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. Yes, and sometimes youll hear the two ways of writing the single vector as contravariant representation and covariant representation. Covariant and contravariant tensors physics forums. Then the prototypical example of a contravariant vector is a. 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. Covariance and contravariance of vectors wikimili, the best. This convention is important and should be kept in mind. Ill start with an imprecise and somewhat misleading illustration to motivate the difference between covariance and contravariance.
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. A generic type parameter that is not marked covariant or contravariant is referred to as invariant. Distinction between covariant and contravariant vectors. This is the main purpose of this article to explore the notion of contravariant and covariant vectors in di.
Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. 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. Covariance and contravariance of vectors wikimili, the. Covariancem gives the covariance matrix for the matrix m. For a given nonorthogonal basis a, b, c, we can always construct an. 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. In mathematics and theoretical physics, covariance and contravariance are concepts used in many areas, generalizing in a sense invariance, i. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc. Note that the rows of s appear as superscripts and the columns appear as subscripts. Twenty years ago, in an article titled \covariance and contravariance. One part are ordinary vector spaces contravariant, the other dual vector spaces covariant. We wish to write the velocity vector at some time t 0.
Net framework 4, variant type parameters are restricted to generic interface and generic. Covariance and contravariance of vectors article about. I like picturing this in 2d, but it works in any number of dimensions. The covariant derivative of the r component in the q direction is the regular derivative plus another term. A given velocity vector for example has whatever direction and magnitude it has, independent of the coordinate system we use to express it. So we say things like distance vectors and velocity vectors are contravariant. 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. Covariance and contravariance occur all over mathematics in vector spaces, differential geometry and so on. First we will explain the distinction between the covariant and contravariant components of vectors, thinking of vectorfields where a vector is defined at a. General relativitycontravariant and covariant indices. 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. If it transforms as coordinates, and therefore in the opposite direction as basis vectors, its a contravariant vector, or just a vector. The relationship between v and v is one of mutual duality, and to.
Within the general scheme for the construction of tensors, a covariant vector is identified with a covariant tensor of valency 1. Covariance m1, m2 gives the covariance matrix for the matrices m1 and m2. Covariance and contravariance of functors, in category theory. Im reading a quora answer on an intuitive explanation of covariantcontravariant components of vectors. 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.
Contravariant components are conventional components that when changing the. This question points out a problem with the terminology. Covariance and contravariance of vectors semantic scholar. 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. 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. Chapter 10 vectors and tensors georgia institute of. 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. 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. Covariance preserves assignment compatibility and contravariance reverses it. Covariancedist, i, j gives the i, j\nullth covariance for the multivariate. Thus in the orthogonal basis, any arbitrary vector v is represented by a diagonal matrix and it is a mathematical privilege.
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. 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. We remark that orthonormality of the bases is nowhere required or even mentioned. Covariance v1, v2 gives the covariance between the vectors v1 and v2. In cartesian space they must be an orthogonal norm preserving matrix. The question is what is the difference between covariance and contravariance. Kolkers answer gives the gory detail, but heres a quick and dirty version. Abstract vector is a physical quantity and it does not depend on any coordinate system. However, in a more thorough and indepth treatment of mechanics, it is. P via the metric or dot or inner product such that for all vectors in tp then is a real number a. What are covariant and contravariant vectors, as intuitive.
People commonly talk about contravariant and covariant vectors and tensors, when they really mean contravariant and covariant components. A brief summary of facts about variance in the common language runtime. The word vector or the phrase contravariant vector is also used to denote a vector field. Covariancem1, m2 gives the covariance matrix for the matrices m1 and m2. 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.
The second is a brief introduction to coordinatefree geometry, which necessitates a discussion of contravariant and covariant vectors. This extends naturally to the components of higher order tensors. Since you asked for an intuitive way to understand covariance and contravariance, i think this will do. When one coordinate system is just a rotation of the other, this. Covariance and contravariance computer science, whether a type system preserves the ordering.
Twenty years ago, in an article titled \ covariance and contravariance. Before moving on to covariant vectors, we must define the notion of a dual basis. Tk mn rp is a mixed tensor with contravariant rank 4 and covariant rank 2. The most important point is that both vi and v i are representations of the one vector v. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. Imagine youve got some sort of coordinate system, so you can imagine measuring everything with n rulers.
Covariance m gives the covariance matrix for the matrix m. A gentle introduction to tensors washington university in. Covariance, in probability theory and statistics, the measure of. The following code demonstrates the difference between assignment compatibility, covariance, and contravariance. The motivating concept ill start with an imprecise and somewhat misleading illustration to motivate the difference between covariance and contravariance. 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. In the ordered pair, an element of is called a contravariant vector. In einstein notation, contravariant components have upper indices as in. When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. Vectors as opposed to dual vectors are said to be contravariant. Examples of vectors with contravariant components include the.
An element of the vector space dual to an dimensional vector space, that is, a linear functional linear form on. If an object transforms in the same way as basis vectors, its a covariant vector also called a 1form. 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. One is the notation we use for vectors written as components, especially the einstein summation notation. Covariance and contravariance of vectors wikipedia. Covariance and contravariance are properties of a mapping function that associates one member of a set with another. Covariance and contravariance computer science wikipedia. Secondly, in the classical approach to differential geometry, it is not bases of the tangent bundle that are the most primitive object, but rather. Physicists use it this way, the terms contravariant and covariant in mathematics are defined for functors and not for tensors. In ndimensional space a tensor of rank n has nn components. This book has been presented in such a clear and easy way that the students will have no difficulty in understanding it. For orthonormal systems we are used to the fact that when a ake.
Vectors, metric and the connection 1 contravariant and. The g term accounts for the change in the coordinates. It is an idea that seems to originate in physics but in fact it is a more a mathematical concept. Tensors, contravariant and covariant components of a.
Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering. A mapping which preserves but reverses a relation is called a contravariant mapping. A basic explanation of the covariance and contravariance of vectors and of tensors too, since vectors are tensors of rank math1math is best. Covariancedist gives the covariance matrix for the multivariate symbolic distribution dist. 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.
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. Change of basis, reciprocal basis vectors, covariant and. In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a. 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. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. Covariance and contravariance in generics microsoft docs. Vector is a physical quantity and it does not depend on any coordinate system. The vector space or linear space, mve4 space, or just space of all kcontravariant, covariant tensors tensors of valence k. Covariancev1, v2 gives the covariance between the vectors v1 and v2. Vectors, metric and the connection 1 contravariant and covariant vectors 1. So, rather than saying contravariant vectors and covariant vectors are both covariant, we can say vectors and oneforms are both lorentz invariant. 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 covariant derivative of the r component in the r direction is the regular derivative. 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. Covariance and contravariance are concepts that relate to the components of the 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 the f basis as 2. 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. Looking for covariance and contravariance of vectors. What are the physical differences between contravariant.
Furthermore a contravariant functor refers to the dual category, which is exactly the opposite of how physicists use it. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Tensors, contravariant and covariant components of a vector. As the torrent of water dried up for the first time in thousands of years, it revealed a horrific sight. Find out information about covariance and contravariance of vectors. So we can apply covariant vectors to contravariant vectors to get a scalar. Thus, vectors in an ndimensional space are contravariant. Planetphysicscovariance and contravariance wikiversity. A contravariant tensor of degree 1, such as the tensor whose components are differentials of the coordinates explanation of covariance and contravariance of vectors. One contravariant, v i is based on the notion of building v by adding together the basis vectors ei. Change of basis, reciprocal basis vectors, covariant and contravariant components of a vector and metric tensor. 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.
A contravariant tensor of degree 1, such as the tensor whose components are differentials of the coordinates explanation of. In mathematical terms, they occur in a foundational way in linear algebra and multilinear algebra, differential geometry and other branches of geometry. These two topics will be then combined to write down the formalism for special relativity. Covariance and contravariance are collectively referred to as variance.634 1218 1067 95 1360 72 1543 913 528 335 867 994 684 891 430 1086 1130 547 984 316 349 49 177 785 1168 337 1257 1462 190 713 261 449 1268 213 1354 1082 948 838 570 866 149