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A vector itself is a geometrical quantity in principle independent (invariant) of the choosen coordinate system. A vector v is given, say, in components vi on a choosen basis ei, related to a coordinate system xi (the basis vectors are tangent vectors to the coordinate grid). On another basis, say , related to a new coordinate system , the same vector v has different components
and
If, for example in a 2-dim Eulidean space, the new basis vectors are rotated to the right with respect to the old basis vectors, than it will appear in terms of the new system the components of the vector look as if the vector was rotated to the left (see figure).
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2 Contravariant transformation 3 Dual properties 4 Co- and contravariant tensor components |
The explicit form of a covariant transformation is best introduced with the transformation properties of the derivative of a function.
Consider a scalar function f (like the temperature in a space) defined on a set of points p, identifiable in a given coordinate system (such a collection is called a manifold). If we adopt a new coordinates system then for each i, the new coordinate can be expressed as function of the original system, so
One can express the derivative of f in new coordinates in terms of the old coordinates, using the chain rule of the derivative, as
A vector can be expressed in terms of basis vectors. For a certain coordinate system, these are taken as the vectors tangent to the coordinate grid.
To illustrate the transformation properties, consider again the set of points p, identifiable in a given coordinate system (manifold).
A scalar function f, that assigns a real number to very point p in this space, is a function of the coordinates . A curve is a one-parameter collection of points c'\', say with curve parameter λ, c(λ). A tangent vector v to the curve is the derivative along the curve with the derivative taken at the point p'' under consideration.
Note that we can see the tangent vector v as an operator
which can be applied to a function
If we adopt a new coordinates system then for each i, the new coordinate can be expressed as function of the original system, so
Let be the basis, tangent vectors in this new coordinates system.
We can express in the new system by applying the chain rule on x. As a function of coordinates we find
the following transformation
The components of a (tangent) vector transform in a different way, called contravariant transformation.
Consider a tangent vector v and call its components
on a basis . On another basis
An example of a contravariant transformation is given by a
differential form df. For f as a function of coordinates , df can be expressed in terms of
.
The differentials dx transform according to the contravariant rule
since
Entities that transform covariant (like basis vectors) and the ones that transform contravariant (like components of a vector and differential forms) are "almost the same" and yet they are different. They have "dual" properties.
What is behind this, is mathematically known as the
dual space that always goes together
with a given linear vector space.
Take any linear vector space T and let
All such linear functions together form a linear
space by themselves. It is called the dual space of T.
One can easily see that, indeed, the sum f+g is again a linear function for linear f and g by applying f+g to a sum v + w.
And that the same holds for scalar multiplication αf.
We can define a basis, called the dual basis in this space in a natural way by taking the set of linear functions mentioned above:
the projection functions. So those functions ω
that produce the number 1 when they are applyed to
one of the basis vector .
For example
There are as many dual 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
With the aid of the section of dual space, a tensor of rank
is simply defined as a real-valued multilinear function of r dual vectors and s vectors in a point p.
So a tensor is defined in a point. It is a linear machine: feed it with vectors and dual vectors and it produces a real number. Since vectors (and dual vectors) are defined coordinate independently, this definition of a tensor is also free of coordinates and doesnot
depend on the choice of a coordinate system.
This is the main importance of tensors in physics.
The notation of a tensor is
Examples of covariant transformation
Derivative of a function transforms covariant
This is the explicit form of the covariant transformation rule.
The notation of a normal derivative with respect to the coordinates sometimes uses a comma, as follows
where the index i is placed as a lower index, because of the covariant transformation.Basis vectors transform covariant
The parallel between the tangent vector and the operator can also be worked out in coordinates
or in terms of operators
where we have written
,
the tangent vectors to the curves which are simply the coordinate grid itself.
which indeed is the same as the covariant transformation for the derivative of a function.Contravariant transformation
we call the components , so
in which
If we express the new components in 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 distinquish them
from the covariant (tangent) vectors, the index is placed on top.Differential form transforms contravariant
Dual properties
be a basis for this space. Consider
a linear real functions defined in this linear space.
If vand w are two vectors in this vector space, than a real function f (with vectors as argument) is called a linear function if both (for any v, w and scalar α)
A simple example is the function which assigns the
value of one of its components (the so called
projection function). It has a vector as argument and assigns a real number, viz. the value of a component. gives a 1 on and zero elsewhere.
Applying this linear function to a vector
, gives (using its linearity)
so just the value of the first coordinate. For this reason
it is called the projection function.as there are basis vectors ,
so the dual space has the same dimension as the linear
space itself. It is "almost the same space",except that the elements of the dual space (called dual vectors) transform contravariant and the elements of the tangent vector space trabsform covariant.
where is a real number. This notation emphasizes the bilinear character of the form.
it is linear in σ since that is a linear function and its is linear in u since that an element of a vector space.Co- and contravariant tensor components
Without coordinates
for dual vectors (differential forms) ρ, σ and tangent vectors .
In the second notation the distiction between vectors and
differential forms is more obvious.