Explain how the components of a general covariant tensor may be converted into those of the equivalent contravariant tensor, and vice versa. In this context the covariant contravariant distinction is unimportant, because the way that the tensors transform under a change of basis just follows from the usual rules of quantum mechanics. Consider the expression j i a i x where j is free index. For a given nonorthogonal basis a, b, c, we can always construct an. 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. Covariant tensor article about covariant tensor by the free. Covariance and contravariance are collectively referred to as variance. Kolkers answer gives the gory detail, but heres a quick and dirty version. The second is a brief introduction to coordinate free geometry, which necessitates a discussion of contravariant and covariant vectors. But one day i decided to learn it, and finally, free my soul. Covariance and contravariance of vectors wikimili, the.
To put it succinctly, tensors are geometrical objects over vector spaces, whose coordinates obey certain laws of transformation under change of basis. When one coordinate system is just a rotation of the other, this distinction is. Covariance and contravariance of vectors wikimili, the free. We may play this game in the euclidean space en with its \dot inner product. Contravariance, covariance, densities, and all that 3 so to speak. Tensors of rank greater than two are easily defined. Introduction to tensor calculus for general relativity mit. Covariance and contravariance are properties of a mapping function that associates one member of a set with another.
Here i describe it in an easy way and hope to help other people to free their soul too. Covariant and contravariant rank2 tensors physics forums. Apr 10, 2018 first, lets start with some introductory notes. A more general riemann space may have a riemann connection rather than a metric tensor but my understanding is that we do no talk about covariant and contravariant vectors and tensors in such a space. Expressing contravariant basis vectors in terms of position vector. In the lexicon of category theory, covariance and contravariance are properties of functors. In ndimensional space a tensor of rank n has nn components.
The gradient g is an example of a covariant tensor, and the differential position d dx is an example of a contravariant tensor. Covariance and contravariance of vectors wikipedia. So, so if we have two tensors, metric tensor and inverse metric tensor, to every contravariant vector, with the use of the metric tensor we can define corresponding covariant vector. The contravariant component of any arbitrary vector a is ai with superscript index and covariant component is a i with subscript index are taken to be understood. These two topics will be then combined to write down the formalism for special relativity. In typescript, argument types are bivariant both covariant and contravariant, which is unsound although now in typescript 2. General relativitycontravariant and covariant indices. One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor. If i have covariant, but multiplying by this, i obtain contravariant vector. Now, going back to our laymens picture of a vector, namely an arrow pointing from some defined origin, we begin to tackle your question. Covariant and contravariant were always one of those complicated concepts that i was running away from. Covariant derivative 22 mar 2012riemannchristoffel curvature tensor.
More compactly, the q are the components of a tensor of type 2. Jul 20, 2017 tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. A gentle introduction to tensors washington university in. Without the use of tensors, a vector is an object that lives in a space with particular structures imposed upon it addition and scalar multiplication. A tensor with only covariant indices explanation of covariant tensor. Then the elements of the fold tensor product are called contravariant tensors or contravariant tensors of valency or order. A generic type parameter that is not marked covariant or contravariant is referred to as invariant. Contravariant and covariant as transforms there is a lot more behind the concepts of contravariant and covariant tensors of any rank than the fact that their basis vectors are mutually orthogonal to each other. Tensors this will be a brief summary of what we have already covered as it applies to tensors, plus a little about. A more general riemann space may have a riemann connection rather than a metric tensor but my understanding is that we do no talk about covariant and contravariant. P is often called the covariant component to distin. An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0. Vectors, metric and the connection 1 contravariant and.
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. It is an idea that seems to originate in physics but in fact it is a more a mathematical concept. Covariance and contravariance in generics microsoft docs. Tensors this will be a brief summary of what we have already covered as it applies. The metric tensor is g ij and is most easily understood when represented by a square matrix.
In this video, i describe the meaning of contravariant and covariant vector components. Covariant tensor article about covariant tensor by the. When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. Covariance and contravariance computer science, whether a type system preserves the ordering.
In rowvector notation, the basis vectors themselves are just i ex 1,0,0 j ey 0,1,0 k ez 0,0,1 1. Since you asked for an intuitive way to understand covariance and contravariance, i think this will do. For other uses of covariant or contravariant, see covariance and contravariance 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 from one coordinate system to another. Mar 23, 2019 in this video, i describe the meaning of contravariant and covariant vector components.
The question is what is the difference between covariance and contravariance. As such, it is a work in progress and many sections will be marked with the. Explain the difference between a covariant tensor and a contravariant tensor, using the metric tensor as an example. Covariance and contravariance of vectors semantic scholar. Besides, there is an obstacle related to definition of observable components of covariant tensors in which the indices are subscripts and of mixed tensors, which have both subscripts and superscripts. Quick question about contravariant and covariant tensors. What demonstrates that the vector is coordinate invariant or basis independent is the fact that covariant and contravariant objects must transform in opposite ways. Tensors can, in general, have rank greater than 2, and. The second is a brief introduction to coordinatefree geometry, which neces sitates a discussion of contravariant and covariant vectors. Then the prototypical example of a contravariant vector is a. But that would be contravariance but again, arrays are not contravariant, they support covariance only.
This video is a nice explanation for people studying relativity, but in general i think that making a big deal about how vectors and tensors. This is almost completely unintelligible, since the space which contains the. Tensors, contravariant and covariant components of a. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. We wish to write the velocity vector at some time t 0. Vectors are simple and wellknown examples of tensors, but there is much more to tensor theory than vectors. Inasmuch as these notes may help others, the reader is free to use, distribute, and modify. Chapter 10 vectors and tensors georgia institute of.
Covariance and contravariance of functors, in category theory. Net framework 4, variant type parameters are restricted to generic interface and generic. A brief summary of facts about variance in the common language runtime. What lets me transform between different coordinate systems is some matrix we typically refer to it as the jacobian, for the general case. Tensors covariant differential and riemann tensor coursera. In cartesian space they must be an orthogonal norm preserving matrix. Jan 04, 2007 if by system tensor you mean metric tensor, then yes, in order to have covariant and contravariant vectors and tensors, you must have a metric tensor such that a i g ij a j. Here are the examples from msdn that i hope will show you what contravariance really means i own these documents now, so if you think something is unclear in the docs, feel free to give me feedback. Tensors, contravariant and covariant components of a vector. Free and dummy indices we have written the components of aas ai. Vectors, contravariant and covariant penn engineering. Tensors are objects in multilinear algebra that can have aspects of both covariance and contravariance. Weinberg 1972, we will introduce geometrical objects in a coordinatefree manner, only.
Covariance and contravariance of vectors 3 for some invertible n. The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern differential geometry uses more sophisticated index free methods to represent tensors. Covariance and contravariance of vectors, in mathematics and theoretical physics. Covariant derivative do the link between 2 expressions. 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. Now that we have talked about tensors, we need to figure out how to classify them. A di erent metric will, in general, identify an f 2v with a completely di erent ef 2v. What are synonyms for covariance and contravariance. Eiffel also got this wrong, making argument types covariant instead of contravariant. For example, for a tensor of contravariant rank 2 and covariant rank 1. For this reason contravariant tensors are sometimes directly defined as multilinear functionals on. What are the physical differences between contravariant and.
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. In coordinatefree language, this means that an orientation on m can be specified by. Id have preferred to answer through a comment, but im not yet allowed to do so. In the pages that follow, we shall see that a tensor may be designated as contravariant, covariant, or mixed, and that the velocity expressed in equation 2 is in its contravariant. 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. 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 the f basis as 2. Vector is a physical quantity and it does not depend on any coordinate system. The difference between these two kinds of tensors is how they transform under a continuous change of coordinates. Ill have a crack at answering some of your questions. 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. Synonyms for covariance and contravariance in free thesaurus. This collection of notes attempts to demystify some of the vagaries of tensor. Introduction to tensors contravariant and covariant vectors rotation in 2space.
The dimension of contravariant vector is the inverse of the covariant vector and hence we expect the behaviour of contravariant vector and covariant vector under coordinate. Why are the contravariant and covariant basis perpendicular. Although the assertion covariance and contravariance refer to how coordinates change under a change of basis seems about right, the next few sentences try to say roughly that vectors in v have contravariant components and vectors in v have covariant components. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc. As mentioned in a previous video, tensors are invariant under coordinate transformations. What is contravariant and covariant vector, tensor in. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. Covariance and contravariance synonyms, covariance and. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. More generally, let be a commutative ring with unit element and a unitary module over. The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation.
The vector space or linear space, mve4 space, or just space of all kcontravariant, covariant tensors tensors of valence k at the point p in a manifold m will be denoted tk mp, with tmp and t. The terms contravariant and covariant apply based on how they are derived. Requirements 1 the derivative of a tensor must be a tensor. Differenceconvert between covariantcontravariant tensors.
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