Covariance and contravariance are terms that refer to the ability to use a more derived type more specific or a less derived type less specific than originally specified. If w is a covariant vector, and v a contravariant vector, then vw is defined as wv. Chapter 10 vectors and tensors in this chapter we explain how a vector space v gives rise to a family of associated tensor spaces, and how mathematical objects such as linear maps or quadratic forms should be understood as being elements of these spaces. Hence acceleration is also a contravariant tensor of rank two. Vectors and tensors in curved space time asaf peer1 may 20, 2015 this part of the course is based on refs. When we derived the condition for a contravariant tensor, we considered a one dimensional curve defined within the manifold by using a single.
This material is covered thoroughly in chapters one and seven in our textbook classical electrodynamics, 2nd ed. Tensors can, in general, have rank greater than 2, and. Sep 14, 2016 variance is a topic that comes up fairly often in type systems and can be a bit confusing the first time you hear it. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. Covariant and contravariant tensors physics forums. Intuitive way to understand covariance and contravariance in. It is an idea that seems to originate in physics but in fact it is a more a mathematical concept.
I understand that action is contravariant and is probably declared as such internal delegate void action t t. Contravariant is a mathematical term with a precise definition in tensor analysis. Introduction using the equivalence principle, we have studied the trajectories of free test particles. Requirements 1 the derivative of a tensor must be a tensor. Aalborg universitet tensor calculus with applications to. Intuitive way to understand covariance and contravariance. This means it is not necessary to transform the expression such as must be done for a tensor in the box on page iii1. The original aspect tensor effectively assumes the role of contravariant metric tensor in the transformed problem in which diffusivity is unity isotropically.
This paper presents tensorbased covariance matrices for object modeling and tracking. The same applies to covariant tensors of rank n or mixed tensor of rank n. It specifies precisely the method direction of projection used to derive the components by projecting the magnitude of the tensor quantity onto the coordinate system being used as the basis of the tensor. Grounding the kaluzaklein fifth dimension in observed. An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0. General relativitycontravariant and covariant indices. Tensorbased covariance matrices for object tracking. An informal discussion on tensor calculus chris tiee abstract. Fact is that covariant doesnt have a wiki description and only 33 questions are tagged with it whereas covariance h. For example, for a tensor of contravariant rank 2 and covariant rank 1. To state this strange behavior in the proper jargon, we allow function types to be covariant in their return type and contravariant in their argument type.
Covariance and contravariance of vectors wikipedia. In that spirit we begin our discussion of rank 1 tensors. For instance, a rank2 contravariant tensor has the transformation property. First we will explain the distinction between the covariant and contravariant components of vectors, thinking of vectorfields where a vector is defined at a point rather than as a. Understanding tensoroperations, covariance, contravariance, in the context of special relativity. Covariance and contravariance of vectors 9 contravariant expression. For other uses of covariant or contravariant, see covariance and contravariance disambiguation. Jul 20, 2017 tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. A polar vector in 3 dimensions 3d is an object which, given a coordinate frame, can be. Dec 08, 2014 this course will continue on patreon at textbook.
Explain the difference between a covariant tensor and a contravariant tensor, using the metric tensor as an example. Id really appreciate it if someone could try to explain the reasoning logic behind this. We are now ready to get serious about electrodynamics. Tensor which depends on two directions one normal to the area and other that of the force on it. I understand that, depending on how something transforms, its either covariant or contravariant and that that determines whether the index is written at the top or bottom. Vectors and covectors in special relativity jim napolitano march 12, 2010 these notes are meant to accompany the course electromagnetic theory for the spring 2010 term at rpi. Covariance of the dirac equation 3 which is called the feynmanslash.
Tensor notation tensor notation in three dimensions. Covariance and contravariance of vectors 5 then the frame f is related to the frame f by the inverse of the jacobian matrix of the coordinate transition. Contravariant and covariant components of a vector in a curvinilinear coordinate system. Covariance and contravariance computer science wikipedia. In cartesian space they must be an orthogonal norm preserving matrix. Covariance and contravariance of vectors semantic scholar. For orthonormal cartesian coordinate systems, the two variance types i. On the other hand, the energymomentum tensor of a scalar field is more naturally written as a covariant tensor 4. Understanding tensoroperations, covariance, contravariance. What are the physical differences between contravariant.
Explain how the components of a general covariant tensor may be converted into those of the equivalent contravariant tensor, and vice versa. In mathematics and theoretical physics, covariance and contravariance are concepts used in many areas, generalizing in a sense invariance, i. By a widely followed convention, covariant indices are written as lower indices, whereas contravariant indices are upper indices. Feb 17, 2010 vector is a physical quantity and it does not depend on any coordinate system. This collection of notes attempts to demystify some of the vagaries of tensor analysis, to emphasize the connections between the old way of doing things and the new. Differenceconvert between covariantcontravariant tensors. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. The differences of zircon titanium thermometry, iuhft value and the covariant relation of rbsr and ba among the two stages of magmatism show that tsona leucogranties originated from more than one melting reactions, they had experienced the change from dehydration melting to fluid fluxed melting, and the reason respond to this transformation may be the start of the ew extension in southern. Vectors, metric and the connection 1 contravariant and.
A contravariant tensor is a tensor having specific transformation properties c. In the process of contraction one contravariant index and one covariant index of a mixed tensor are set equal and the repeated index is summed over, the result is a tensor of rank lower by two than the original tensor. Covariance and contravariance are terms that are used in different ways in the theory of objectoriented programming and they sound advanced and difficult but in fact the idea they they encapsulate is very, very simple. Jun 15, 2011 similarly, contravariant vectors can be thought of as scalarvalued linear functions of one covariant vector each, with the following definition. Now that we have talked about tensors, we need to figure out how to classify them. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering. We have developed a beautiful, geometric system for describing the coordinates in terms of which electrodynamics must be formulated for the speed of light to be an invariant. Variance is a topic that comes up fairly often in type systems and can be a bit confusing the first time you hear it. Or, in indices, a tangent vector is by definition a vector that is a linear combination of the coordinate partials. Writing vector or tensor equations in generalized coordinate systems is a process familiar to students in classical mechanics. Then the prototypical example of a contravariant vector is a.
Abstract vector is a physical quantity and it does not depend on any coordinate system. The contravariant components are those which transform as follows e. One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor. Tensors can, in general, have rank greater than 2, and often do. The definition of contravariant tensor of rankis a simple generalization. The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern. The valence of a tensor is the number of variant and covariant terms, and in einstein notation, covariant components have lower indices, while contravariant components have upper indices. Browse other questions tagged specialrelativity metrictensor coordinatesystems tensorcalculus covariance or ask your own question. Contravariant components the components of a vector in two dimensions are defined in the literature in relation to a change of coordinates from x,y to x,y, say. He does a nice job in the video showing that a tensor can be thought of as an array of numbers, but how is this formal definition coming into play. Introduction to the essentials of tensor calculus 6 the invariant measure of volume is easily constructed as. Secondly, if a tensor is a ultimately a map to the reals, where is that evident in this video. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. Chapter 10 vectors and tensors georgia institute of.
One might presume in view of dirac theory that the fivedimensional g and g in the. Thus contravariant vectors often called simply vectors are the simplest kind of contravariant tensor. Covariant definition of covariant by the free dictionary. The mnemonic for producer extends, consumer super pecs, from the book effective java by joshua bloch gives an easy way to remember when to use covariance and contravariance.
This is something im having a hard time wrapping my head around. Introduction to tensors contravariant and covariant vectors rotation in 2space. Id have preferred to answer through a comment, but im not yet allowed to do so. The second is a brief introduction to coordinatefree geometry, which necessitates a discussion of contravariant and covariant vectors. An interface or delegate type can have both covariant and contravariant type parameters.
We have developed a group of coordinate transformations that preserves. Similarly, contravariant vectors can be thought of as scalarvalued linear functions of one covariant vector each, with the following definition. When you see the feynman slash, you must recognize that it is a 4. Covariant and contravariant 4vector in special relativity. First of all, remember that the reason of having covariant or contravariant tensors is because you want to represent the same thing in a different coordinate system.
Spaces with contravariant and covariant affine connections and. Vectors, metric and the connection 1 contravariant and covariant vectors 1. Since you asked for an intuitive way to understand covariance and contravariance, i think this will do. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. In order to change a contravariant vector into a oneform or viseversa so that we can compute the inner product, we use the metric tensor as a lowering or raising operator.
Unlike the traditional vectorbased or matrixbased object representation, this method represents an object. Tensors, contravariant and covariant components of a. When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. In mathematical terms, they occur in a foundational way in linear algebra and multilinear algebra, differential geometry and other branches of geometry, category theory and. Twenty years ago, in an article titled \covariance and contravariance. Covariance and contravariance in generics microsoft docs.
Having in mind the analogy to coordinate differentials and derivative of a scalar, take a look at this picture, which i think will help to make it clearer. The expansion in orthonormal basis is mathematically simple. However, i dont understand why an action is covariant. Vector is a physical quantity and it does not depend on any coordinate system. On a manifold, a tensor field will typically have multiple indices, of two sorts. Tensors, contravariant and covariant components of a vector. Here, each vector y j of the f basis is a linear combination of the vectors x i of the f basis, so that contravariant transformation a vector v in v is expressed uniquely as a linear combination of the elements of. We present here a brief summary of tensor notation in three dimensions simply to refresh the memory of the reader and provide a smooth introduction to the relativistic tensor notation which follows. Note that in chapter ii it was shown that the last two expressions have the. In ndimensional space a tensor of rank n has nn components. To examine the transformation properties of a contravariant tensor, first consider a tensor of rank 1 a vector 1 for which 2 now let, then any set of quantities which transform according to 3.
But in many physical situations we have to choose an nonorthogonal basis or oblique coordinate system. Twenty years ago, in an article titled \ covariance and contravariance. I think covariant should be merged into covariance and be a tag synonym for it. In typescript, argument types are bivariant both covariant and contravariant, which is unsound although now in typescript 2. We wish to write the velocity vector at some time t 0. Covariant derivative 22 mar 2012riemannchristoffel curvature tensor. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc. Covariant and contravariant components of a vector in. Planetphysicscovariance and contravariance wikiversity. For what its worth i cleaned up a document i wrote a while ago which tries to give a slightly lesssucky overview of covariance and contravariance without getting tangled in the whole differential geometry of it. What links here related changes upload file special pages permanent link page information wikidata item cite this page. The algebraic operation by which the rank of a mixed tensor is lowered by 2 is known as contraction. Today, i read some articles about covariance, contravariance and invariance in java. I read the english and german wikipedia article, and some other blog posts and articles from ibm.
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