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Co/Contra-variant Transformations

The co/contra-variant nature of vector coordinates has been treated as an elementary characterisation in Tensor Analysis. For example in the classic text translated from the 1966 Russian 3rd Edition and published by Dover, the coordinate space is first introduced with oblique bases, with basis vectors in subscript indices (e_i = ${\frac {\partial \;}{\partial {x}^{i}}}$) and coordinates in superscript indices (xⁱ) , and the motivation for this convention is deferred to later with the following passage as quoted :

"These designations of the components of a vector stem from the fact that the direct transformation of the covariant components involves the coefficients α^k_i' of the direct transformation, that is A'_i= α^k_i'A_k. while the direct transformation of the contravariant components involve the coefficients α^i'_k of the inverse transformation A^i' = α^i'_kA^k."

In the first instance, suppose f is a function over vector space, one can express the scalar derivative components of f in new coordinates in terms of the old coordinates using the chain rule and get ${\frac {\partial f\;}{\partial {x'\,}^{i}}}=\sum _{j}{\frac {\partial f\;}{\partial {x}^{j}}}\;{\frac {\partial {x}^{j}\;}{\partial {x'\,}^{i}}}$. Direct differentiation of the coordinate values produces a transformation where the transformed (new) bases equal the rate of change of the old bases with respect to the new coordinates, times the old bases. To paraphrase, transforms as change of OLD bases times the OLD, (transform directly).

In the second case where the components are not coordinates but some derivative of the coordinate such that vⁱ = dxⁱ/dλ, when we perform a change of bases, for each new coordinate component (i), xⁱ, it fixes relative to independent scalar components (j), by the chain rule: $v'^{i}={\frac {dx^{i}}{dx^{j}}}{\frac {dx^{j}}{d\lambda }}={\frac {\partial x'^{i}}{\partial x^{j}}}v^{j}$, namely, that the new bases equal the rate of change of the new coordinates with respect to the old coordinates, times the old bases. To paraphrase, transforms as change of NEW bases times the OLD (transform inversely).

Algebraic Characterisation

The co/contra-variance of a transformation is an algebraic property, and the designation cited above also applies in a generalised differential geometry setting where in a Category C, with vector spaces V, W belonging to the Category C, a covariant Functor L maps the set of homomorphisms Hom(V,W) to Hom(LV,LW), whereas a contravariant Functor L' maps Hom(V,W) to Hom(L'W,L'V). Again notice the transformation domain's push forward direction in the covariant, and pull-back direction in the contravariant case. This construct extends the covariant differential to Manifolds such as Vector Bundles and their Connections, for example by Parallel Transport extending covariant derivatives to Vector Fields over Manifolds by affinely connecting tangent vectors from one Tangent bundle on the manifold to neighbouring fibres along a 'curve'.(see for example:covariant derivative).

The diagram following illustrates one such curve across three manifolds, from around the cone in conical chart X, then around the cylinder through cylindrical chart X' and finally along the rectilinear plane through rectangular chart X". The force along that force transmission path barring friction loss or geometrical distortion of medium is preserved by parallel transport along the curve.

Displacement of force vectors through change of coordinates from Conical coordinates X to Cylindrical coordinates X' and Rectangular coordinates X", by Parallel transport firstly of torques $\omega$_z, $\omega$_x in coincident parallel planes along coordinate system overlap connection line (zx) and consequently to force f_y along coordinate system overlap connection line (yx) respectively. The force / torque vector (magnitude and direction) at each connection point is invariant of coordinates.

Invariance

Metric Invariance is a geometric concept indicating how a physical property does not change with arbitrary coordinate frames. For example, in (early) tensor analysis elasticity, namely Young's Modulus, should not change simply because we measure it with tensile testing of a rectangular object (a bar) or cylindrical one (a rod). Invariance refers to the fact that elasticity is an independent physical property which is only recovered through the determinant of stress/strain tensors - (summation over all indices), irrespective whether in cylindrical or rectangular coordinates and the transform and its adjoint commute.

Other references to invariance exist - of say parallelism structures on fibre bundles where the covariant derivative can be recovered unchanged across bundle charts under parallel transport.

An earlier (1931) reference to Invariance is given as an attribute of scalar or vector quantities of physical or geometric nature that should be meaningfully remain unchanged with change of coordinate system, and even more instructive is the precedence of Invariance in relation to co/contra-variant transformation property; viz: "Predetermination of the invariance of certain quantities is a basic aid in the development of transformation theory.".

Examples of geometric or physical invariants given in the work cited, include:

"The magnitude of a fixed vector, the scalar products of two and three fixed vectors, the divergence of a fixed vector, the work done by a force in a displacement, and the energy stored per unit volume in a strained elastic medium are examples of scalar invariants. In addition to the fixed vector, the vector product of two or three fixed vectors, the gradient of a scalar function, and the curl of a vector point function are examples of vector invariants."

The subsequent transformation theory developed in the cited work reduces to the following simple rules:

"Reciprocal differentials and the unitary vectors are said to transform 'co-gradiently'; on the other hand the differentials of coordinates and reciprocal unitary vectors transform 'contra-gradiently'".

Other examples motivated from Physics of variance and invariance as a neutral alternative follow.

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