Relative tensor calculus and the tensor time derivative

Abstract

A relative tensor calculus is formulated for expressing equations of mathematical physics. A tensor time derivative operator ▽ ^a_b is defined which operates on tensors λ^ia...ib. Equations are written in a rigid, flat, inertial or other coordinate system a, altered to relative tensor notation, and are thereby expressed in general flowing coordinate systems or materials b, c, d, .... Mirror tensor expressions for ▽ ^a_b λ^ic...id and ▽ ^a_b λ_ic...id exist in a relative geometry G if and only if a rigid coordinate system a exists in G, where ▽ ^a_b λ^ic = λ ^ic_,0c + λ^kev ^aic_ckc + λ ^ic_kc v ^ckc_b , ▽_jcλ^ic = λ ^ic_,jc + λ^kcΓ ^ie_{jc kc} , and v ^aic_b is the velocity of b relative to a with components in c. These operators are convenient in theoretical analyses and can be incorporated into machine programs for the numerical solution of physical problems.