Naming of contravariant vector field and covariant vector field

Question

I don’t know why contravariant and covariant vector field are named as such. contravariant literally means going against changing, or changing in the opposite way, covariant literally means changing with something in the same way.

For example, vector field and its dual are (in classical terminology) seems contravariant and covariant vector field of the first order. I don’t see how vector field and its dual bear the literal meanings of the two words. Wiki says that they have something to do with their relations to basis vector, but I still can’t figure it out. Is there any reason or historical explanation for that naming?

—

I see so

1. covariant vector is an element in dual space, it’s transformation of or action on vector, but (for first order tensor this transformation of vector is inner product so) it can also be regarded as a transposed vector, and visualized in the same space (but it’s not in the same space) as the upenn link does. The key is to recognize the covariant vector in the above $e^j e_i$ Is not ‘vector‘ $e^j$ but $e^j$ as a transformation or transposed vector (but we need to realize that only when it act on a vector it’s covariant vector).

2. there are two dual: Space of covariant vector, dual space. Its basis can be any proper set of covariant vectors; dual vector, the special basis of dual space $e^j$ that satisfy that its action on basis of vector space produces $\delta_{i, j}$.

3. formally contravariant means inverse of basis transformation matrix: after changing basis by transformation matrix A, the same vector (=components * basis) equals (components transformed by inverse transformation $A^{-1}$ times $A$ transformed basis, i.e. the new basis $e’_i$ ), the result is easier to understand from perspective of matrix operation as insertion of $A^{-1}A$. For dual basis it’s similar. Transformation $B$ of dual basis that satisfies ‘duality’ requirement needs to be inverse $A^{-1}$ of A too,—that is $\delta_{i, j}= e^j e_i = e’^j e’_i = (e^j B)(A e_i)$, where $e’^j , e’_i$ are transformed basis, so $B= A^{-1}$—so transformation of components of dual basis (the components can be easily understood if we view covariant vector as a transposed vector) seems to be just $A$, in this sense it’s covariant.

• Jackozee Hakkiuz: Is the difference between a vector and its components clear to you? Both vectors and dual vectors are invariant. What is covariant and contravariant are their components with respect to a change of basis.
• Charlie Chang: Yes, I understand that. But what’s the component of the dual of a vector? There is an explanation here: seas.upenn.edu/~amyers/DualBasis.pdf. So basis {$e_i$} of a vector and basis {$e^i$} of covector or dual vector satisfy $e_ie^j=\delta_{i,j}$ which means basis vector of different index together determines the direction of a basis vector of covector and basis vector of the same index alone determines the size of the basis vector of covector. So if basis of vector shrinks the vector components will be inflates, the covector or dual vector’s components will shrink
• Charlie Chang: if basis of vector rotates the vector ‘regarded’ as turple of components will rotates in the opposite direction, (the covector or dual vector ‘viewed’ as turple of components will rotates in what direction?) so that roughly fits the literal meaning of the two words. It can also be understood by shrinking and rotating a reference system (axis and units) for a physical system.
• Jackozee Hakkiuz: Does this answer your question?
• Charlie Chang: Yes. Thanks. Another article arxiv.org/pdf/1603.01660.pdf#page17 says a vector can be both contravariant and covariant using different indexes, that confuses me a bit, but I think basically the above explanation makes sense to me, particularly in the case of tangent space and it’s dual space (cotangent space) and their sessions (in classical terminology) contravariant vector field and covariant vector field.

Answer 1

The notions "contravariant" and "covariant" originally apply to vectors in a vector space $V$ and its dual $V^*$. This transfers of course to vector fields on a manifold $M$. A contravariant vector field is a section $M \to TM$ of the tangent bundle, a covariant vector field is a section $M \to T^*M$ of the cotangent bundle.

Now see my answer to Why is tensor from a vector space covariant, not contravariant?