See also Covariance and contravariance of vectors

In physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, a covariant transformation is a rule (specified below), that describes how certain physical entities change under a change of coordinate system

Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

.
In particular the term is used for vectors and tensor

Tensor

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

s. 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 a covariant transformation is a contravariant transformation. In order that a vector should be invariant under a coordinate transformation, its components must 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

Invariant (physics)

In mathematics and theoretical physics, an invariant is a property of a system which remains unchanged under some transformation.-Examples:In the current era, the immobility of polaris under the diurnal motion of the celestial sphere is a classical illustration of physical invariance.Another...

  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 e_i, 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
(in the so called Einstein notation

Einstein notation

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...

 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 transforming covariant, it must be that the (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 anti-clockwise with respect to the old basis vectors, then it will appear in terms of the new system that the componentwise representation of the vector look as if the vector was rotated in the opposite direction, i.e. clockwise (see figure).

----

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 basis vectors are e_x and e_y. In another coordinate system (dashed and red), the new basis vectors are tangent vectors in the radial direction and perpendicular to it. These basis vectors are indicated in red as e_r and e_φ. They appear rotated anticlockwise with respect to the first basis. The covariant transformation here is thus an anticlockwise rotation.

If we view the vector v with e_φ pointed upwards, its representation in this frame appears rotated to the right. The contravariant transformation is a clockwise rotation.
.
----

The derivative of a function transforms covariantly

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

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

). If we adopt a new coordinates system then for each i, the original coordinate can be expressed as function of the new system, so
One can express the derivative of f in new coordinates in terms of the old coordinates, using the chain rule

Chain rule

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

 of the derivative, as
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 covariantly

A vector can be expressed in terms of basis vectors. For a certain coordinate system, we can choose the vectors tangent to the coordinate grid. This basis is called the coordinate basis.

To illustrate the transformation properties, consider again the set of points p, identifiable in a given coordinate system where (manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

).
A scalar function f, that assigns a real number to every 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

Tangent vector

A tangent vector is a vector that is tangent to a curve or surface at a given point.Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....

 v as an operator (the Directional derivative

Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P in the direction of V...

)
which can be applied to a function
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.

If we adopt a new coordinates system then for each i, the old coordinate can be expressed as function of the new 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

Chain rule

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

 on x. As a function of coordinates we find
the following transformation
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
on a basis . On another basis
we call the components , 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.

Differential forms transform contravariantly

An example of a contravariant transformation is given by a
differential form

Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

 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

Dual properties

Entities that transform covariantly (like basis vectors) and the ones that transform contravariantly (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

Dual space

In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...

 that always goes together with a given linear vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

.

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.

All such scalar-valued linear functions together form a vector space, 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, and that the same holds for scalar multiplication αf.

Given a basis 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. So those functions ω
that produce the number 1 when they are applied to
one of the basis vector .
For example
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.

There are as many dual basis vectors
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 covariant and the elements of the tangent vector space transform contravariantly.

Sometimes an extra notation is introduced where the real
value of a linear function σ on a tangent vector
u is given as
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 it is linear in u since that is an element of a vector space.

Without coordinates

With the aid of the section of dual space, a tensor

Tensor

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

 of type (r,s)
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 independent of coordinate system, 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
for dual vectors (differential forms) ρ, σ and tangent vectors .
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 and

The numbers 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 covariant tensor
contravariant tensor

For a mixed co- and contravariant tensor of rank 2 mixed co- and contravariant tensor