Covariant transformation

From Academic Kids

Template:Mergewith Template:Mergewith In mathematics, a covariant transformation is a rule (specified below), that describes how certain physical entities change under a change of coordinate system. In particular the term is used for vectors and tensors. The transformation that describes the new basis vectors in terms of the old basis, 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 the covariant transformation is called the contravariant transformation. Vectors transform according to the covariant rule, but the components of a vector 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 summation over all indices of a product with the same lower and upper indices, are invariant to a transformation.

A vector itself is a geometrical quantity in principle independent (invariant) of the chosen coordinate system. A vector v is given, say, in components vi on a chosen basis ei, related to a coordinate system xi (the basis vectors are tangent vectors to the coordinate grid). On another basis, say <math>{\mathbf e}'_i<math>, related to a new coordinate system <math>{x'\;}^i<math>, the same vector v has different components <math>{\mathbf v}'\,^i<math> and

<math> {\mathbf v} = \sum_i v^i {\mathbf e}_i = \sum_i {v'\;}^i {\mathbf e}'_i

<math> (in the so called Einstein notation the summation sign is often omitted, implying summation over the same upper and lower indices occurring in a product). With v as invariant and the <math>{\mathbf e}_i<math> transforming covariant, it must be that the <math>v^i<math> (the set of numbers identifying the components) transform in a different way, the inverse called the contravariant transformation rule.

If, for example in a 2-dim Euclidean space, the new basis vectors are rotated to the right with respect to the old basis vectors, then it will appear in terms of the new system that the components of the vector look as if the vector was rotated to the left (see figure).

 curve in polar coordinates
A vector v is described in a given coordinate grid (black lines) on a basis which are the tangent vectors to the (here rectangular) coordinate grid. The components are <math>v_x<math> and <math>v_y<math>. In another coordinate system (dashed and red), the new basis are tangent vectors in the radial direction and perpendicular to it. They appear rotated to the right with respect to the first basis. The covariant transformation here is a rotation to the right. The components in red are indicated as <math>v_r<math> and <math>v_\phi<math>. If we view the new components with <math>v_r<math> pointed upwards, it appears as if the components are rotated to the left. The contravariant transformation is a rotation to the left.


Examples of covariant transformation

Derivative of a function transforms covariant

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 <math>x^i,\; i=0,1,...<math> (such a collection is called a manifold). If we adopt a new coordinates system <math>{x'\,}^i, i=0,1,...<math> then for each i, the new coordinate <math>{x'\,}^i<math> can be expressed as function of the original system, so <math>{x'\,}^i({x}^j), j=0,1,...<math> One can express the derivative of f in new coordinates in terms of the old coordinates, using the chain rule of the derivative, as

    \frac{\partial f\;}{\partial {x'\,}^i} = \sum_j
                \frac{\partial f\;}{\partial {x}^j} \;
                \frac{\partial {x}^j\;}{\partial {x'\,}^i}

<math> 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

    f_{,i} \equiv \frac{\partial f\;}{\partial x^i} 

<math> where the index i is placed as a lower index, because of the covariant transformation.

Basis vectors transform covariant

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 <math>x^i, i=0,1,...<math> (manifold). A scalar function f, that assigns a real number to every point p in this space, is a function of the coordinates <math>f\;(x^0,x^1,...)<math>. 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 <math>df/d\lambda<math> 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

<math> {\mathbf v}[f] \equiv \frac{df}{d\lambda}= \frac{d\;\;}{d\lambda} f(c(\lambda))

<math> The parallel between the tangent vector and the operator can also be worked out in coordinates

<math> {\mathbf v}[f] = \sum_i \frac{dx^i}{d\lambda} \frac{\partial f}{\partial x^i}

<math> or in terms of operators <math>\partial/\partial x^i<math>

   {\mathbf v} = \frac{dx^i}{d\lambda} \frac{\partial \;\;}{\partial x^i} = \frac{dx^i}{d\lambda} {\mathbf e}_i

<math> where we have written <math>{\mathbf e}_i = \partial/\partial x^i<math>, the tangent vectors to the curves which are simply the coordinate grid itself.

If we adopt a new coordinates system <math>{x'\,}^i, \;i=0,1,...<math> then for each i, the new coordinate <math>{x'\,}^i<math> can be expressed as function of the original system, so <math>x^i({x'\,}^j), j=0,1,...<math> Let <math>{\mathbf e}'_i = {\partial\;}/{\partial {x'\,}^i}<math> be the basis, tangent vectors in this new coordinates system. We can express <math>{\mathbf e}_i<math> in the new system by applying the chain rule on x. As a function of coordinates we find the following transformation

    {\mathbf e}'_i = \frac{\partial\;\;\;}{\partial {x'\,}^i} =
                \frac{\partial x^i\;}{\partial {x'\,}^i}
                \frac{\partial\;\;\;}{\partial x^i} =
                \frac{\partial x^i\;\;}{\partial {x'\,}^i\;} 
                {\mathbf e}_i

<math> which indeed is the same as the covariant transformation for the derivative of a function.

Contravariant transformation

The components of a (tangent) vector transform in a different way, called contravariant transformation. Consider a tangent vector v and call its components <math>v^i<math> on a basis <math>{\mathbf e}_i<math>. On another basis <math>{\mathbf e}'\,_i<math> we call the components <math>{v'\,}^i <math>, so

    {\mathbf v} =  \sum_i v^i {\mathbf e}_i  = 
     \sum_i {v'\,}^i  {\mathbf e}'\,_i

<math> in which

<math> v^i = \frac{dx^i}{d\lambda\;} \;\mbox{ and }\;
        {v'\,}^i = \frac{d{x'\,}^i}{d\lambda\;\;}    

<math> If we express the new components in the old ones, then

        {v'\,}^i = \frac{d{x'\,}^i}{d\lambda\;\;} =
                \frac{\partial {x'\,}^i}{\partial x^j}
                \frac{dx^j}{d\lambda\;\;} =
                \frac{\partial {x'\,}^i}{\partial x^j} {\mathbf v}^j

<math> 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.

Differential form transforms contravariant

An example of a contravariant transformation is given by a differential form df. For f as a function of coordinates <math>x^i<math>, df can be expressed in terms of <math> dx^i<math>. The differentials dx transform according to the contravariant rule since

<math> d{x'\,}^i = \sum_j \frac{\partial {x'\,}^i}{\partial {x}^j\;\;} {dx}^j


Dual properties

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 <math>{\mathbf e}_i<math> 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)

<math> f({\mathbf v}+{\mathbf w}) = f({\mathbf v}) + f({\mathbf w}) <math>
<math> f(\alpha {\mathbf v}) = \alpha f({\mathbf v})<math>

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, the value of a component.

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 applied to one of the basis vector <math>{\mathbf e}_i<math>. For example <math>{\omega}^0<math> gives a 1 on <math>{\mathbf e}_0<math> and zero elsewhere. Applying this linear function <math>{\omega}^0<math> to a vector <math>{\mathbf v} =v^i {\mathbf e}_i<math>, gives (using its linearity)

    \omega^0({\mathbf v}) = \omega^0(v^i {\mathbf e}_i) = 
    v^i \omega^0({\mathbf e}_i) = v^0

<math> so just the value of the first coordinate. For this reason it is called the projection function.

There are as many dual basis vectors <math>\omega^i<math> as there are basis vectors <math>{\mathbf e}_i<math>, 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 transform covariant.

Sometimes an extra notation is introduced where the real value of a linear function σ on a tangent vector u is given as

<math> \sigma [{\mathbf u}] \equiv <\sigma, {\mathbf u}>

<math> where <math><\sigma, {\mathbf u}><math> 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

With the aid of the section of dual space, a tensor of rank <math>(^r_s)<math> 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 does not depend on the choice of a coordinate system. This is the main importance of tensors in physics.

The notation of a tensor is

<math> T(\sigma, \ldots ,\rho, {\mathbf u}, \ldots, {\mathbf v})

\;\mbox{ or as }\; { T^{\sigma \ldots \rho} }_{ {\mathbf u} \ldots {\mathbf v}} <math> for dual vectors (differential forms) ρ, σ and tangent vectors <math>{\mathbf u}, {\mathbf v}<math>. In the second notation the distinction between vectors and differential forms is more obvious.

With coordinates

Because a tensor depends linearly on its arguments, it is completely determined if one knows the values on a basis <math>\omega^i \ldots \omega^j<math> and <math> {\mathbf e}_k \ldots {\mathbf e}_l<math>

<math> T(\omega^i,\ldots,\omega^j, {\mathbf e}_k \ldots {\mathbf e}_l) =
    {T^{i\ldots j}}_{k\ldots l}

<math> The numbers <math>{T^{i\ldots j}}_{k\ldots l}<math> 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. For a tensor of rank 2, we can easily verify that

    A'_{i j} = \frac{\partial x^l}{\partial {x'\,}^i}
                      \frac{\partial x^m}{\partial {x'\,}^j} A_{l m}

<math> covariant tensor

    {A'\,}^{i j} = \frac{\partial {x'\,}^i}{\partial x^l}
                \frac{\partial {x'\,}^j}{\partial x^m} A^{l m}

<math> contravariant tensor

For a mixed co- and contravariant tensor of rank 2

    {A'\,}^i_j= \frac {\partial {x'\,}^i} {\partial x^l}
                \frac {\partial x^m}      {\partial {x'\,}^j} A^l_m

<math> mixed co- and contravariant tensor


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