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Suppose I have a velocity field [itex] V^a [/itex] which looks like [itex] < \partial x / \partial t , \partial y / \partial t , \partial z / \partial t > [/itex] and I use the metric tensor to change it into a covector field [tex] V_a = g_{ab}V^a. [/tex] Does it then represent the same quantity or [itex] < \partial t / \partial x, \partial t / \partial y , \partial t / \partial z > [/itex] instead? I ask because long ago I read that vectors (like velocity) — when expressed as a set of partial derivatives — have spacial components in their numerators while covectors (like gradient) have them in their denominators. This would suggest - in this case at least - that the covector V would represent time per distance traveled instead of distance traveled per time.