Editing Covariant transformation (section)

{{Short description|Physics concept}}
{{Multiple issues|
{{refimprove|date=December 2018}}
{{Lead too long|date=November 2019}}
}}
In [[physics]], a '''covariant transformation''' is a rule that specifies how certain entities, such as [[Vector (geometric)|vector]]s or [[tensor]]s, change under a [[change of basis]].<ref>{{Cite book |last=Fleisch |first=Daniel A. |year=2011 |title=A Student's Guide to Vectors and Tensors | chapter=Covariant and contravariant vector components}}</ref> The transformation that describes the new [[Basis (linear algebra)|basis vectors]] as a linear combination of the old basis vectors is ''defined'' as a '''covariant transformation'''. Conventionally, indices identifying the basis vectors are placed as '''lower indices''' and so are all entities that transform in the same way. The inverse of a covariant transformation is a '''[[Covariance and contravariance of vectors|contravariant]] transformation'''. Whenever a vector should be ''invariant'' under a change of basis, that is to say it should represent the same geometrical or physical object having the same magnitude and direction as before, its ''components'' must transform according to the contravariant rule. Conventionally, indices identifying the components of a vector are placed as '''upper indices''' and so are all indices of entities that transform in the same way. The sum over pairwise matching indices of a product with the same lower and upper indices is [[Invariant (physics)|invariant]] under a transformation.

A vector itself is a geometrical quantity, in principle, independent (invariant) of the chosen basis. A vector '''v''' is given, say, in components ''v''<sup>''i''</sup> on a chosen basis '''e'''<sub>''i''</sub>. On another basis, say '''e'''′<sub>''j''</sub>, the same vector '''v''' has different components ''v''′<sup>''j''</sup> and
<math display="block"> \mathbf{v} = \sum_i v^i {\mathbf e}_i  = \sum_j {v'\,}^j \mathbf{e}'_j.</math>
As a vector, '''v''' should be invariant to the chosen coordinate system and independent of any chosen basis, i.e. its "real world" direction and magnitude should appear the same regardless of the basis vectors. If we perform a change of basis by transforming the vectors '''e'''<sub>''i''</sub> into the basis vectors '''e'''′<sub>''j''</sub>, we must also ensure that the components ''v''<sup>''i''</sup> transform into the new components ''v''′<sup>''j''</sup> to compensate.

The needed transformation of '''v''' is called the '''contravariant transformation''' rule.

<div style="float:left; border:1px solid #aaa; padding:3px; margin-right:1em; text-align:left">
<gallery widths="200px" heights="200px"> 
  Image:Transformation2polar_basis_vectors.svg|A vector '''v''', and local tangent basis vectors {{nowrap|{'''e'''<sub>x</sub>, '''e'''<sub>y</sub>} }} and {{nowrap|{'''e'''<sub>r</sub>, '''e'''<sub>φ</sub>} }}.
  <!-- Image:Transformation2polar.svg|Components (''v''<sup>x</sup>,''v''<sup>y</sup>) and (''v''<sup>r</sup>,''v''<sup>φ</sup>) of '''v'''. -->
  Image:Transformation2polar contravariant vector.svg|Coordinate representations of '''v'''.
</gallery>
</div>
In the shown example, a vector <math display="inline">\mathbf{v} = \sum_{i \in \{x,y\} } v^i {\mathbf e}_i  = \sum_{j \in \{r,\phi\}} {v'\,}^j \mathbf{e}'_j</math> is described by two different coordinate systems: a rectangular coordinate system (the black grid), and a radial coordinate system (the red grid).  Basis vectors have been chosen for both coordinate systems: '''e'''<sub>x</sub> and '''e'''<sub>y</sub> for the rectangular coordinate system, and '''e'''<sub>r</sub> and '''e'''<sub>φ</sub> for the radial coordinate system. The radial basis vectors '''e'''<sub>r</sub> and '''e'''<sub>φ</sub> appear rotated anticlockwise with respect to the rectangular basis vectors '''e'''<sub>x</sub> and '''e'''<sub>y</sub>. The '''covariant transformation,''' performed to the basis vectors, is thus an anticlockwise rotation, rotating from the first basis vectors to the second basis vectors.

The coordinates of '''v''' must be transformed into the new coordinate system, but the vector '''v''' itself, as a mathematical object, remains independent of the basis chosen, appearing to point in the same direction and with the same magnitude, invariant to the change of coordinates. The contravariant transformation ensures this, by compensating for the rotation between the different bases. If we view '''v''' from the context of the radial coordinate system, it appears to be rotated more clockwise from the basis vectors '''e'''<sub>r</sub> and '''e'''<sub>φ</sub>. compared to how it appeared relative to the rectangular basis vectors '''e'''<sub>x</sub> and '''e'''<sub>y</sub>. Thus, the needed contravariant transformation to '''v''' in this example is a clockwise rotation.{{clear}}