Vector contravariant

APPENDIX - SETMMARY OF VECTOR AND TENSOR ANALYSTS 8.2.4 Covariant and contravariant vectors... 

Contravariant vectors, those that transform like the coordinate... 

Although it is customary to refer to covariant and contravariant vectors, this may be misleading. Any vector can be described in terms of its contravariant or its covariant components with equal validity. There is no reason other than numerical simplicity for the preference of one set of components over the other. 

The fractional coordinates in crystallography, referred to direct and reciprocal cells respectively, are examples of covariant and contravariant vector components. 

Since both sides of Eq. (33) transform as contravariant vectors, the minimum energy path is invariant to any coordinate transformation. From this point of view, the minimum energy path together with the stationary points can be regarded as a fundamental characteristics of an elementary process. 

Whence, the operator -L/ acting on the vector field of the annihilation operators satisfies the transformation law for contravariant vectors.33 Making the analogous observations as before now for the annihilation operators, the action of the transformation operator on the annihilation operators can formally be written in the usual form of the transformation law for contravariant vectors,33... 

In words, the transformation operator transforms a covariant vector into a covariant vector [cf. Eq. (54)], but the transformation operator transforms a contravariant vector into a contravariant rank 1 tensor that is not a traditional vector. Since Lrs is antisymmetric, the rank 1 contravariant tensor in Eq. (55) can be converted into a vector by interchanging indices, which results in a minus sign. However, in cases in which there is no ambiguity, the covariant and contravariant indices will be collimated to make the notation more compact. 

Equation (4) implies that (detA) = 1, detA = 1. It follows that A is a non-singular matrix, with A = g Ag. Along with contravariant vectors we associate covariant vectors (covectors) a such that... 

By this means the theory of the underlying space may be reduced to the simultaneous affine geometry of this set of affine spaces. The tensors provide a suitable aid for treatment of this simultaneous-affine theory. As first example we take contravariant vectors or contravariant tensors of first rank. That is a geometrical object that contains four components in each coordinate system... 

A contravariant vector is defined by analogy with a covariant one. It is only necessary to assume the particular transformation law... 

It is because of this difference in behaviour between the null component and the others in the case of a covariant and a contravariant vector, that the projective tensor calculus is non-trivial. Otherwise we could think that these tensors simply arise from the combination of affine tensors. In fact, a decomposition into affine tensors happens instead, but it happens differently for covariant and contravariant tensors. 

The behave under gauge transformation like the components of a projective contravariant vector. 

Because our homogeneous coordinates behave like the components of a contravariant vector under coordinate transformation we may interpret the equation = A as characreristic of one and only one point in each tangent space. The index can here be arbitrary but different from zero, i.e. the function A is of the form... 

We know that the Z are components of a contravariant vector of index -1. We next consider the projective derivative... 

Considered as functions of q the Z° are contravariant vectors of index — 1. It therefore follows from the considerations above that the G transform according to the formulae... 

A projective contravariant vector A (a ) of index 0 describes a specific point in each five-dimensional associated space,... 

A straight line through the origin of each five-dimensional associated space is also determined by an affine contravariant vector V x). Their points satisfy the equations... 

The differentials of the coordinates x, ..., x and the differential of the gauge variable transform exactly like the components of a contravariant vector ... 

It is convenient to introduce the Einstein summation convention for products of covariant and contravariant vectors, whereby... 

Covariant and contravariant vectors can be interconverted with use of the metric tensor ri v, given by... 

In contrast to the three-dimensional situation of nonrelativistic mechanics there are now two kinds of vectors within the four-dimensional Minkowski space. Contravariant vectors transform according to Eq. (3.36) whereas covariant vectors transform acoording to Eq. (3.38) by the transition from IS to IS. The reason for this crucial feature of Minkowski space is solely rooted in the structure of the metric g given by Eq. (3.8) which has been shown to be responsible for the central structure of Lorentz transformations as given by Eq. (3.17). As a consequence, the transposed Lorentz transformation A no longer represents the inverse transformation A . As we have seen, the inverse Lorentz transformation is now more involved and given by Eq. (3.25). 

Both sides of eq.(18) are contravariant vectors and change according to the same rule. Mathematically, a solution of eq.(18) yields a curve invariant from the actual coordinate system. 

Many texts use a more complicated notation involving a metric and covariant and contravariant vectors, and the sign of the scalar product is the opposite of the one given here. The more complicated notation is not necessary for the developments in this book. 

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