Functor - Wikipedia: "There is a convention, now widely disparaged but still in use, which perversely refers to "vectors"—i.e, vector fields, elements of the space of sections {\displaystyle \Gamma (TM)} of a tangent bundle {\displaystyle TM} —as "contravariant" and to "covectors"—i.e., 1-forms, elements of the space of sections {\displaystyle \Gamma (T^{*}M)} of a cotangent bundle {\displaystyle T^{*}M} —as "covariant". This terminology originates in physics, and its rationale has to do with the position of the indices ("upstairs" and "downstairs") in expressions such as {\displaystyle x^{i}=\Lambda _{j}^{i}x^{j}} for {\displaystyle \mathbf {x} '={\boldsymbol {\Lambda }}\mathbf {x} } or {\displaystyle \omega _{i}=\Lambda _{i}^{j}\omega _{j}} for {\displaystyle {\boldsymbol {\omega }}'={\boldsymbol {\omega }}{\boldsymbol {\Lambda }}^{T}.} In this formalism it is observed that the coordinate transformation symbol {\displaystyle \Lambda _{i}^{j}} (representing the matrix {\displaystyle {\boldsymbol {\Lambda }}^{T}} ) acts on the basis vectors "in the same way" as on the "covector coordinates": {\displaystyle \mathbf {e} _{i}=\Lambda _{i}^{j}\mathbf {e} _{j}} —whereas it acts "in the opposite way" on the "vector coordinates" (but "in the same way" as on the basis covectors: {\displaystyle \mathbf {e} ^{i}\Lambda _{j}^{i}\mathbf {e} ^{j}} ). This terminology is perverse because it is the covectors that have pullbacks in general and are thus contravariant, whereas vectors in general are covariant since they can be pushed forward. See also Covariance and contravariance of vectors.
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