For example, the tensor, a mixed tensor of rank 5, contravariant of order 3 and covariant of order 2, is defined by the following relation note the pattern of the indices in the successive fractions. Tensors, contravariant and covariant components of a vector. Covariance and contravariance of vectors 3 for some invertible n. Vectors are simple and wellknown examples of tensors, but there is much more to tensor theory than vectors. What are the physical differences between contravariant and. To put it succinctly, tensors are geometrical objects over vector spaces, whose coordinates obey certain laws of transformation under change of basis. 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 form. The difference between these two kinds of tensors is how they transform under a continuous change of coordinates. The following code demonstrates the difference between assignment compatibility, covariance, and contravariance. We may play this game in the euclidean space en with its \dot inner product. Mar 23, 2019 in this video, i describe the meaning of contravariant and covariant vector components.
Chapter 10 vectors and tensors georgia institute of. It is an idea that seems to originate in physics but in fact it is a more a mathematical concept. The metric tensor is g ij and is most easily understood when represented by a square matrix. 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.
As mentioned in a previous video, tensors are invariant. Covariance and contravariance fabulous adventures in coding. Covariance and contravariance are collectively referred to as variance. This is the main purpose of this article to explore the notion of contravariant and covariant vectors in di.
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. We see that a general tensor f, contravariant of degree m. Tensors are objects in multilinear algebra that can have aspects of both covariance and contravariance. Covariance and contravariance of vectors wikipedia. But in many physical situations we have to choose an nonorthogonal basis or oblique coordinate system.
In the lexicon of category theory, covariance and contravariance are properties of functors. More generally, let be a commutative ring with unit element and a unitary module over. Covariance and contravariance of vectors wikimili, the best. Vectors, metric and the connection 1 contravariant and. Contravariant and covariant vectors and tensors so that we can get ready for a discussion of special relativity, we need to take up a distinction between di erent kinds of vectors and tensors based on abstract geometry. The expansion in orthonormal basis is mathematically simple.
In this video, i describe the meaning of contravariant and covariant vector components. The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. Covariance and contravariance of vectors semantic scholar. The precise rule 11 12 is that the return types of all methods in the interface must be valid covariantly and all the method parameter types. 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. One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor. A gentle introduction to tensors washington university in. Planetphysicscovariance and contravariance wikiversity. We wish to write the velocity vector at some time t 0. A generic type parameter that is not marked covariant or contravariant is referred to as invariant. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis.
Since you asked for an intuitive way to understand covariance and contravariance, i think this will do. Such a map is what is usually described in the literature when talking about tensors. Conversion of mixed tensors into mixed tensors and into covariant or contravariant ones 0 i wanted to know of book suggestions that can help me overcome my fear of indices. 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. Tensors of rank greater than two are easily defined. Physicists use it this way, the terms contravariant and covariant in mathematics are defined for functors and not for tensors. The terms contravariant and covariant apply based on how they are derived.
For this reason contravariant tensors are sometimes directly defined as multilinear functionals on. Notice how this relationship is the opposite direction of the covariant one. In ndimensional space a tensor of rank n has nn components. Requirements 1 the derivative of a tensor must be a tensor. 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. Higher order tensors are in principle handled similarly, but they may be expressed with mixed coordinate types i. Then the elements of the fold tensor product are called contravariant tensors or contravariant tensors of valency or order. 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. 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. Furthermore a contravariant functor refers to the dual category, which is exactly the opposite of how physicists use it. 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. People commonly talk about contravariant and covariant vectors and tensors, when they really mean contravariant and covariant components. This illustrates a subtle and slightly unfortunate design choice in the method type inference algorithm, which of course was designed long before covariance and contravariance were added to the language.
Vector is a physical quantity and it does not depend on any coordinate system. A device which can compare two animals can also compare two tigers, but a device which can compare two tigers cannot necessarily compare any two animals. Oct 16, 2007 i have been wanting for a long time to do a series of articles about covariance and contravariance which i will shorten to variance for the rest of this series. This question points out a problem with the terminology. Covariant and contravariant rank2 tensors physics forums. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc. The fact that frog is convertible to animal implies that action is convertible to action. If we use the symbol a to denote a vector, and a b to denote the inner product between. A brief summary of facts about variance in the common language runtime.
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. Covariance and contravariance in generics microsoft docs. The tensor itself is independent of the coordinate system we are using. Net framework 4, variant type parameters are restricted to generic interface and generic. Covariance preserves assignment compatibility and contravariance reverses it. Jul 21, 2017 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. A mapping which preserves but reverses a relation is called a contravariant mapping. A di erent metric will, in general, identify an f 2v with a completely di erent ef 2v. The vector space or linear space, mve4 space, or just space of all kcontravariant, lcovariant tensors tensors of valence. Thus in this example we have say components cij k fui,uj,u k. Covariant and contravariant tensors physics forums. As mentioned in a previous video, tensors are invariant under coordinate transformations.
Vectors, contravariant and covariant penn engineering. The principle of general covariance we want to understand how the laws of physics, beyond those governing freelyfalling particles described by the geodesic equation, adapt to the curvature of spacetime. That is, a parameter that was declared covariant must not occur in any contravariant positions where a position is contravariant if it occurs under an odd number of contravariant type constructors. 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. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. Expressions for lengths, areas and volumes of objects in the vector space can then be given in terms of tensors with covariant and contravariant indices. It is a tensor contravariant of degree 2, and covariant of degree 1. Under simple expansions and contractions of the coordinates, the reciprocity is exact. Jul 20, 2017 tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. 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.
Introduction to tensors contravariant and covariant vectors rotation in 2space. 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. 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. One part are ordinary vector spaces contravariant, the other dual vector spaces covariant. The gradient g is an example of a covariant tensor, and the differential position d dx is an example of a contravariant tensor.
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. In cartesian space they must be an orthogonal norm preserving matrix. Cliquer ici pour le pdf en francais the foolish purpose of this post is to tackle the concept of tensor, while trying to keep it accessible to the widest audience possible. For a given nonorthogonal basis a, b, c, we can always construct an. I understand what a tensor is on a very elementary level that a tensor is a generalization of a matrix in the same way that a matrix is a generalization of a vector but one thing that i dont understand is contravariance and covariance.
Well, the above idea of a tensor is actually a bit of a cheat. The velocity vector in equation 3 corresponds to neither the covariant nor contravari. Vectors, metric and the connection 1 contravariant and covariant vectors 1. Even though we use the terms contravariant tensor or covariant tensor, what we are really referring to are the components of a tensor, and not to the tensor itself. A given velocity vector for example has whatever direction and magnitude it has, independent of the coordinate system we use to express it. I have been wanting for a long time to do a series of articles about covariance and contravariance which i will shorten to variance for the rest of this series. Covariant derivative 22 mar 2012riemannchristoffel curvature tensor. Covariance and contravariance computer science wikipedia. 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. General relativitycontravariant and covariant indices. With the notion of contravariant and covariant components of a vector, we make nonorthogonal basis to behave like orthonormal basis. Kolkers answer gives the gory detail, but heres a quick and dirty version. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. For example, for a tensor of contravariant rank 2 and covariant rank 1.
This is almost completely unintelligible, since the space which contains the. I dont know what the difference between the two is, and i dont know why that distinction matters. Tensors can, in general, have rank greater than 2, and. 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.
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