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. Covariance and contravariance of vectors, in mathematics and theoretical physics covariance and contravariance of functors, in category theory covariance and contravariance computer science, whether a type system preserves the ordering. Nov 26, 2015 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. As mentioned in a previous video, tensors are invariant.
General relativitycontravariant and covariant indices. So, rather than saying contravariant vectors and covariant vectors are both covariant, we can say vectors and oneforms are both lorentz invariant. Imagine youve got some sort of coordinate system, so you can imagine measuring everything with n rulers. Planetphysicscovariance and contravariance wikiversity. Covariance and contravariance occur all over mathematics in vector spaces, differential geometry and so on. One is the notation we use for vectors written as components, especially the einstein summation notation.
P via the metric or dot or inner product such that for all vectors in tp then is a real number a. The motivating concept ill start with an imprecise and somewhat misleading illustration to motivate the difference between covariance and contravariance. In the ordered pair, an element of is called a contravariant vector. The vector space or linear space, mve4 space, or just space of all kcontravariant, covariant tensors tensors of valence k. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc. The relationship between v and v is one of mutual duality, and to. The most important point is that both vi and v i are representations of the one vector v. As the torrent of water dried up for the first time in thousands of years, it revealed a horrific sight. People commonly talk about contravariant and covariant vectors and tensors, when they really mean contravariant and covariant components. Covariance and contravariance are properties of a mapping function that associates one member of a set with another. Tensors, contravariant and covariant components of a vector. The coordinate notation for a covariant vector is particularly simple if one chooses in and socalled dual bases in and in, that is, bases such that where is the kronecker symbol. A gentle introduction to tensors washington university in.
Twenty years ago, in an article titled \covariance and contravariance. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. Vectors, metric and the connection 1 contravariant and covariant vectors 1. Or, in indices, a tangent vector is by definition a vector that is a linear combination of the coordinate partials. In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a. Tensors, contravariant and covariant components of a. Note that the rows of s appear as superscripts and the columns appear as subscripts. Although tensors are applied in a very broad range of physics and math. The second is a brief introduction to coordinatefree geometry, which necessitates a discussion of contravariant and covariant vectors.
Vectors as opposed to dual vectors are said to be contravariant. If it transforms as coordinates, and therefore in the opposite direction as basis vectors, its a contravariant vector, or just a vector. Kolkers answer gives the gory detail, but heres a quick and dirty version. In mathematical terms, they occur in a foundational way in linear algebra and multilinear algebra, differential geometry and other branches of geometry. The g term accounts for the change in the coordinates. Covariancev1, v2 gives the covariance between the vectors v1 and v2.
Covariance preserves assignment compatibility and contravariance reverses it. We wish to write the velocity vector at some time t 0. Covariance and contravariance computer science wikipedia. These two topics will be then combined to write down the formalism for special relativity. If an object transforms in the same way as basis vectors, its a covariant vector also called a 1form. 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. 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 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. Covariance m gives the covariance matrix for the matrix m. Thus, vectors in an ndimensional space are contravariant. 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. A mapping which preserves but reverses a relation is called a contravariant mapping.
Covariancedist gives the covariance matrix for the multivariate symbolic distribution dist. One part are ordinary vector spaces contravariant, the other dual vector spaces covariant. The question is what is the difference between covariance and contravariance. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components.
When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. 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. Chapter 10 vectors and tensors georgia institute of. Secondly, in the classical approach to differential geometry, it is not bases of the tangent bundle that are the most primitive object, but rather. It is an idea that seems to originate in physics but in fact it is a more a mathematical concept. Now, a position vector is represented by either a row or column of numbers row vector or column vector. Then the prototypical example of a contravariant vector is a. Covariance and contravariance of vectors wikipedia. An element of the vector space dual to an dimensional vector space, that is, a linear functional linear form on. Covariance and contravariance in generics microsoft docs.
The following code demonstrates the difference between assignment compatibility, covariance, and contravariance. 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. Covariance and contravariance of vectors, in mathematics and theoretical physics. Change of basis, reciprocal basis vectors, covariant and. I have used both in undergraduate math and physics classes and i can recite you the formal definitions variance with change of basis transformations or their inverses, but i am struggling a bit with how to think of covariant and contravariant vectors on an intuitive level, or how 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. These topics are usually encountered in fundamental mathematics courses. 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. Abstract vector is a physical quantity and it does not depend on any coordinate system. This question points out a problem with the terminology.
This is the main purpose of this article to explore the notion of contravariant and covariant vectors in di. Covariancedist, i, j gives the i, j\nullth covariance for the multivariate. Vector is a physical quantity and it does not depend on any coordinate system. In einstein notation, contravariant components have upper indices as in. Covariance, in probability theory and statistics, the measure of. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. Im reading a quora answer on an intuitive explanation of covariantcontravariant components of vectors. Examples of vectors with contravariant components include the. Vectors, metric and the connection 1 contravariant and. If we have a coordinate system with straight coordinate axes, the geometric explanation given is that a vector s covariant components in such a system will be perpendicular projections on the axes, whereas its contravariant components will be parallel projections. This extends naturally to the components of higher order tensors.
However, in a more thorough and indepth treatment of mechanics, it is. What are covariant and contravariant vectors, as intuitive. One contravariant, v i is based on the notion of building v by adding together the basis vectors ei. Ill start with an imprecise and somewhat misleading illustration to motivate the difference between covariance and contravariance. Consider a coordinate system with its origin located at a point p in region r and with unitary base vectors then any vector a can be expressed with respect to this coordinate system in terms of its base vectors as.
The covariant derivative of the r component in the q direction is the regular derivative plus another term. Likewise, vectors whose components are contravariant push forward under smooth mappings, so the operation assigning the space of contravariant vectors to a smooth manifold is a covariant functor. Looking for covariance and contravariance of vectors. A basic explanation of the covariance and contravariance of vectors and of tensors too, since vectors are tensors of rank math1math is best.
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. This book has been presented in such a clear and easy way that the students will have no difficulty in understanding it. In mathematics and theoretical physics, covariance and contravariance are concepts used in many areas, generalizing in a sense invariance, i. Before moving on to covariant vectors, we must define the notion of a dual basis. A generic type parameter that is not marked covariant or contravariant is referred to as invariant. For a given nonorthogonal basis a, b, c, we can always construct an.
Covariance v1, v2 gives the covariance between the vectors v1 and v2. 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. So we say things like distance vectors and velocity vectors are contravariant. A brief summary of facts about variance in the common language runtime. Yes, and sometimes youll hear the two ways of writing the single vector as contravariant representation and covariant representation. A given velocity vector for example has whatever direction and magnitude it has, independent of the coordinate system we use to express it. Covariancem1, m2 gives the covariance matrix for the matrices m1 and m2. In cartesian space they must be an orthogonal norm preserving matrix. Furthermore a contravariant functor refers to the dual category, which is exactly the opposite of how physicists use it. What are the physical differences between contravariant.
When one coordinate system is just a rotation of the other, this. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. This convention is important and should be kept in mind. Covariance and contravariance of vectors article about. The word vector or the phrase contravariant vector is also used to denote a vector field. For orthonormal systems we are used to the fact that when a ake. A contravariant tensor of degree 1, such as the tensor whose components are differentials of the coordinates explanation of covariance and contravariance of vectors. Change of basis, reciprocal basis vectors, covariant and contravariant components of a vector and metric tensor.
Covariancem gives the covariance matrix for the matrix m. The vectors are orthogonal g 1g2 0 and can be are related to the standard orthonormal polar base vectors via g1 er and g2 re 1. So we can apply covariant vectors to contravariant vectors to get a scalar. Tk mn rp is a mixed tensor with contravariant rank 4 and covariant rank 2. Covariance and contravariance of functors, in category theory. In this video, i describe the meaning of contravariant and covariant vector components. Find out information about covariance and contravariance of vectors. They derive from a very general idea based on observing what happens when you make a change the result usually goes in the same direction as the change, i. A contravariant tensor of degree 1, such as the tensor whose components are differentials of the coordinates explanation of. Covariance and contravariance are collectively referred to as variance. Covariance and contravariance of vectors wikimili, the. Covariance and contravariance of vectors semantic scholar. Covariant and contravariant tensors physics forums.
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 wikimili, the best. Thus in the orthogonal basis, any arbitrary vector v is represented by a diagonal matrix and it is a mathematical privilege. In ndimensional space a tensor of rank n has nn components. Although the geometrical view of vectors is fundamental, in practice it is often easier to convert vectors to a set of numbers.
The covariant derivative of the r component in the r direction is the regular derivative. Physicists use it this way, the terms contravariant and covariant in mathematics are defined for functors and not for tensors. Net framework 4, variant type parameters are restricted to generic interface and generic. Covariance and contravariance computer science, whether a type system preserves the ordering. 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. Contravariant components are conventional components that when changing the. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering.
Since you asked for an intuitive way to understand covariance and contravariance, i think this will do. 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. Within the general scheme for the construction of tensors, a covariant vector is identified with a covariant tensor of valency 1. Twenty years ago, in an article titled \ covariance and contravariance. Examples of contravariant vectors include the position of an object relative to an observer, or any derivative of position with respect to time, including velocity, acceleration, and jerk. Distinction between covariant and contravariant vectors. I like picturing this in 2d, but it works in any number of dimensions.1376 1034 1520 1363 399 1286 55 138 160 1508 1115 1349 1031 1263 184 1548 859 231 483 61 953 1133 469 1267 784 1192 1372 825 1494 1066 513 1402 1235 735 412 1064 297 487 1527 601 186 1234 534 231 87 63 751 1335 1063 1396 114