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vector field on a manifold
"an assignment of a vector to each point on the manifold"
"In differential geometry, a vector is an operator that takes directional derivatives of manifold functions at its anchor point. The direction and magnitude of the vector are the direction and scale factor of the directional derivative."
"the vector field is an operator that takes a real-valued manifold function and a manifold point and produces a number"
vector field is a *geometric object*; it is independent of any one coordinate system
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