Consider the holonomic basis defined in Box 26.1 . Using that the tangent vector for a curve can be written \(t=t^{\mu} e_{\mu}=\left(d x^{\mu} / d \lambda\right) e_{\mu}\), show that

Thus, \(g_{\mu v}=e_{\mu}(x) \cdot e_{v}(x)\) and the components of the metric tensor are defined by the scalar products of the coordinate-dependent basis vectors. \(* * *\)

Transcribed Image Text:

ds . ds ds = guy dxdx" = (e ev)dx" dx".