What is Covariant vectors: Definition and 15 Discussions

In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis.
In physics, a basis is sometimes thought of as a set of reference axes. A change of scale on the reference axes corresponds to a change of units in the problem. For instance, by changing scale from meters to centimeters (that is, dividing the scale of the reference axes by 100), the components of a measured velocity vector are multiplied by 100. Vectors exhibit this behavior of changing scale inversely to changes in scale to the reference axes and consequently are called contravariant. As a result, vectors often have units of distance or distance with other units (as, for example, velocity has units of distance divided by time).
In contrast, covectors (also called dual vectors) typically have units of the inverse of distance or the inverse of distance with other units. An example of a covector is the gradient, which has units of a spatial derivative, or distance−1. The components of covectors change in the same way as changes to scale of the reference axes and consequently are called covariant.
A third concept related to covariance and contravariance is invariance. An example of a physical observable that does not change with a change of scale on the reference axes is the mass of a particle, which has units of mass (that is, no units of distance). The single, scalar value of mass is independent of changes to the scale of the reference axes and consequently is called invariant.
Under more general changes in basis:

A contravariant vector or tangent vector (often abbreviated simply as vector, such as a direction vector or velocity vector) has components that contra-vary with a change of basis to compensate. That is, the matrix that transforms the vector components must be the inverse of the matrix that transforms the basis vectors. The components of vectors (as opposed to those of covectors) are said to be contravariant. Examples of vectors with contravariant components include the position of an object relative to an observer, or any derivative of position with respect to time, including velocity, acceleration, and jerk. In Einstein notation, contravariant components are denoted with upper indices as in







{\displaystyle \mathbf {v} =v^{i}\mathbf {e} _{i}}
(note: implicit summation over index "i")
A covariant vector or cotangent vector (often abbreviated as covector) has components that co-vary with a change of basis. That is, the components must be transformed by the same matrix as the change of basis matrix. The components of covectors (as opposed to those of vectors) are said to be covariant. Examples of covariant vectors generally appear when taking a gradient of a function. In Einstein notation, covariant components are denoted with lower indices as in









{\displaystyle \mathbf {e} _{i}(\mathbf {v} )=v_{i}.}
Curvilinear coordinate systems, such as cylindrical or spherical coordinates, are often used in physical and geometric problems. Associated with any coordinate system is a natural choice of coordinate basis for vectors based at each point of the space, and covariance and contravariance are particularly important for understanding how the coordinate description of a vector changes by passing from one coordinate system to another.
The terms covariant and contravariant were introduced by James Joseph Sylvester in 1851 in the context of associated algebraic forms theory. Tensors are objects in multilinear algebra that can have aspects of both covariance and contravariance.
In the lexicon of category theory, covariance and contravariance are properties of functors; unfortunately, it is the lower-index objects (covectors) that generically have pullbacks, which are contravariant, while the upper-index objects (vectors) instead have pushforwards, which are covariant. This terminological conflict may be avoided by calling contravariant functors "cofunctors"—in accord with the "covector" terminology, and continuing the tradition of treating vectors as the concept and covectors as the coconcept.

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  1. D

    I Transforming Contra & Covariant Vectors

    Hi. The book I am using gives the following equations for the the Lorentz transformations of contravariant and covariant vectors x/μ = Λμν xν ( 1 ) xμ/ = Λμν xv ( 2 ) where the 2 Lorentz transformation matrices are the inverses of each other. I am trying to get equation 2...
  2. saadhusayn

    I Transformation of covariant vector components

    Riley Hobson and Bence define covariant and contravariant bases in the following fashion for a position vector $$\textbf{r}(u_1, u_2, u_3)$$: $$\textbf{e}_i = \frac{\partial \textbf{r}}{\partial u^{i}} $$ And $$ \textbf{e}^i = \nabla u^{i} $$ In the primed...
  3. Dyatlov

    I Worked example on a covariant vector transformation

    Hello. I would like to check my understanding of how you transform the covariant coordinates of a vector between two bases. I worked a simple example in the attached word document. Let me know what you think.
  4. G

    Different formulations of the covariant EM Lagrangian

    Homework Statement I'm reading through A. Zee's "Quantum Field Theory in a nutshell" for personal learning and am a bit confused about a passage he goes through when discussing field theory for the electromagnetic field. I am well versed in non relativistic quantum mechanics but have no...
  5. M

    I Trying to understand covariant tensor

    I am taking a course on GR and trying to understand Tensor calculus. I think I understand contravariant tensor (transformation of objects such as a vector from one frame to another) but I am having a hard time with covariant tensors. I looked into the Wikipedia page...
  6. M

    Covariant and contravariant basis vectors /Euclidean space

    I want ask another basic question related to this paper - http://www.tandfonline.com/doi/pdf/10.1080/16742834.2011.11446922 If I have basis vectors for a curvilinear coordinate system(Euclidean space) that are completely orthogonal to each other(basis vectors will change from point to point)...
  7. S

    Contravariant and covariant vectors

    I know if the number of coordinates are same in both the old and new frame then A.B=A`.B` . But if the number of coordinates are not same in both old and new frame then A.B=0 means that both the vectors A and B are perpendicular. Why is it so that if the number of coordinates of both the frames...
  8. putongren

    Change of Basis, Covariant Vectors, and Contravariant Vector

    I'm having trouble understanding those concepts in the title. Can someone explain those concepts in an easy to understand manner? Please don't refer me to a wikipedia page. I know some linear algebra and multi-variable calculus. Thank you.
  9. N

    Derivatives of contravariant and covariant vectors

    Can someone explain why the derivative with respect to a contravariant coordinate transforms as a covariant 4-vector and the derivative with respect to a covariant coordinate transforms as a contravariant 4-vector.
  10. K

    Contravariant and Covariant Vectors

    I remember I have read somewhere that contravariant/covariant vectors correspond to polar/axial vectors in physics, respectively. Examples for polar/axial vectors are position, velocity,... and angular momentum, torque,..., respectively. Is this right? Can I prove that, say, any axial...
  11. I

    Physical intrepretation of contra-variant and covariant vectors?

    Hey all, I starting to study QED along with a slew with other materials. (I read in the QED book and when I don't understand a reference I go to Jackson's E&M and work some problems out, it has been beneficial thus far!) Most of the topics are not too far fetched but I am struggling to...
  12. T

    Requesting Clear Description of Contravariant vs Covariant vectors

    Ok, so here's my problem. I just graduated with a mathematics degree and am going full force into a physics graduate program. I'm taking a course called mathematical methods for physicists, in which the first subject is tensors. Everyone else seems to be comfortable with the material, but me...
  13. G

    Contravariant and covariant vectors transformations

    Hi all, I am new to General Relativity and I started with General Relativity Course on Youtube posted by Stanford (Leonard Susskind's lectures on GR). So first thing to understand is transformation of covariant and contravariant vectors. Before I can understand a transformation, I would...
  14. T

    General Relativity - Contravariant and Covariant Vectors: aaargghh

    Homework Statement I know this is an easy question, I just can't seem to grasp what I am actually doing: Let M be a manifold. Let Va be contravariant, and Wa be covariant. Show that \mu=VaWa Homework Equations (couldn't get Latex to work consistently, sorry) (1) V 'a = (dx 'a /...
  15. F

    Covariant vectors vs reciprocal vectors

    If there is a contravariant vector v=aa+bb+cc with a reciprocal vector system where [abc]v=xb×c+ya×c+za×b would the vector expressed in the reciprocal vector system be a covariant vector? Is there any connection between the reciprocal vector system of a covariant vector and a...