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A vector field is a pointwise assignment of vectors(arrows).

Tensors are defined as $\mathbb{R}$-valued multilinear maps, and a tensor field is a pointwise assignment of tensors.

Also, a vector space $V$ is isomorphic to it's dual space $V^*$(thus, a vector can be realized as a linear operator too).

Is a tensor a generalization of a vector in the sense that a tensor is a multilinear map, and a vector can be seen as an operator(since V $\cong$ V*)? And thus, vector fields are just assignments of '1-tensors', and differential forms assignments of 'k-tensors'?

Avi123
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    The isomorphism $V \cong V^*$ depends on some additional structure on $V$ (a basis, a metric, some nondegenerate bilinear form, etc.) So vector fields and 1-forms are dual but not identical. – Matthew Leingang Mar 30 '24 at 17:00
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    Usually, one first defines tangent bundle, then cotangent bundle (and their sections) and only then general tensor bundles and their sections (tensor fields). Thus, tensor fields are generalizations of vector fields and covector fields. – Moishe Kohan Mar 30 '24 at 17:04
  • @MatthewLeingang oh yes, I missed that. – Avi123 Mar 30 '24 at 17:19
  • @MoisheKohan my question is about the nature of the generalization. – Avi123 Mar 30 '24 at 17:27

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