{\displaystyle M} At each point p ∈ M there is a vector space TpM, called the tangent space, consisting of all tangent vectors to the manifold at the point p. A metric tensor at p is a function gp(Xp, Yp) which takes as inputs a pair of tangent vectors Xp and Yp at p, and produces as an output a real number (scalar), so that the following conditions are satisfied: A metric tensor field g on M assigns to each point p of M a metric tensor gp in the tangent space at p in a way that varies smoothly with p. More precisely, given any open subset U of manifold M and any (smooth) vector fields X and Y on U, the real function, The components of the metric in any basis of vector fields, or frame, f = (X1, ..., Xn) are given by[3], The n2 functions gij[f] form the entries of an n × n symmetric matrix, G[f]. In 1+1 dimensions, suppose we observe that a free-falling rock has \(\frac{dV}{dT}\) = 9.8 m/s 2. @x 0. G Covariant derivative of determinant of the metric tensor. DefMetric@1,metrich@-a,-bD,cd, 8"È","D"<,InducedFrom® 8metricg,n<,PrintAs->"h"D. DefMetric::old : There are already metrics 8metricg

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