[1] The transformation that describes the new basis vectors as a linear combination of the old basis vectors is defined as a covariant 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.
A vector itself is a geometrical quantity, in principle, independent (invariant) of the chosen basis.
A vector v is given, say, in components vi on a chosen basis ei.
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 ei into the basis vectors e′j, we must also ensure that the components vi transform into the new components v′j to compensate.
The radial basis vectors er and eφ appear rotated anticlockwise with respect to the rectangular basis vectors ex and ey.
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 er and eφ.
compared to how it appeared relative to the rectangular basis vectors ex and ey.
Consider a scalar function f (like the temperature at a location in a space) defined on a set of points p, identifiable in a given coordinate system
One can express the derivative of f in old coordinates in terms of the new coordinates, using the chain rule of the derivative, as This is the explicit form of the covariant transformation rule.
To illustrate the transformation properties, consider again the set of points p, identifiable in a given coordinate system
, the tangent vectors to the curves which are simply the coordinate grid itself.
be the basis, tangent vectors in this new coordinates system.
, so in which If we express the new components in terms of the old ones, then This is the explicit form of a transformation called the contravariant transformation and we note that it is different and just the inverse of the covariant rule.
In order to distinguish them from the covariant (tangent) vectors, the index is placed on top.
An example of a contravariant transformation is given by a differential form df.
Take any vector space T. A function f on T is called linear if, for any vectors v, w and scalar α: A simple example is the function which assigns a vector the value of one of its components (called a projection function).
It has a vector as argument and assigns a real number, the value of a component.
for T, we can define a basis, called the dual basis for the dual space in a natural way by taking the set of linear functions mentioned above: the projection functions.
Each projection function (indexed by ω) produces the number 1 when applied to one of the basis vectors
Sometimes an extra notation is introduced where the real value of a linear function σ on a tangent vector u is given as where
This notation emphasizes the bilinear character of the form.
Since vectors and dual vectors may be defined without dependence on a coordinate system, a tensor defined in this way is independent of the choice of a coordinate system.
In the second notation the distinction between vectors and differential forms is more obvious.
Because a tensor depends linearly on its arguments, it is completely determined if one knows the values on a basis
are called the components of the tensor on the chosen basis.
If we choose another basis (which are a linear combination of the original basis), we can use the linear properties of the tensor and we will find that the tensor components in the upper indices transform as dual vectors (so contravariant), whereas the lower indices will transform as the basis of tangent vectors and are thus covariant.