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Vector-fields are vector valued functions which define a vector at each point in space. Examples of the vector field include the electric field and the velocity of a fluid.
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Misunderstanding of lowering indexes using Euclidian metric
One may define a vector field in $R$ and see how its components transform under a basis transformation.
$ v= v^{u}\partial _{u} $
In principle, the components transform as contravariant such that t …