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
v
=
v
i
e
i
{\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
e
i
(
v
)
=
v
i
.
{\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.
I have been given the following problem:
The covariant vector field is:
\(v_{i}\) = \begin{matrix} x+y\\ x-y\end{matrix}What are the components for this vector field at (4,1)?
\(v_{i}\) = \begin{matrix} 5\\ 3\end{matrix}
Now I can use this information to solve the...
Hi everyone,
I am having a little trouble with the difference between a covariant vector and contravariant vector. The examples that I come across say that an example of a contravariant vector is velocity and that a contravariant must contra-vary with a change of basis to compensate.
So...
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...
Homework Statement
B is a third order tensor. Show that B^{ij}_{i} is a contravariant vector.
The Attempt at a Solution
Well... I just thought about a simple solution but I don't think I'm right. But anyways.
Considering B^{ij}_{i}. If I raise the index i: g^{ij}B^{ij}_{i} = B^{ijj}
And...
Can the covariant components of a vector, v, be thought of as v multiplied by a matrix of linearly independent vectors that span the vector space, and the contravariant components of the same vector, v, the vector v multiplied by the *inverse* of that same matrix?
thinking about it like that...
I'm slightly confused by the difference between covariant and contravariant 4-vectors and how they transform under Lorentz boosts. I'm aware that x_{\mu} = (-x^0 ,x^1, x^2, x^3) = (x_0 ,x_1, x_2, x_3), but when I do a Lorentz transform of the covariant vector, it seems to transform exactly like...
Hi
I am trying to learn about covariant and contravariant vectors and derivatives. The videos I have been watching talk about displacement vector as the basis for contravariant vectors and gradient as the basis for covariant vectors. Can somone tlel me the difference between displacemement...
Homework Statement
This is me doing some independent study on Tensors because I eventually hope to understand General Relativity.
My question is about the following equation which describe hoe the components of a displacement vector transform when there is a change in the coordinate system...
In their article [Integrals in the theory of electron correlations, Annalen der Physik 7, 71] L.Onsager at el. write:
By resolving the vector \vec{s} into its contravariant components in the oblique coordinate system formed by the vectors \vec{q} and \vec{Q} it is possible to reduce the region...
Hello
i know how to derive the components of acceleration in other coordinates like spherical
start here :
http://up.iranblog.com/images/0mbwuclckbu51bxt8jfa.jpg
and at last we have :
http://up.iranblog.com/images/geotowiaxdya2s6ewxk.jpg
also , i know that acceleration is a contravariant...
Homework Statement
Show that \frac{{\partial}g^c{^d}}{{\partial}g_a{_b}}=-\frac{1}{2}(g^a{^c}g^b{^d}+g^b{^c}g^a{^d})
Homework Equations
The Attempt at a Solution
It seems like it should be simple, but I just do not see how to come up with the above solution. This is what I am coming...
I am aware that the following operation:
mathbf{M}_{ij} \delta_{ij}
produces
mathbf{M}_{ii} or mathbf{M}_jj
However, if we have the following operation:
mathbf{M}_{ij} \delta^i{}_j
will the tensor M be transformed at all?
Thank you for your time.
I'm pretty comfortable with special relativity, and at least familiar with the principles of the general theory, but recently I've tried to learn SR using tensors. It is my first foray into this branch of mathematics. I understand they're handy because they represent invariant objects, but the...
Hey everyone, I am reading a Schaum's Outline on Tensor Calculus and came to something I can't seem to understand. I'm admittedly young to be reading this but so far I've understood everything except this. My question is: what is the difference between a contravariant tensor and a covariant...
I realize this is a "simple" mathematical exercise, in theory, but I'm having a lot of trouble finding some algorithmic way to do it. The problem is this: I want to expand the contravariant metric tensor components g^{\mu\nu} in terms of the covariant metric tensor g_{\mu\nu}. The first order...
Dear friends,
while reading about schwarzschild geometry, I learned that E=-p_0 and L=p_{\phi} are constant along a geodesic or are constant of motion. I further read that p^0=g^{00}p_0=m(1-2M/r)^{-1}E and p^{\phi}=g^{\phi\phi}p_{\phi}=m(1/r^2)L, which I can see depends on radius r. This made...
Homework Statement
Consider the two-dimensional space given by
ds^2 = e^y dx^2 + e^x dy^2
Calculate the covariant and contravariant components of the metric tensor for this
spacetime.
The Attempt at a Solution
Are the covariant components just e^y and e^x with the...
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 /...
The title says it all, basically I'm trying to figure out what the difference is between the two tensors (levi-civita) that are 3rd rank. Do they expand out in matrix form differently?
Hi,
How can we represent covariant and contravariant vectors on curved spacetime diagrams?
How can we draw these vectors on a spacetime diagram?
Contravariant vectors are really vectors,
therefore we can represent them on the diagram with directed line elements.
Covariant vectors are...
My understanding of differential forms is that, given the map
\phi : V \to V^{*}
If f \in V^{*} is a 0-form (a real-valued function on V^{*} ), then \phi defines the function \phi^{*} f on V
where \phi^{*}: F^{0}(V^{*}) \to F^{0}(V)
as the function such that...
we have studied in Tensor's analysis that there are two kinds of tensors that usually used in transformation. one is Contravariant & covariant. what is the difference between them and and why they are same for Rectangular coordinates?
Homework Statement
I have a few introductory problems dealing with proofs of tensor properties, and one about a transformation from rectangular to spherical coordinates. If someone has the time and inclination to help out this week, I can email you the specific problem set. (I'd prefer not...
I often meet the question whether the (physical) space is 'covariant' or 'contravariant'.
I once replied to that question with: Space is space. The COMPONENTS of a tensor are covariant/contravariant if the basis is CHOSEN TO BE contravariant/covariant. As far as I know tensors the...
Hi, everyone
I was playing with the coordinate transformations and metric tensors to get a feeling of how it all behaves, and got stuck with some basic problem I am hoping you can help me with.
So, I have defined a coordinate system (s,t), with the s axis going along the x-axis in the...
Are potentials appearing in the Maxwell equations the components of a contravariant vector or a covariant vector?
Let us be specific. metric is (+,-,-,-) . Let us write the potentials which appear in the Maxwell equations as \Phi and \vec{A}=(A_x,A_y,A_z)
Is it then the case that...
I'm having some trouble breaking into tensors. What is specifically bewildering me is contravariant vs. covariant indices. Could someone please explain this to me (or link me)? I barely understand what each one means in its own right, let alone the differences between the two.
Note: The derivatives are partial.
I've seen the coordinate transformation equation for contravariant vectors given as follows,
V'a=(dX'a/dXb)Vb
What I don't get is the need for two indices a and b. Wouldn't it be adequate to just write the equation as follows?
V'a=(dX'a/dXa)Va...
...of the four-momentum vector.
Why is the energy of a particle identified with p0 instead of p0? Is there a theoretical basis for this, or was it simply observed that p0 is conserved in a larger set of circumstances?
Homework Statement
Let e_{i} with i=1,2 be an orthonormal basis in two-dimensional Euclidean space ie. the metric is g_{ij} = \delta _{ij}. In the this basis the vector v has contravariant components v^{i} = (1,2). Consider the new basis
e_{1}^{'} = 5e_{1} - 2e_{2}
e_{2}^{'} = 3e_{1} - e_{2}...
Hi everyone;
I'm new to both PF and GR, so please bear with me if I'm not being very clear, or using standard syntax and such. Here is my question.
Given a vector v^a, the covariant derivative is defined as v^a _;b = v^a _,b + v^c GAMMA^a _bc.
(here I'm using ^ before upper indices and _...
Homework Statement
Can someone explain/help me prove the formulas
\vec{e_a}' = \frac{ \partial{x^b}}{\partial x'^a} \vec{e_b}
\vec{e^a}' = \frac{ \partial{x'^a}}{\partial x^b} \vec{e^b}
I do not understand why the partial derivative flip?
Homework Equations
The Attempt at a Solution
Dear Fellows,
Do anyone have an idea of whether there must be a system tensor in order to be able to transform from the covariant form of a certain tensor to its contravariant one?
This is a bit important to get rigid basics about tensors.
Schwartz Vandslire...
Does anyone know the physical (or historical) basis for the terms covariant and contravariant?
I'm guessing a particular class of mapping always tranforms components (of ..?) in exactly two different ways, so I'm wondering what the mappings are (Change of coordinate charts? Lorentz...
I read some books and see that the definition of covariant tensor and contravariant tensor.
Covariant tensor(rank 2)
A'_ab=(&x_u/&x'_a)(&x_v/&x'_b)A_uv
Where A_uv=(&x_u/&x_p)(&x_u/&x_p)
Where p is a scalar
Contravariant tensor(rank 2)
A'^uv=(&x'^u/&x^a)(&x'^v/&x^b)A^ab
Where A^ab=dx_a...
So much has been talking about covariant derivative. Anyone knows about contravariant derivative? What is the precise definition and would that give rise to different \Gamma^{k}_{i,j} and other concepts? :rolleyes:
Since there are some equations in my question. I write my question in the following attachment. It is about the covariant derivative of a contravariant vector.
Thank you so much!
I'm starting to learn differential geometetry on my own, but I'm having a little trouble figuring out the difference between covariant and contravariant vector fields. It seems that contravariant fields are just the normal vector fields they introduced in multivariable calculus, but if so, I...
I'm going to be completely unambiguous on this: the problem I am about to ask is an assigned homework problem so please, do NOT simply just reply with the answer. I have no intention to cheat.
That said, the question I have is with regards to problem 12.55 in Griffith's Intro to...
I have a little question. I hope someone can help me.
When we learn the theory of relativity and its formalism, we'll meet concepts : covariant and contravariant, such as covariant vector, covariant tensor...
I wonder that why we need to use the concepts ? What are advantages of them ?
I...