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Covariance and contravariance of vectors - this is a copy of the page from 2009 May 12

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For other uses of "covariant" or "contravariant", see covariance and contravariance.

Definition

In

.

In

k, then a linear functional f is a function from V to k which is linear:

${\displaystyle f({\vec {v}}+{\vec {w}})=f({\vec {v}})+f({\vec {w}})}$ for all ${\displaystyle {\vec {v}},{\vec {w}}\in V}$

${\displaystyle f(a{\vec {v}})=af({\vec {v}})}$ for all ${\displaystyle {\vec {v}}\in V,a\in k}$

The set of all linear functionals from V to k, Hom_k(V,k), is itself a k-vector space. This space is called the dual space of V, or sometimes the algebraic dual space, to distinguish it from the continuous dual space. It is often written V^*, or V^′ when the field k is understood.

This means that in a matrix form, a vector times a covector yields the scalar quantity that is the intersection between the vector expressed in the column, and the covector expressed by the row.

The distinction is particularly important for computations with tensors, which often have mixed variance. This means that they have both covariant and contravariant components, or both vectors and covectors. A tensor's valence is number of variant and covariant terms.

Using Einstein notation, covariant components have lower indices, while contravariant components have upper indices.

When one chooses coordinates on a vector space ${\displaystyle V\,}$, for concreteness say Euclidean n-space ${\displaystyle \mathbf {R} ^{n}}$, both vectors and covectors can be written as an n-tuple of numbers, ${\displaystyle (x_{1},\dots ,x_{n})}$, but if one changes the basis, they transform differently. Vectors are called contravariant vectors, while covectors are called covariant vectors.

Given a basis for a vector space, a transform ${\displaystyle T\colon V\to V}$ is represented by a matrix ${\displaystyle M\,}$, while the dual transform ${\displaystyle T^{*}\colon V^{*}\leftarrow V^{*}}$ is represented by the transpose ${\displaystyle M^{t}\,}$, and the inverse dual transform ${\displaystyle T^{*}\colon V^{*}\to V^{*}}$ is represented by the inverse of the transpose ${\displaystyle (M^{t})^{-1}\,}$ (equivalently, transpose of the inverse); duality reverses direction (it is a

), hence the need for the inverse to reverse direction. Thus vectors transform as ${\displaystyle M\,}$, while covectors transform via ${\displaystyle (M^{t})^{-1}\,}$ .

These matrices agree if and only if ${\displaystyle M\,}$ is an orthogonal matrix, in which case covariant and contravariant vectors transform identically.

Context

Both special relativity (Lorentz covariance) and general relativity (general covariance) use covariant basis vectors.

Systems of

are contravariant in the variables.

A major potential cause of confusion is that this duality of covariance/contravariance intervenes every time discussion of a vector or tensor quantity is represented by its components. This causes discussion in the mathematics and physics literature often apparently to be using opposite conventions. It is not the convention that differs, but whether an intrinsic or component-wise description is the primary way of thinking of quantities. As the names suggest, covariant quantities are thought of as moving or transforming forwards, while contravariant quantities transform backwards. This depends on whether one is using a fixed background—a fact that switches the point of view.

Informal usage: invariance

One can contrast covariance and contravariance (transforming in a particular way) with

.

In common

(nor are they in any sense covariant either), and thus proper use should indicate what the invariance is in respect to.

Similar informal usage is sometimes seen with respect to quantities like

energy-momentum tensor

, but one might occasionally see language referring to the covariant mass, meaning the length of the momentum four-vector.

Rules of covariant and contravariant transformation

Vectors are covariant, and covectors are contravariant, but the components of vectors are contravariant and the components of covectors are covariant.'' (This is in conflict with another section on this page, http://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors#Definition, "Vectors are called contravariant vectors, while covectors are called covariant vectors," and another page! "As stated [in http://en.wikipedia.org/wiki/Curvilinear_coordinates#Covariant_basis], contravariant vectors are vectors with contravariant components...)" See Einstein notation for details.

This is a frequently confused point.

In tensor representation, a vector ${\displaystyle \mathbf {A} }$ can be expressed as the sum of the products of each of its components times the basis vector belonging to that component in two ways (repeated indices are assumed to sum according to the

):

${\displaystyle \mathbf {A} =a^{i}\mathbf {e} _{i}=a_{i}\mathbf {e} ^{i}}$

where ${\displaystyle a^{i}\,}$ are called the contravariant components of ${\displaystyle \mathbf {A} }$, ${\displaystyle a_{i}\,}$ are called the covariant components of ${\displaystyle \mathbf {A} }$, ${\displaystyle \mathbf {e} _{i}}$ are covariant basis vectors, and ${\displaystyle \mathbf {e} ^{i}}$ are contravariant basis vectors if and only if these transform from coordinates ${\displaystyle x'^{i}\,}$ to coordinates ${\displaystyle x^{i}\,}$ (where ${\displaystyle x^{i}\,}$ are differentiable functions of ${\displaystyle x'^{i}\,}$, and vice versa) according to the rules:

${\displaystyle a^{i}=a'^{j}{\partial x^{i} \over \partial x'^{j}},}$

${\displaystyle a_{i}=a'_{j}{\partial x'^{j} \over \partial x^{i}},}$

${\displaystyle \mathbf {e} ^{i}=\mathbf {e'} ^{j}{\partial x^{i} \over \partial x'^{j}},}$

${\displaystyle \mathbf {e} _{i}=\mathbf {e'} _{j}{\partial x'^{j} \over \partial x^{i}},}$

where the primed components and basis vectors represent A in the coordinates ${\displaystyle x'^{i}\,}$:

${\displaystyle \mathbf {A} =a'^{i}\mathbf {e'} _{i}=a'_{i}\mathbf {e'} ^{i}.}$

We could also compute the inverse relations:

${\displaystyle a'^{i}=a^{j}{\partial x'^{i} \over \partial x^{j}},}$

${\displaystyle a'_{i}=a_{j}{\partial x^{j} \over \partial x'^{i}},}$

${\displaystyle \mathbf {e'} ^{i}=\mathbf {e} ^{j}{\partial x'^{i} \over \partial x^{j}},}$

${\displaystyle \mathbf {e'} _{i}=\mathbf {e} _{j}{\partial x^{j} \over \partial x'^{i}},}$

which is only possible if the determinant of the matrices formed by the components of ${\displaystyle \partial x^{i}/\partial x'^{j}}$ and ${\displaystyle \partial x'^{j}/\partial x^{i}}$ are non-zero. The determinant of the matrix formed by ${\displaystyle \partial x^{i}/\partial x'^{j}}$ is called the Jacobian ${\displaystyle J}$ of the transformation, which must be non-zero to provide a complete set of transformation laws.

Note that the matrices formed by all of the above partial derivative transformations can be generated as the inverse, transpose, and transpose of the inverse of the matrix formed by the components of ${\displaystyle \partial x^{i}/\partial x'^{j}}$. The key property of the tensor representation is the preservation of invariance in the sense that vector components which transform in a covariant manner (or contravariant manner) are paired with basis vectors that transform in a contravariant manner (or covariant manner), and these operations are inverse to one another according to the transformation rules. Substituting the transformation rules for the definition of ${\displaystyle \mathbf {A} }$ gives:

${\displaystyle \mathbf {A} =a^{i}\mathbf {e} _{i}=a'_{j}{\partial x'^{j} \over \partial x^{i}}\mathbf {e'} ^{j}{\partial x^{i} \over \partial x'^{j}}=a'_{j}\mathbf {e'} ^{j}}$

where the partial derivative terms cancel one another since they must be inverse to one another. This illustrates what is meant by invariance. A similar relation holds for all vectors (or higher-order tensors), allowing them to be written in the manner described above. Using the transformation rules can also show that: ${\displaystyle \mathbf {e} ^{i}\cdot \mathbf {e} _{j}=\delta _{j}^{i}}$, where ${\displaystyle \delta _{j}^{i}}$ is 1 if ${\displaystyle i=j}$ and 0 otherwise.

Note that in this kind of system the basis vectors are not generally of unit length, nor are covariant basis vectors necessarily parallel to their contravariant basis vectors (if the coordinates are non-orthogonal).

The above figure illustrates how the contravariant and covariant representations would be plotted in terms of components on a 2D curvilinear non-orthogonal grid for a generic vector${\displaystyle \mathbf {A} }$. Note that the sum of either pair of vectors yields the same vector. Also note that the covariant basis vectors are parallel to their respective coordinate lines while the contravariant basis vectors are orthogonal to the directions of the other coordinate lines.

There are many other useful properties of the tensor representation. If we take the dot product of ${\displaystyle \mathbf {A} =a^{i}\mathbf {e} _{i}=a_{k}\mathbf {e} ^{k}}$ and ${\displaystyle \mathbf {e} _{j}}$ then we obtain:

${\displaystyle a_{j}=a^{i}g_{ij}=a^{i}(\mathbf {e} _{i}\cdot \mathbf {e} _{j})}$

where ${\displaystyle g_{ij}}$ is the covariant metric tensor. The dot product of ${\displaystyle \mathbf {A} =a^{k}\mathbf {e} _{k}=a_{j}\mathbf {e} ^{j}}$ and ${\displaystyle \mathbf {e} ^{i}}$ likewise gives:

${\displaystyle a^{i}=a_{j}g^{ij}=a_{j}(\mathbf {e} ^{i}\cdot \mathbf {e} ^{j})}$

where ${\displaystyle g^{ij}}$ is the contravariant metric tensor. This gives two useful results: 1) the covariant (or contravariant) components of a vector can be recovered by taking the dot product of that vector and the covariant (or contravariant) basis vectors, and 2) the covariant and contravariant components are related by the

, and that the above relations require that ${\displaystyle g^{ij}}$ and ${\displaystyle g_{ij}}$ are inverse to one another.

We note that the tensor representation is not restricted to vectors, but can be used on higher-order tensors where each covariant or contravariant component transforms individually according to the rules described above. For example, we could transform a so-called mixed tensor of the form:

${\displaystyle b_{j}^{i}={b'}_{l}^{k}{\partial x^{i} \over \partial x'^{k}}{\partial x'^{l} \over \partial x^{j}}}$

by successively applying the transformation rules to each index according to whether it is covariant (lowered) or contravariant (raised).

Dual basis

Given a basis ${\displaystyle e_{1},\dots ,e_{n}}$ of a vector space V, there is a unique dual basis ${\displaystyle e^{1},\dots ,e^{n}}$ of the dual space, which is determined by requiring

${\displaystyle \mathbf {e} ^{i}\cdot \mathbf {e} _{j}=\delta _{j}^{i}}$.

Euclidean R³

If e¹, e², e³ are contravariant

of R³ (not necessarily orthogonal nor of unit norm) then the covariant basis vectors of their reciprocal system are:

${\displaystyle \mathbf {e} _{1}={\frac {\mathbf {e} ^{2}\times \mathbf {e} ^{3}}{\mathbf {e} ^{1}\cdot (\mathbf {e} ^{2}\times \mathbf {e} ^{3})}};\qquad \mathbf {e} _{2}={\frac {\mathbf {e} ^{3}\times \mathbf {e} ^{1}}{\mathbf {e} ^{2}\cdot (\mathbf {e} ^{3}\times \mathbf {e} ^{1})}};\qquad \mathbf {e} _{3}={\frac {\mathbf {e} ^{1}\times \mathbf {e} ^{2}}{\mathbf {e} ^{3}\cdot (\mathbf {e} ^{1}\times \mathbf {e} ^{2})}}.}$

Note that even if the e_i and eⁱ are not orthonormal, they are still by this definition mutually orthonormal:

${\displaystyle \mathbf {e} ^{i}\cdot \mathbf {e} _{j}=\delta _{j}^{i}.}$

Then the contravariant coordinates of any vector v can be obtained by the dot product of v with the contravariant basis vectors:

${\displaystyle q^{1}=\mathbf {v} \cdot \mathbf {e} ^{1};\qquad q^{2}=\mathbf {v} \cdot \mathbf {e} ^{2};\qquad q^{3}=\mathbf {v} \cdot \mathbf {e} ^{3}.}$

Likewise, the covariant components of v can be obtained from the dot product of v with covariant basis vectors, viz.

${\displaystyle q_{1}=\mathbf {v} \cdot \mathbf {e} _{1};\qquad q_{2}=\mathbf {v} \cdot \mathbf {e} _{2};\qquad q_{3}=\mathbf {v} \cdot \mathbf {e} _{3}.}$

Then v can be expressed in two (reciprocal) ways, viz.

${\displaystyle \mathbf {v} =q_{i}\mathbf {e} ^{i}=q_{1}\mathbf {e} ^{1}+q_{2}\mathbf {e} ^{2}+q_{3}\mathbf {e} ^{3}}$

or

${\displaystyle \mathbf {v} =q^{i}\mathbf {e} _{i}=q^{1}\mathbf {e} _{1}+q^{2}\mathbf {e} _{2}+q^{3}\mathbf {e} _{3}.}$

Combining the above relations, we have

${\displaystyle \mathbf {v} =(\mathbf {v} \cdot \mathbf {e} _{i})\mathbf {e} ^{i}=(\mathbf {v} \cdot \mathbf {e} ^{i})\mathbf {e} _{i}}$

and we can convert from covariant to contravariant basis with

${\displaystyle q_{i}=\mathbf {v} \cdot \mathbf {e} _{i}=(q^{j}\mathbf {e} _{j})\cdot \mathbf {e} _{i}=(\mathbf {e} _{j}\cdot \mathbf {e} _{i})q^{j}}$

and

${\displaystyle q^{i}=\mathbf {v} \cdot \mathbf {e} ^{i}=(q_{j}\mathbf {e} ^{j})\cdot \mathbf {e} ^{i}=(\mathbf {e} ^{j}\cdot \mathbf {e} ^{i})q_{j}.}$

The indices of covariant coordinates, vectors, and tensors are subscripts. If the contravariant basis vectors are

then they are equivalent to the covariant basis vectors, so there is no need to distinguish between the covariant and contravariant coordinates, and all indices are subscripts.

What 'contravariant' means

Contravariant is a

of the tensor.

Another method is used to derive covariant tensor components. When performing tensor transformations it is critical that the method used to map to the coordinate systems in use be tracked so that operations may be applied correctly for accurate, meaningful results.

In two dimensions, for an oblique rectilinear coordinate system, contravariant coordinates of a directed line segment (in two dimensions this is termed a vector) can be established by placing the origin of the coordinate axis at the tail of the vector. Parallel lines are placed through the head of the vector. The intersection of the line parallel to the x¹ axis with the x² axis provides the x² coordinate. Similarly, the intersection of the line parallel to the x² axis with the x¹ axis provides the x¹ coordinate.

By definition, the oblique, rectilinear, contravariant coordinates of the point P above are summarized as: xⁱ = (x¹, x²)

Notice the superscript; this is a standard nomenclature convention for contravariant tensor components and should not be confused with the subscript, which is used to designate covariant tensor components.

Is there a fundamental difference in the way contravariant and covariant components can be used, or could one simply interchange them everywhere? The answer is that in curved spaces, or in curved coordinate systems in flat space (e.g.

that can be immediately integrated to yield xⁱ, whilst the covariant components of the same differential, dx_i are not in general perfect differentials; the integrated change depends on the path. In the example of cylindrical coordinates, the radial and z components are the same in covariant and contravariant form, but the covariant component of the differential of angle round the z axis is r²dθ and its integral depends on the path.

Using the definition above, the contravariant components of a position vector vⁱ, where i = {1, 2}, can be defined as the differences between coordinates (or position vectors) of the head and tail, on the same coordinate axis. Stated in another way, the vector components are the projection onto an axis from the direction parallel to the other axis.

So, since we have placed our origin at the tail of the vector,

vⁱ = ( (x¹ − 0), (x² − 0 ) )

vⁱ = (x¹, x²)

This result is generalized into n-dimensions. Contravariance is a fundamental concept or property within tensor theory and applies to tensors of all ranks over all manifolds. Since whether tensor components are contravariant or covariant, how they are mixed, and the order of operations all impact the results it is imperative to track for correct application of methods.

In more modern terms, the transformation properties of the covariant indices of a tensor are given by a pullback; by contrast, the transformation of the contravariant indices is given by a pushforward (differential).

Use in tensor analysis

In

, a covariant vector varies more or less reciprocally to a corresponding contravariant vector. Expressions for lengths, areas and volumes of objects in the vector space can then be given in terms of tensors with covariant and contravariant indices. Under simple expansions and contractions of the coordinates, the reciprocity is exact; under affine transformations the components of a vector intermingle on going between covariant and contravariant expression.

On a

with the metric tensor. Contravariant indices can be gotten by contracting with the (matrix) inverse of the metric tensor. Note that in general, no such relation exists in spaces not endowed with a metric tensor. Furthermore, from a more abstract standpoint, a tensor is simply "there" and its components of either kind are only calculational artifacts whose values depend on the chosen coordinates.

The explanation in geometric terms is that a general tensor will have contravariant indices as well as covariant indices, because it has parts that live in the tangent bundle as well as the cotangent bundle.

A contravariant vector is one which transforms like ${\displaystyle {\frac {dx^{\mu }}{d\tau }}}$, where ${\displaystyle x^{\mu }\!}$ are the coordinates of a particle at its proper time ${\displaystyle \tau \!}$. A covariant vector is one which transforms like ${\displaystyle {\frac {\partial \phi }{\partial x^{\mu }}}}$, where ${\displaystyle \phi \!}$ is a scalar field.

Algebra and geometry

In

.

In

, while this does not make the same sense for a field of tangent vectors.

Covariant and contravariant components transform in different ways under coordinate transformations. By considering a coordinate transformation on a manifold as a map from the manifold to itself, the transformation of covariant indices of a tensor are given by a pullback, and the transformation properties of the contravariant indices is given by a pushforward.

See also

• covariant transformation

External links

• Weisstein, Eric W. "Covariant Tensor". MathWorld.