Differential geometry for physicists and engineers, mathematical engineering. Covariance and contravariance of vectors, in mathematics and theoretical physics. Distinction between covariant and contravariant vectors. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc. Thus in the orthogonal basis, any arbitrary vector v is represented by a diagonal matrix and it is a mathematical privilege. Reading weinbergs gravitation and cosmology, i came across the sentence p. This means that they have both covariant and contravariant components, or both vectors and dual vectors. What are the physical differences between contravariant. The gradient of a scalar field is a covariant vector field. Covariance and contravariance of vectors infogalactic. In physics, a basis is sometimes thought of as a set of reference axes. However, sather supported both covariance and contravariance.
General relativitycontravariant and covariant indices. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. If you want a simple explanation that applies to any computer language, here it is. In this video, i describe the meaning of contravariant and covariant vector components. The important exception of multiplication of vectors will be dealt with shortly. The distinction between covariant and contravariant vectors is particularly important for computations with tensors, which often have mixed variance.
Covariant and contravariant basis vectors euclidean space. In ndimensional space a tensor of rank n has nn components. With covariant and contravariant vectors defined, we are now ready to extend our analysis to tensors of arbitrary rank. Understanding tensoroperations, covariance, contravariance, in the context of special relativity. Examples of vectors with contravariant components include the position of an object relative to an observer, or any derivative of position with respect to time, including velocity, acceleration, and jerk. So we can apply covariant vectors to contravariant vectors to get a scalar. If you encounter the use of the contravariance and covariance in say physics or math then it might not be obvious that the difference is related to changes in inputs contravariant or outputs covariant, but if you dig deep enough you will find. 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 think of them geometrically. In cartesian space they must be an orthogonal norm preserving matrix. 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 mathematical terms, they occur in a foundational way in linear algebra and multilinear algebra, differential geometry and other branches of geometry, category theory and. The valence or type of a tensor is the number of variant and covariant terms. Jul 20, 2017 tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Covariance and contravariance occur all over mathematics in vector spaces, differential geometry and so on.
A change of scale on the reference axes corresponds to a change of units in the problem. Contravariant basis vectors are typically concocted in some way to support transformation invariance. As mentioned in a previous video, tensors are invariant. Net framework 4, variant type parameters are restricted to generic interface and generic. An element of the vector space dual to an dimensional vector space, that is, a linear functional linear form on. Thus standard vectors are contravariant with respect.
Furthermore a contravariant functor refers to the dual category, which is exactly the opposite of how physicists use it. Covariance v1, v2 gives the covariance between the vectors v1 and v2. Vectors and covectors in special relativity jim napolitano march 12, 2010 these notes are meant to accompany the course electromagnetic theory for the spring 2010 term at rpi. Covariance and contravariance of vectors semantic scholar. A quantity having magnitude only is called scalar and a quantity with. Jan 30, 2016 for me it is confusing to read that whether a vector is covariant or contravariant depends on a condition of orthogonality or lack thereof. A contravariant transformation is the inverse of a covariant transformation.
Covariance, in probability theory and statistics, the measure of how much two random. Intuitive way to understand covariance and contravariance in. Abstract vector is a physical quantity and it does not depend on any coordinate system. Covariance and contravariance computer science, whether a type system preserves the ordering. Covariance and contravariance in generics microsoft docs. 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. For other uses of covariant or contravariant, see covariance and.
Or, in indices, a tangent vector is by definition a vector that is a linear combination of the coordinate partials. Suppose we are given a system of equations u 1 u 1 x, y, z u 2 u 2 x, y, z. In the lexicon of category theory, covariance and contravariance are properties of functors. Covariance and contravariance of vectors wikimili, the. Examples of contravariant vectors include displacement, velocity and acceleration. All that i managed to glean from it is that contravariant vectors e. We wish to write the velocity vector at some time t 0. In physics they are important to the treatment of vectors and other quantities, such as tensors, that have physical meaning but are not scalars. A given velocity vector for example has whatever direction and magnitude it has, independent of the coordinate system we use to express it. Covariance and contravariance are collectively referred to as variance.
In the classical theory of invariants was called the covariant of. First of, i have read many explanations on so and blogs about covariance and contravariance and a big thanks goes out to eric lippert for producing such a great series on covariance and contravariance however i have a more specific question that i am trying to get my head around a little bit. In typescript, argument types are bivariant both covariant and contravariant, which is unsound although now in typescript 2. A brief summary of facts about variance in the common language runtime. What are contravariant and covariant components of a. Voitsekhovskii originator, which appeared in encyclopedia of mathematics isbn 1402006098. For every contravariant index there is an a transformation and for every covariant index there is a b transformation. Appendix a relations between covariant and contravariant bases. Covariance m gives the covariance matrix for the matrix m. 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. A gentle introduction to tensors boaz porat department of electrical engineering technion israel institute of technology.
Covariance m1, m2 gives the covariance matrix for the matrices m1 and m2. A generic type parameter that is not marked covariant or contravariant is referred to as invariant. Well get to how to manipulate tensors and their indices a little later. 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. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. 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.
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. 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 the meantime, there are already some common mistakes to avoid. Before moving on to covariant vectors, we must define the notion of a dual basis. A gentle introduction to tensors washington university in. Since you asked for an intuitive way to understand covariance and contravariance, i think this will do. 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. Covariancedist gives the covariance matrix for the multivariate symbolic distribution dist. Discusses covariance and contravariance in generic and nongeneric delegates and provides a list of variant generic delegates in the. Both special relativity lorentz covariance and general relativity general covariance use covariant basis vectors. A brief look at applications of differential geometry and the concept of contravariant and covariant components of a vector. Intuitive way to understand covariance and contravariance.
People commonly talk about contravariant and covariant vectors and tensors, when they really mean contravariant and covariant components. This collection of notes attempts to demystify some of the vagaries of tensor. Ill start with an imprecise and somewhat misleading illustration to motivate the difference between covariance and contravariance. 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. Vectors, metric and the connection 1 contravariant and. See also the lecture notes vectors and dual vectors pdf. Contravariant and covariant components of a vector in a curvinilinear coordinate system. It need to be expanded in some basis for practical calculation and its. This question points out a problem with the terminology. In multilinear algebra and tensor analysis, covariance and contravariance describe how the.
The question is what is the difference between covariance and contravariance. The components of vectors as opposed to those of covectors are said to be contravariant. Covariant and contravariant tensors physics forums. One part are ordinary vector spaces contravariant, the other dual vector spaces covariant. A case that was particularly often considered is when is a form in one single contravariant vector. Covariant and contravariant 4vector in special relativity. 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 second is a brief introduction to coordinatefree geometry, which necessitates a discussion of contravariant and covariant vectors. Then the prototypical example of a contravariant vector is a. Tensors were introduced by professor gregorio ricci of university of padua italy in 1887 primarily as extension of vectors. 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. An informal discussion on tensor calculus chris tiee abstract. Covariance and contravariance of vectors wikimili, the free encyclopedia wikimili, the free encyclopedia. Covariant derivative 22 mar 2012riemannchristoffel curvature tensor.
This article was adapted from an original article by m. Covariance and contravariance of vectors wikimili, the best. Tensors, contravariant and covariant components of a vector. You may have heard of covariance and contravariance and wondered what they were all about. 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. Twenty years ago, in an article titled \covariance and contravariance. This is important, because when we move to systems where the basis vectors are no longer constants, the gradient operation does not need to be modified in any way. Covariance and contravariance are properties of a mapping function that associates one member of a set with another. The mathematics of spacetime university of maryland. In the ordered pair, an element of is called a contravariant vector. The strain tensor with respect to rotations between cartesian frames is contravariant of rank 1 and covariant of rank 1. Chapter 10 vectors and tensors georgia institute of.
This collection of notes attempts to demystify some of the vagaries of tensor analysis, to emphasize the connections between the old way of doing things and the new. Kolkers answer gives the gory detail, but heres a quick and dirty version. Change of basis, reciprocal basis vectors, covariant and. Is it foolish to distinguish between covariant and contravariant vectors. A mapping which preserves but reverses a relation is called a contravariant mapping. Covariance and contravariance of functors, in category theory. Vectors and tensors in curved space time asaf peer1 may 20, 2015 this part of the course is based on refs. Requirements 1 the derivative of a tensor must be a tensor. 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. The terms contravariant and covariant apply based on how they are derived. You can add vectors, but you cant add vectors and scalars. Physicists use it this way, the terms contravariant and covariant in mathematics are defined for functors and not for tensors. Over the years, of course, as the higher powers of abstraction in mathematics gained maturity, and the.
Vectors, metric and the connection 1 contravariant and covariant vectors 1. Although it is true that a covariant vector transforms one way, and a contravariant vector transforms another way, i find that to be an extremely unsatisfactory way to define which vectors are co or contravariant, since it fails to explain why one vector. These two topics will be then combined to write down the formalism for special relativity. This material is covered thoroughly in chapters one and seven in our textbook classical electrodynamics, 2nd ed. For contravariant vectors or more correctly vectors with contravariant components, the matrix that transforms the vector components must be the inverse of the matrix that transforms the basis vectors. I found the examples of contravariant vectors as position vector, velocity, acceleration etc. Change of basis, reciprocal basis vectors, covariant and contravariant components of a vector and metric tensor. Covariancem1, m2 gives the covariance matrix for the matrices m1 and m2. Planetphysicscovariance and contravariance wikiversity. Covariancev1, v2 gives the covariance between the vectors v1 and v2. The contravariant basis vector gk of the curvilinear coordinate of uk at the point. Although tensors are applied in a very broad range of physics and math. In mathematics and theoretical physics, covariance and contravariance are concepts used in many areas, generalizing in a sense invariance, i. Introduction using the equivalence principle, we have studied the trajectories of free test particles.
Id have preferred to answer through a comment, but im not yet allowed to do so. Covariancem gives the covariance matrix for the matrix m. We do so by generalizing the cartesiantensor transformation rule, eq. Suppose that the form corresponds in this manner to the tensor and that the form corresponds to its covariant. Calling convention for overridden methods are covariant with out parameters and return values, and contravariant with normal parameters with the mode in. Covariance and contravariance computer science wikipedia. Harris, in mathematics for physical science and engineering, 2014. Covariance and contravariance of vectors wikipedia. Covariancedist, i, j gives the i, j\nullth covariance for the multivariate symbolic. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Within the general scheme for the construction of tensors, a covariant vector is identified with a covariant tensor of valency 1. 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. When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. So, rather than saying contravariant vectors and covariant vectors are both covariant, we can say vectors and oneforms are both lorentz invariant.
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