The General Leg Calculus

Abstract

We now proceed to merge the approaches of the Ricci and Cartan calculus given in the two previous chapters into a single unified theory. Our point of departure is the observation that given an arbitrary vector ξ in E₃, we may regard it either as an element ξ of T _P having contravariant components ξ^r, or as an element ξ* of T ^* _P having covariant components ξ^r. The choice — if one is indicated — may be dictated by geometrical or physical considerations. However, since the spaces T _P and T ^* _P are isomorphic, the choices are algebraically equivalent, and the transitions between them are given by the rules

$$ {T_P} \to T_P^*\,:\,{\xi ^r} \mapsto \,{\xi _r} = \,{g_{rs}}{\xi ^s} $$

((1.1))

$$ T_P^* \to \,{T_P}:\,{\xi _r} \mapsto \,{\xi ^r} = \,{g^{rs}}{\xi _s} $$

((1.2))

where the components of the metric tensors g_rs and g^rs satisfy

$$ {g^{rt}}{g_{st}} = \delta _s^r. $$

((1.3))

.