Definition talk:Coordinate Function

Subscripts for coordinate functions are incompatible with Einstein notation, a widely used and highly useful notation for differential geometry. For example in this case, if $e_i$ is a basis in $\R^n$, then $\kappa = \kappa^i \, e_i$, where one assumes that the "covariant" and "contravariant" indicies are summed from $1$ to the dimension of the manifold. Lower indicies are used for cobasis vectors, so $e^i \cdot \kappa = e^i \cdot \left( { \kappa^j \, e_j}\right) = \kappa^j \, e^i \cdot e_j = \kappa^j \, \delta^i_j = \kappa^i \equiv \mathrm{pr}^i \circ \kappa$. Hence $e^i \cdot = \mathrm{pr}^i$ and the upper case index is consistent. The full benefit of Einstein notation shows itself when dealing with higher order tensors. Hence I suggest sticking to the notation common in differential geometry (possibly with a rigorous explaination of such). --Geometry dude (talk) 22:44, 17 September 2014 (UTC)


 * Hence what I said somewhere else. All development of a particular field of knowledge benefits from starting at the bottom and working up. --prime mover (talk) 05:03, 18 September 2014 (UTC)