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
Velocity gradient analysis, the dimensions of velocity gradient can be determined. The dimensions of velocity are formula_4 , and the dimensions of distance are formula_5. Since the velocity gradient can be expressed as formula_6. Therefore, the velocity gradient has the same dimensions as this ratio, i.e. formula_7. In 3 dimensions, the gradient, formula_8 , of the velocity formula_9 is a second-order tensor which can be expressed as the matrix L: formula_11 can be decomposed into the sum of a symmetric matrix formula_12 and a skew-symmetric matrix formula_13 as follows formula_12 is called the strain rate tensor and describes the rate of stretching