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Tangent vectors as velocity vectors of curves
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We define velocity at c of a curve as the pushforward of the vector d/dt at c. In finding the local coordinate expression of it in, we see that this definition correspond to the usual definition of velocity of curves (as a vector instead of as a derivation). Every tangent vector is the velocity vector at 0 of some curve (in coordinate, just take the line through the origin and the vector). This viewpoint allows us to see tangent vectors as directional derivatives: they act on germ f by taking velocity of image curve under f. This also allows us to compute differential of a map F in a new way: the image of a velocity vector under differential of F is just the velocity of the image curve.
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