What is the difference between contravariant and covariant tensors, and why do they transform differently under a change of coordinates. Tensors, contravariant and covariant components of a. When first learning this material i could apply the formulae but was very confused by what the concepts intuitively meant, and could not discern the difference between the esoteric paths termed covariant and contravariant. First we will explain the distinction between the covariant and contravariant components of vectors, thinking of vectorfields where a vector is defined at a.
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. One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor. Physicists use it this way, the terms contravariant and covariant in mathematics are defined for functors and not for tensors. Understanding the difference between co and contra variant vectors. The terms contravariant and covariant apply based on how they are derived. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. A covariant relationship is indicated since the arrows travel in the same direction. Relationship between covariantcontravariant basis vectors. It is an idea that seems to originate in physics but in fact it is a more a mathematical concept. As adjectives the difference between covariant and invariant is that covariant is category theory of a functor which preserves composition while invariant is not varying. Vectors are simple and wellknown examples of tensors, but there is much more to tensor theory than vectors. The difference between these two kinds of tensors is how they transform under a continuous change of coordinates.
Introduction to tensors contravariant and covariant vectors rotation in 2space. Covariance and contravariance computer science wikipedia. One part are ordinary vector spaces contravariant, the other dual vector spaces covariant. What are the physical differences between contravariant. Ill start with an imprecise and somewhat misleading illustration to motivate the difference between covariance and contravariance.
Could someone explain, in as much detail as they are willing to give, how the contravariant tensors can be transformed into covariant tensors. Tensors of rank greater than two are easily defined. On an elementary level, if a is an m x n matrix then it defines a function r n r m while its transpose matrix a t is n x m and defines a function in the opposite direction r m r n. 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. 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 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. The question is what is the difference between covariance and contravariance.
General relativitycontravariant and covariant indices. As mentioned in a previous video, tensors are invariant under. On a generalization of the index notation for absolute. Net framework 4, variant type parameters are restricted to generic interface and generic. Jul 20, 2017 tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. So that we can get ready for a discussion of special relativity, we need to take up a distinction.
I, too, have a concern about the clarity of the difference between covariant and contravariant transformations. To put it succinctly, tensors are geometrical objects over vector spaces, whose coordinates obey certain laws of transformation under change of basis. For more information, please see the wikipedia page on covariance and contravariance. In tensor analysis, a covariant vector varies more or less reciprocally to a corresponding contravariant vector. An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0. Since you asked for an intuitive way to understand covariance and contravariance, i think this will do. 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 indexfree methods to represent tensors. Tensors, contravariant and covariant components of a vector. I understand that contravariant tensors are like tangents and covariant are as normal vectors to the surface.
Ultimately the distinction between covariance and contravariance comes from the distinction between a vector space and its dual space. A gentle introduction to tensors washington university in. I get that the contravariant has an upper index and the covariant has a lower index, but i dont really have an intuitive feel for how they are different. As mentioned in a previous video, tensors are invariant under coordinate transformations. With the notion of contravariant and covariant components of a vector, we. Physics stack exchange is a question and answer site for active researchers, academics and students of physics. 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. Covariance and contravariance are collectively referred to as variance. Tensors can, in general, have rank greater than 2, and.
Indeed, a vector is a tensor of rank one, and a scalar is a tensor of. Covariant and contravariant tensors physics forums. Chapter 10 vectors and tensors georgia institute of. Covariance and contravariance of vectors 3 for some invertible n. An inner product is between elements of the same vector space and not between two vector spaces, it is not how it is defined. Mar 23, 2019 in this video, i describe the meaning of contravariant and covariant vector components. Now that we have talked about tensors, we need to figure out how to classify them. This extends naturally to the components of higher order tensors. A brief summary of facts about variance in the common language runtime. Covariance and contravariance of vectors wikipedia. In that spirit we begin our discussion of rank 1 tensors. 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.
The relationship between v and v is one of mutual duality, and to. 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. There are as many explanations, examples, and mathematical descriptions for covariance and contravariance as you care to explore, but all of this information adds to the confusion. The contravariant tensor conjugate to gij is denoted by gij satisfy 5. Course notes tensor calculus and differential geometry. As adjectives the difference between covariant and invariant is that covariant is category theory of a functor which. So there is actually no difference between a contravariant and covariant vector, there is a difference between a contravariant and covariant basis as is shown in arxiv. However, in a more thorough and indepth treatment of mechanics, it is. Here, we abstain from giving an exact definition of even and odd permutations.
This post is intended to give a developerfocused explanation of the terms without diving into the category theory behind them too much. 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 position vector. Learning tensors for the first time, but im kind of stuck on the difference between covariant and contravariant this is for a relativity course, if that matters. Appendix a relations between covariant and contravariant bases. This blog post is also part of the fp complete haskell syllabus and part of our haskell training. Is it foolish to distinguish between covariant and. In this video, i describe the meaning of contravariant and covariant vector components. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc. Covariance preserves assignment compatibility and contravariance reverses it. A generic type parameter that is not marked covariant or contravariant is referred to as invariant.
A contravariant vector is a tangent vector and a covariant vector is technically a linear functional on the collection of tangent vectors a covector. Covariance and contravariance in generics microsoft docs. It is necessary in order to really understand what they are made of and why they. 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.
A contravariant tensor in other words a vector, transform oppositely contra to the way basis vectors transform, while a covariant tensor or dual vector. In general direction vector like velocity vector is contravariant vector and dual vector like gradient e. 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. The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern. Kolkers answer gives the gory detail, but heres a quick and dirty version. This article shows you why they are important concepts for software developers and how you can leverage variance when you design your code. Prerequisites from linear algebra linear algebra forms the skeleton of tensor calculus and differential geometry. Covariance and contravariance describes how a tensor transforms with respect to dilation, not a rotation or other lorentz transformation. 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. The discussion of rotations should come before covariance and contravariance. Then the prototypical example of a contravariant vector is a. Covariant and contravariant bases are dual to one another and are physics nomenclature for constructs that arise in differential geometry.
More compactly, the q are the components of a tensor of type 2. These topics are usually encountered in fundamental mathematics courses. Doubts on covariant and contravariant vectors and on double tensors. The key to the trransformation is having a metric around, i. The gradient g is an example of a covariant tensor, and the differential position d dx is an example of a contravariant tensor. 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 of vectors semantic scholar.
When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. 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. Now that we have a notion for the difference between the transformation of a vector and the. The problem here is that physicists often need to use differential geometry for example, for relativity long before they have seen a proper course on differential geometry. In equation 29 components of covariant vector dual covary with change of coordinate basis and hence the name covariant vector. Of course, if you search the web, the descriptions, the mathematics, and the applications of these terms can seem bewildering. 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. If we use the symbol a to denote a vector, and a b to denote the inner product between. Contravariant tensors of any rank are derived from the derivative of the coordinate axes covariant tensors of any rank are derived from the gradient which are also related to how the tensor s components transform between different coordinate systems. If you are using cartesian coordinates with a euclidean metric, its difficult to notice the difference between contravariant and covariant mathematically, there is a. This means, an interface instance that has method with more derived return types than originally specified covariance or that has methods with less derived parameter types contravariance. 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.
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. Covariance and contravariance are properties of a mapping function that associates one member of a set with another. A disadvan tage of the notation is that, in the study of general properties of tensors, equations such as the transformation relations, the expressions for covariant derivatives, or the integrability condi. So, rather than saying contravariant vectors and covariant vectors are both covariant, we can say vectors and oneforms are both lorentz invariant. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Planetphysicscovariance and contravariance wikiversity. A di erent metric will, in general, identify an f 2v with a completely di erent ef 2v.
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