What is Contravariant: Definition and 94 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. yucheng

    Contravariant derivative of a tensor field in terms of generalized coordinates?

    1. The laplacian is defined such that $$ \vec{\nabla} \cdot \vec{\nabla} V = \nabla_i \nabla^i V = \frac{1}{\sqrt{Z}} \frac{\partial}{\partial Z^{i}} \left(\sqrt{Z} Z^{ij} \frac{\partial V}{\partial Z^{j}}\right)$$ (##Z## is the determinant of the metric tensor, ##Z_i## is a generalized...
  2. A

    I Deriving Contravariant Form of Levi-Civita Tensor

    The covariant form for the Levi-Civita is defined as ##\varepsilon_{i,j,k}:=\sqrt{g}\epsilon_{i,j,k}##. I want to show from this definition that it's contravariant form is given by ##\varepsilon^{i,j,k}=\frac{1}{\sqrt{g}}\epsilon^{i,j,k}##.My attemptWhat I have tried is to express this tensor...
  3. AdvaitDhingra

    B Regarding Contravarient Vector Transformations

    Hello, I have a question regarding the contravarient transformation of vectors. So the formula: V'n = dx'n / dxm Vm So in words, the nth basis vector in the ' frame of reference over the mth (where m is the summation term) basis vector in the original frame of reference times the mth...
  4. J

    I Is the Wavefunction a Contravariant Component?

    I've heard that the wavefunction as a function of x has units of square root of inverse distance, but I haven't heard an intuitive description of why this is aside from that the math works out when you integrate to get the probability. But aside from the math working out, I'm hoping to get a...
  5. W

    I Trouble understanding contravariant transformations for vectors

    Hey, so I've been studying some math on my own and I'm really confused by this one bit. I understand what contravariant components of a vector are, but I don't understand the ways in which they transform under a change of coordinate system. For instance, let's say we have two coordinate...
  6. E

    B A covariant vs contravariant vector?

    We have a basis {##\mathbf{e}_1##, ##\mathbf{e}_2##, ##\dots##} and the corresponding dual basis {##\mathbf{e}^1##, ##\mathbf{e}^2##, ##\dots##}. I learned that a vector ##\vec{V}## can be expressed in either basis, and the components in each basis are called the contravariant and covariant...
  7. S

    I Transformation of the contravariant and covariant components of a tensor

    I have read many GR books and many posts regarding the title of this post, but despite that, I still feel the need to clarify some things. Based on my understanding, the contravariant component of a vector transforms as, ##A'^\mu = [L]^\mu~ _\nu A^\nu## the covariant component of a vector...
  8. D

    I Covariant and contravariant tensors

    Is there a purpose of using covariant or contravariant tensors other than convenience or ease in a particular coordinate system? Is it possible to just use one and stick to one? Also what is the meaning of mixed components used in physics , is there a physical significance in choosing one over...
  9. mishima

    Write ∇u with covariant components and contravariant basis

    The first part I'm fairly sure is just the regular gradient in polar coordinates typically encountered: $$\nabla u= \hat {\mathbf e_r} \frac {\partial u} {\partial r} + \hat {\mathbf e_\theta} \frac 1 r \frac {\partial u} {\partial \theta}$$ or in terms of scale factors: $$=\sum \hat...
  10. A

    I Contravariant Vector Transformation in Spherical Polar Coordinates

    In a spherical polar coordinate system if the components of a vector given be (r,θ,φ)=1,2,3 respectively. Then the component of the vector along the x-direction of a cartesian coordinate system is $$rsinθcosφ$$. But from the transformation of contravariant vector...
  11. A

    I Contravariant derivative of tensor of rank 1

    If we have two sets of coordinates such that x1,x2...xn And y1,y2,...ym And if any yi=f(x1...,xn)(mutually dependent). Then dyi=(∂yi/∂xj)dxj Again dyi/dxk=(∂2yi/∂xk∂xj)dxj+∂yi/∂xk Is it the contravariant derivative of a vector?? Or in general dAi/dxk≠∂Ai/∂xk
  12. Prez Cannady

    A Einstein Field Equations: Covariant vs Contravariant

    Depending on the source, I'll often see EFE written as either covariantly: $$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8 \pi GT_{\mu\nu}$$ or contravariantly $$R^{\alpha\beta} - \frac{1}{2}Rg^{\alpha\beta} = 8 \pi GT^{\alpha\beta}$$ Physically, historically, and/or pragmatically, is there a...
  13. George Keeling

    Covariant coordinates don't co-vary

    Homework Statement I am studying co- and contra- variant vectors and I found the video at youtube.com/watch?v=8vBfTyBPu-4 very useful. It discusses the slanted coordinate system above where the X, Y axes are at an angle of α. One can get the components of v either by dropping perpendiculars...
  14. Gene Naden

    I Contravariant first index, covariant on second, Vice versa?

    I am working through a derivation of the Dirac matrix transformation properties. I have a tensor for the Lorentz transformation that is covariant on the first index and contravariant on the second index. For the derivation, I need vice versa, i.e. covariant on the second index and contravariant...
  15. T

    I Finding Covariant and Contravariant components of Cylin Coor

    I'm going through Introduction_to_Tensor_Calculus by Wiskundige_Ingenieurstechnieken. I want to find the covariant and contravariant components of the cylindrical cooridnates. Ingerniurstechnieken sets tangent vectors as basis (E1, E2, E3). He also sets the normal vectors as another basis (E1...
  16. JTC

    A Understanding the Dual Basis and Its Directions

    Please help. I do understand the representation of a vector as: vi∂xi I also understand the representation of a vector as: vidxi So far, so good. I do understand that when the basis transforms covariantly, the coordinates transform contravariantly, and v.v., etc. Then, I study this thing...
  17. F

    I Factors for contravariant components with transported vector

    I am currently coding a small application that reproduces the transport of a vector along a geodesic on a 2D sphere. Here's a capture of this application : You can see as pink vectors the vectors of curvilinear coordinates and in cyan the transported vector. The transport of vector along...
  18. S

    Contravariant Four-gradient ESN in Wikipedia appears wrong

    Homework Statement I am self studying relativity. In Wikipedia under the four-gradient section, the contravariant four-vector looks wrong from an Einstein summation notation point of view. https://en.wikipedia.org/wiki/Four-vector Homework Equations It states: E0∂0-E1∂1-E2∂2-E3∂3 = Eα∂α...
  19. BiGyElLoWhAt

    I Covariant derivative of a contravariant vector

    This is (should be) a simple question, but I'm lost on a negative sign. So you have ##D_m V_n = \partial_m V_n - \Gamma_{mn}^t V_t## with D_m the covariant derivative. When trying to deduce the rule for a contravariant vector, however, apparently you end up with a plus sign on the gamma, and I'm...
  20. S

    I Why are contravariant and covariant vectors important in general relativity?

    1) I read different texts on Contravariant , Covariant vectors. 2) Contravariant - they say is like vector . Covariant is like gradient From what I see they have those vector spaces because it eventually helps get scalar out of it if we multiply contravariant by covariant Also Contravariant...
  21. S

    Understand Contravariant Transformations b/w Coordinate Systems

    I am trying to make sure that I have a proper understanding of contravariant transformations between coordinate systems. The contravariant transformation formula is: Vj = (∂yj/∂xi) * Vi where Vj is in the y- frame of reference and Vi is in the x-frame of reference. Einstein summation...
  22. 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)...
  23. 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...
  24. joneall

    Units if conversion between covariant/contravariant tensors

    I am still at the stage of trying to assimilate contravariant and covariant tensors, so my question probably has a simpler answer than I realize. A covariant tensor is like a gradient, as its units increase when the coordinate units do. A contravariant tensor's components decrease when the...
  25. B

    Covariant & Contravariant Components

    Homework Statement This is really 3 questions in one but I figure it can be grouped together: 1. The vector A = i xy + j (2y-z2) + k xz. is in rectangular coordinates (bold i,j,k denote unit vectors). Transform the vector to spherical coordinates in the unit vector basis. 2. Transform the...
  26. 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.
  27. G

    A question about covariant representation of a vector

    Homework Statement Hi I am reviewing the following document on tensor: https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf Homework Equations In the middle of page 27, the author says: Now, using the covariant representation, the expression $$\vec V=\vec V^*$$...
  28. N

    Tensors with both covariant and contravariant components

    Hey all, I'm just starting into GR and learning about tensors. The idea of fully co/contravariant tensors makes sense to me, but I don't understand how a single tensor could have both covariant AND contravariant indices/components, since each component is represented by a number in each index...
  29. G

    Fundamental and contravariant representations

    The invariant of SL(2,C) is proven to be invariant under the action of the group by the following \epsilon'_{\alpha\beta} = N_{\alpha}^{\rho}N_{\beta}^{\sigma}\epsilon_{\rho\sigma}=\epsilon_{\alpha\beta}detN=\epsilon_{\alpha\beta} The existence of an invariant of this form (with two indices...
  30. L

    Difference Between Covariant & Contravariant Vectors Explained

    Can someone explain to me what is the difference between covariant and contravariant vectors ? Thank You
  31. T

    Why Do Covariant and Contravariant Vectors Use Opposite Indexing?

    I am reading a notes about tensor when I came across this which the notes did not elaborate more on it. As a result I don't quite understand why. Here it is : " Note that we mark the covariant basis vectors with an upper index and the contravariant basis vectors with a lower index. This may...
  32. putongren

    Covariant and Contravariant Tensors

    Not sure where to post this thread. That being said, can someone explain to me simply what covariant and contravariant tensors are and how covariant and contravariant transformation works? My understanding of it from googling these two mathematical concepts is that when you change the basis of...
  33. B

    Lorentz Transform on Covariant Vector (Lahiri QFT 1.5)

    Homework Statement Given that ##x_\mu x^\mu = y_\mu y^\mu## under a Lorentz transform (##x^\mu \rightarrow y^\mu##, ##x_\mu \rightarrow y_\mu##), and that ##x^\mu \rightarrow y^\mu = \Lambda^\mu{}_\nu x^\nu##, show that ##x_\mu \rightarrow y_\mu = \Lambda_\mu{}^\nu x_\nu##. Homework Equations...
  34. Math Amateur

    MHB The Contravariant Functor Hom_R( _ , X)

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 3.1 on categories and need help in understanding the contravariant functor \text{Hom}_R(\_, X) as described in Bland, Example 13 in Ch. 3: Categories (page 76). Example 13 in Ch. 3 reads as follows...
  35. N

    GR vs SR: Reconciling Contravariant & Covariant Vector Components

    I am trying to reconcile the definition of contravariant and covariant components of a vector between Special Relativity and General Relativity. In GR I understand the difference is defined by the way that the vector components transform under a change in coordinate systems. In SR it seems...
  36. electricspit

    Tensors: switching between mixed and contravariant components

    I'm working on the electromagnetic stress-energy tensor and I've found this in a book by Landau-Lifshitz: T^{i}_{k} = -\frac{1}{4\pi} \frac{\partial A_{\ell}}{\partial x^{i}} F^{k\ell}+\frac{1}{16\pi}\delta^{k}_{i} F_{\ell m} F^{\ell m} Becomes: T^{ik} = -\frac{1}{4\pi}...
  37. 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.
  38. A

    QFT (derivative the covariant and contravariant fields)

    Hi, please help me .. How can I derivative covariant and contravariant fields? as in the attached picture Thanks.. http://www.gulfup.com/?tNXcaN w.r.t alpha
  39. S

    Contravariant and covariant indices

    When we write contravariant and covariant indices, for example for the Lorentz transformation, does it matter if we write \Lambda^\mu\,_\nu or \Lambda^\mu_\nu ? i.e. if the \nu index is to the right of the \mu or they are at the same place with respect to left-right?
  40. 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...
  41. J

    Covariant and Contravariant Coordinate

    Hellow everybody! A simples question: is it correct the graphic representation for covariant (x₀, y₀) and contravariant (x⁰, y⁰) coordinates of black vector?
  42. E

    How Would Physics Change Without Covariant and Contravariant Tensors?

    If the notions of covariant and contravariant tensors were not introduced,what would happen?E.g. what form will the Einstein E.q. Guv=8πTuv be changed into ?
  43. L

    I understand energy-momentum tensor with contravariant indices, where

    I understand energy-momentum tensor with contravariant indices, where I think I get T^{αβ}, but how do I derive the same result for T_{αβ}? Why are the contravariant vectors simply changed to covariant ones, and why does it work in Einstein's equation?
  44. P

    Covariant Versus Contravariant Vectors

    I'm confused about the difference between a contravariant and covariant vector. Some books and articles seem to say that there really is no difference, that a vector is a vector, and can be written in terms of contravariant components associated with a particular basis, or can be written in...
  45. L

    Covariant and Contravariant Components of a Vector

    I think this may be a simple yes or no question. I am currently reading a book Vector and Tensor Analysis by Borisenko. In it he introduces a reciprocal basis \vec{e_{i}} (where i=1,2,3) for a basis \vec{e^{i}} (where i is an index, not an exponent) that may or may not be orthogonal...
  46. bcrowell

    Newtonian force as a covariant or contravariant quantity

    I recently came across a very cool book called Div, Grad, and Curl are Dead by Burke. This is apparently a bit of a cult classic among mathematicians, not to be confused with Div, Grad, Curl, and All That. Burke was killed in a car accident before he could put the book in final, publishable...
  47. S

    Covariant and contravariant vector

    Will anyone help me to under stand the covariant and contravariant vector ? And can anyone show me the derivation of dr=(dr/dx)dx + (dr/dy)dy + (dr/dz)dz
  48. T

    Covariant and Contravariant components in Oblique System

    Homework Statement In the oblique coordinate system K' defined in class the position vector r′ can be written as: r'=a\hat{e'}_{1}+b\hat{e'}_{2} Are a and b the covariant (perpendicular) or contravariant (parallel) components of r′? Why? Give an explanation based on vectors’ properties...
  49. 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...