inverse of this mass in the equation above as the tensor This tensor expresses the change in group velocity due to a change in crystal momentum. Its
Angular velocity two-dimensional case above, one may define the angular velocity vector as: where "θ" is the angle between r and v. In terms of the cross product, this is: From this, one can recover the tangential velocity as: If a point rotates with angular velocity formula_28 in a coordinate frame formula_29 which itself rotates with an angular velocity formula_30 with respect to an external frame formula_31, we can define formula_32 to be the composite angular velocity vector of the point with respect to formula_31. This operation coincides with usual addition of vectors, and it gives angular velocity the algebraic structure of