Here are the examples from msdn that i hope will show you what contravariance really means i own these documents now, so if you think something is unclear in the docs, feel free to give me. Physicists use it this way, the terms contravariant and covariant in mathematics are defined for functors and not for tensors. Covariance and contravariance of vectors wikimili, the. As mentioned in a previous video, tensors are invariant under coordinate transformations. So, rather than saying contravariant vectors and covariant vectors are both covariant, we can say vectors and oneforms are both lorentz invariant. Covariance and contravariance of vectors 1 covariance and contravariance of vectors for other uses of covariant or contravariant, see covariance and contravariance disambiguation. And so many physicists have the sloppy tendency to. However, in physics, we tend do ignore the basis vectors and only look at the components. According to the representation theory of the lorentz group, these quantities are built out of scalars, fourvectors, fourtensors, and spinors. When y r, the elements of lx,r are called linear functionals on x or dual vectors or covariant vectors. The same notion appears in quantum mechanics as ket and bra vectors and we compare the two equivalent situation via the completeness relation.
Im starting to learn some of the basics of covariant and contravariant vectors. Intuition behind covariant and contravariant vectors. 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. It is an idea that seems to originate in physics but in fact it is a more a mathematical concept. 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. Similarly, the set of numbers f with downstairs indices is called the components of a covariant vector. Feb 17, 2010 with the notion of contravariant and covariant components of a vector, we make nonorthogonal basis to behave like orthonormal basis. The expansion in equation 17 or in equation 18 similar to the orthogo. Requirements 1 the derivative of a tensor must be a tensor. However, you will read all over the literature that vectors are contravariant.
An orthonormal basis is selfdual, there no distinction between. 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. A vector v red represented by tangent basis vectors yellow, left. Contravariant and covariant components of a vector in a curvinilinear coordinate system. However, sather supported both covariance and contravariance. Now, the only question i have left is, what is gained by talking about contravariance and covariance, rather than just vector spaces and their dual spaces.
Furthermore a contravariant functor refers to the dual category, which is exactly the opposite of how physicists use 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. 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. The relationship between v and v is one of mutual duality, and to. As we can see contravariant vectors and covariant vectors are very different from each other and we want to. The same rules work for a general transformation of the basis vectors but it is much easier to understand if you first restrict your attention to a simple change of scale. Covariance and contravariance computer science wikipedia. 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.
Understanding the covariant derivative and its relation to parallel transport. An interface or delegate type can have both covariant and contravariant type parameters. Thus, contravariant vectors are elements of v and covariant vectors are elements of v. Covariant and contravariant basis vectors euclidean space. Variance is a topic that comes up fairly often in type systems and can be a bit confusing the first time you hear it. Tensors, contravariant and covariant components of a vector. The second is a brief introduction to coordinatefree geometry, which necessitates a discussion of contravariant and covariant vectors. In terms of the x coordinate system the contravariant components of p are. What are the physical differences between contravariant.
Mar 23, 2019 in this video, i describe the meaning of contravariant and covariant vector components. Further, the domain of the v i is the whole of the range of x. Ill start with an imprecise and somewhat misleading illustration to motivate the difference between covariance and contravariance. The covariant components are the result of a projection, so they are images of a linear form applied to the vector. Relationship between covariantcontravariant basis vectors. 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. Ill start with an imprecise and somewhat misleading illustration to motivate the. One is called the contravariant vector or just the vector, and the other one is called the covariant vector or dual vector or onevector.
To understand in detail how the representations of vectors in different coordinate systems are related to each other, consider the displacement vector p in a flat 2dimensional space shown below. Confusions about covariant and contravariant vectors. Covariance and contravariance of vectors wikimili, the free encyclopedia wikimili, the free encyclopedia. As far as i understand per erics explanation is that covariance and contravariance are both adjectives that describe a transformation. Transformation of contravariantcovariant components. 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. Covariant and contravariant vectors request pdf researchgate. Covariant and contravariant tensors physics forums. The valence or type of a tensor is the number of variant and covariant terms. An informal discussion on tensor calculus chris tiee abstract.
Covariance and contravariance in generics microsoft docs. Discusses covariance and contravariance in generic and nongeneric delegates and provides a list of variant generic delegates in the. A contravariant tensor in other words a vector, transform oppositely contra to the way basis vectors transform, while a covariant tensor or dual vector transforms in he same way as basis vectors. The basis set for dual vectors enables any dual vector. When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. Abstract vector is a physical quantity and it does not depend on any coordinate system. A physical quantity is said to be lorentz covariant if it transforms under a given representation of the lorentz group.
The terms contravariant and covariant apply based on how they are derived. With the notion of contravariant and covariant components of a vector, we make nonorthogonal basis to behave like orthonormal basis. Im a little confused about the difference between a covariant and a contravariant basis vector. By contrast, the vectors in x are called contravariant vectors. But that would be contravariance but again, arrays are not contravariant, they support covariance only. These two topics will be then combined to write down the formalism for special. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. A contravariant tensor of degree 1, such as the tensor whose components are differentials of the coordinates explanation of contravariant vector.
The valence or type of a tensor gives the number of covariant and contravariant component indices. In typescript, argument types are bivariant both covariant and contravariant, which is unsound although now in typescript 2. Covariant transformation is that which preserves the order of types and contravariant transformation is one that reverses it. This means that they have both covariant and contravariant components, or both vectors and dual vectors.
Vector is a physical quantity and it does not depend on any coordinate system. As shown on figure 1, the dual basis vectors are perpendicular to all basis vectors with a di erent index, and the scalar product of the dual basis vector with the basis vector of the same index is unity. Calling convention for overridden methods are covariant with out parameters and return values, and contravariant with normal parameters with the mode in. Even if that sentence does not make sense right now, it will soon, so bear with me. Vectors, metric and the connection 1 contravariant and. General relativitycontravariant and covariant indices. In terms of the x coordinate system the contravariant components of p are x 1, x 2 and the covariant components are x 1, x 2. Thus, contravariant vectors are elements of v and covariant. 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. Covariance and contravariance are collectively referred to as variance. The components of vectors as opposed to those of covectors are said to be contravariant. A brief summary of facts about variance in the common language runtime. Once again looking at the gradient example, the more closely spaced the level surfaces are, the faster f is increasing, and so the larger the gradient covector. Kolkers answer gives the gory detail, but heres a quick and dirty version.
The following code demonstrates the difference between assignment compatibility, covariance, and contravariance. Tensors, contravariant and covariant components of a. A contravariant vector field v on m associates with each chart x a collection of n smooth realvalued coordinate functions v i of the n variables x 1, x 2. What are the physical differences between contravariant and. Coordinatization of vectors for sophomore linear algebra math 2010. 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. Doubts on covariant and contravariant vectors and on double tensors. For every contravariant index there is an a transformation and for every covariant index there is a b transformation. Vectors, metric and the connection 1 contravariant and covariant vectors 1.
One is the notation we use for vectors written as components, especially the einstein summation notation. Aug 06, 2019 one part are ordinary vector spaces contravariant, the other dual vector spaces covariant. The contravariant basis vector gk of the curvilinear coordinate of uk at the point. There the product of the balance of momentum and velocity two contravariant vectors appeared. 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. A generic type parameter that is not marked covariant or contravariant is referred to as invariant.
The strain tensor with respect to rotations between cartesian frames is contravariant of rank 1 and covariant of rank 1. 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. Lorentz covariance has two distinct, but closely related meanings. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis. Request pdf covariant and contravariant vectors vector is a physical quantity and it does not depend on any coordinate system. The first is that vectors whose components are covariant called covectors or 1forms actually pull back under smooth functions, meaning that the operation assigning the space of covectors to a smooth manifold is actually a contravariant functor. Appendix a relations between covariant and contravariant bases. Relations between covariant and contravariant bases. This means that they have components that are both covariant and contravariant. Covariance and contravariance of vectors wikipedia. 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. 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. There are lots of articles, all over the place about the distinction between covariant vectors and contravariant vectors after struggling through many of them, i think im starting to get the idea.
The set of all linear maps from x to y, denoted lx,y, is itself a vector space. The transformation that describes the new basis vectors as a linear combination of the old basis vectors is defined as a covariant transformation. And does covariance and contravariance of vectors have any relation to the contravariant functor mapping a vector space v to its dual v. Then the prototypical example of a contravariant vector is a. 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. Lets work in the three dimensions of classical space forget time, relativity, four vectors etc. Covariant derivative 22 mar 2012riemannchristoffel curvature tensor.
All that i managed to glean from it is that contravariant vectors e. In cartesian space they must be an orthogonal norm preserving matrix. 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 question is what is the difference between covariance and contravariance. Covariance and contravariance are properties of a mapping function that associates one member of a set with another. The basis set for dual vectors enables any dual vector p to be written. One part are ordinary vector spaces contravariant, the other dual vector spaces covariant. The basis for these names will be explained in the next section, but at this stage it is just a name used to distinguish two types of vector. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. The distinction between covariant and contravariant vectors is particularly important for computations with tensors, which often have mixed variance.
Chapter 10 vectors and tensors georgia institute of. The equation and points i made above do not distinguish between vectors and dual vectors and i chose a arbitrary basis metric to make my point. Covariance preserves assignment compatibility and contravariance reverses it. Contravariant vector article about contravariant vector. Suppose we are given a system of equations u 1 u 1 x, y, z u 2 u 2 x, y, z.
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. This covariant contravariant distinction doesnt seem to mean much right now because they are the same for this vector v. In ndimensional space a tensor of rank n has nn components. For me it is confusing to read that whether a vector is covariant or contravariant depends on a condition of orthogonality or lack thereof. 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. We wish to write the velocity vector at some time t 0.
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