The terms contravariant and covariant apply based on how they are derived. Covariance and contravariance describes how a tensor transforms with respect to dilation, not a rotation or other lorentz transformation. Tensors can, in general, have rank greater than 2, and. If both r and s are nonzero then its called a mixed tensor. A gentle introduction to tensors washington university in. Note that in chapter ii it was shown that the last two expressions have the. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Intuitive way to understand covariance and contravariance in.
This collection of notes attempts to demystify some of the vagaries of tensor. For more information, please see the wikipedia page on covariance and contravariance. I dont know what the difference between the two is, and i dont know why that distinction matters. Appendix a relations between covariant and contravariant bases. I, too, have a concern about the clarity of the difference between covariant and contravariant transformations. Since you asked for an intuitive way to understand covariance and contravariance, i think this will do. Introduction using the equivalence principle, we have studied the trajectories of free test particles. The study was limited to the operators on the spaces of symmetric and either contravariant or covariant secondorder. Is it basically a matrix 6 x 1, with the elements being a random number from the interval 0, 1. Abstract vector is a physical quantity and it does not depend on any coordinate system. Contravariant indices can be gotten by contracting with the matrix inverse of. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. When the manifold is equipped with a metric, covariant and contravariant indices become very closely related to oneanother. In cartesian space they must be an orthogonal norm preserving matrix.
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. A change of scale on the reference axes corresponds to a change of units in the problem. Chapter 10 vectors and tensors georgia institute of. All samples ive seen so far was just casting some object into. In the pages that follow, we shall see that a tensor may be designated as contravariant, covariant, or mixed, and that the velocity expressed in equation 2 is in its contravariant form. The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern. An interface or delegate type can have both covariant and contravariant type parameters. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. Calling convention for overridden methods are covariant with out parameters and return values, and contravariant with normal parameters with the mode in. For what its worth i cleaned up a document i wrote a while ago which tries to give a slightly lesssucky overview of covariance and contravariance without getting tangled in the whole differential geometry of it.
Jul 20, 2017 tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Covariant formulation of the tensor algebra of nonlinear. We are now ready to get serious about electrodynamics. Why even use contravariant and covariant tensors why not just stick to one, why are they mixed. The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. Conversion of mixed tensors into mixed tensors and into covariant or contravariant ones 0 i wanted to know of book suggestions that can help me overcome my fear of indices. Vectors and tensors in curved space time asaf peer1 may 20, 2015 this part of the course is based on refs. When the covariant form of maxwells equations are applied to a rotating reference frame, a choice must be made to work with either a covariant electromagnetic tensor f or a contravariant electromagnetic tensor f we argue that which tensor one chooses is ultimately dictated by whether one chooses to express the electric and magnetic fields in terms of a vector basis or in terms of. This material is covered thoroughly in chapters one and seven in our textbook classical electrodynamics, 2nd ed. The valence or type of a tensor gives the number of covariant and contravariant component indices.
It follows that the unique, natural definition of material tensor rate is the convective timederivative or lie timederivative lie and engel, 1888 along the. Understanding tensoroperations, covariance, contravariance, in the context of special relativity. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc. 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. Vectors, metric and the connection 1 contravariant and. This procedure naturally generalizes to multilinear maps with different numbers of covariant and contravariant arguments and also shows where the transforms like a tensor bit comes from. Covariance and contravariance of vectors wikimili, the.
However, sather supported both covariance and contravariance. An informal discussion on tensor calculus chris tiee abstract. Understanding tensoroperations, covariance, contravariance. This post is intended to give a developerfocused explanation of the terms without diving into the category theory behind them too much. Covariant formulation of the tensor algebra of nonlinear elasticity. Indeed, a vector is a tensor of rank one, and a scalar is a tensor of rank zero. 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. In the video it is said that the contravariant vectors are the vectors that i get through adding together the basevectors each basevector multiplied by a certain amount. When one coordinate system is just a rotation of the other, this distinction is. Covariance and contravariance computer science wikipedia. It gives me great pleasure to write the foreword to dr. 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. Vectors, metric and the connection 1 contravariant and covariant vectors 1. 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.
Then the prototypical example of a contravariant vector is a. Contravariant and covariant components of a vector in a curvinilinear coordinate system. An element of the vector space dual to an dimensional vector space, that is, a linear functional linear form on. We have developed a group of coordinate transformations that preserves. Mar 23, 2019 in this video, i describe the meaning of contravariant and covariant vector components. The tensor product of the covariant and contravariant permutation pseudo tensors is a sixthorder tensor. In this video, i describe the meaning of contravariant and covariant vector components. Similarly, contravariant vectors can be thought of as scalarvalued linear functions of one covariant vector each, with the following definition. Requirements 1 the derivative of a tensor must be a tensor. Contravariant indices can be turned into covariant indices by contracting with the metric tensor.
This blog post is also part of the fp complete haskell syllabus and part of our haskell training. We wish to write the velocity vector at some time t 0. That is a nice brief generalization of what tensor can represent but it gives definitions based on undefined terms. For instance, a rank2 contravariant tensor has the. The gradient g is an example of a covariant tensor, and the differential position d dx is an example of a contravariant tensor. The difference between these two kinds of tensors is how they transform under a continuous change of coordinates.
Planetphysicscovariance and contravariance wikiversity. 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. Vectors and covectors in special relativity jim napolitano march 12, 2010. The velocity vector in equation 3 corresponds to neither the covariant nor contravari. Covariance of the dirac equation 3 which is called the feynmanslash. Are contravariant and covariant vectors the same when the base is an orthonormalbasis. General relativitycontravariant and covariant indices. Transformation of contravariantcovariant components. I know the author as a research scholar who has worked with me for several years. Within the general scheme for the construction of tensors, a covariant vector is identified with a covariant tensor of valency 1.
In ndimensional space a tensor of rank n has nn components. Spectral decomposition of a 4thorder covariance tensor. Contravariant is the way to express that a container. Kolkers answer gives the gory detail, but heres a quick and dirty version. With the notion of contravariant and covariant components of a vector, we make nonorthogonal basis to behave like orthonormal basis. Second, tensor theory, at the most elementary level, requires only. An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0. Thus contravariant vectors often called simply vectors are the simplest kind of contravariant tensor. The statement of the covariance paradigm extends naturally to material tensors at displaced configurations, the push of tensors being defined by invariance as detailed in section 3. 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. 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.
For example, for a tensor of contravariant rank 2 and covariant rank 1. Suppose we are given a system of equations u 1 u 1 x, y, z u 2 u 2 x, y, z. Covariance and contravariance of vectors 1 covariance and contravariance of vectors for other uses of covariant or contravariant, see covariance and contravariance disambiguation. 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. The gradient of a scalar field is a covariant vector field. To make its covariance manifest, we define a 4current 16. One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor. Intuitive way to understand covariance and contravariance. Covariant derivative 22 mar 2012riemannchristoffel curvature tensor. I understand what a tensor is on a very elementary level that a tensor is a generalization of a matrix in the same way that a matrix is a generalization of a vector but one thing that i dont understand is contravariance and covariance.
This is obtained from the basic contravariant field strength tensor by the substitutions. The discussion of rotations should come before covariance and contravariance. This means it is not necessary to transform the expression such as must be done for a tensor in the box on page iii1. Apr 10, 2018 first, lets start with some introductory notes. 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 to the widest audience possible. Vectors, metric and the connection 1 contravariant and covariant 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. Nazrul islams book entitled tensors and their applications. Now that we have talked about tensors, we need to figure out how to classify them. Covariance and contravariance of vectors semantic scholar. Tensors, contravariant and covariant components of a vector. If we use the symbol a to denote a vector, and a b to denote the inner product between. If w is a covariant vector, and v a contravariant vector, then vw is defined as wv. Covariance and contravariance of vectors wikipedia. When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. As mentioned in a previous video, tensors are invariant. In order to change a contravariant vector into a oneform or viseversa so that we can compute the inner product, we use the metric tensor as a lowering or raising operator. Covariance and contravariance in scala knoldus blogs.
As mentioned in a previous video, tensors are invariant under coordinate transformations. Covariance and contravariance of vectors wikimili, the free encyclopedia wikimili, the free encyclopedia. We have developed a beautiful, geometric system for describing the coordinates in terms of which electrodynamics must be formulated for the speed of light to be an invariant. A guide on tensors is proposed for undergraduate students in physics or engineering that ties directly to vector calculus in orthonormal coordinate systems.
A gentle introduction to tensors boaz porat department of electrical engineering technion israel institute of technology. 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. Covariance and contravariance in generics microsoft docs. This means that they have components that are both covariant and contravariant. What is contravariant and covariant vector, tensor in.
For instance, a rank2 contravariant tensor has the transformation property. A vector v red represented by tangent basis vectors yellow, left. There are no definitions of covariant and contravariant in. To get an intuitive idea of the difference between invariance, covariance, and contravariance, suppose we have an aquarium tank filled with water, and we define rectangular cartesian coordinates x,y,z to identify each point in the tank. 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 physics, a basis is sometimes thought of as a set of reference axes. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. What are the physical differences between contravariant. More compactly, the q are the components of a tensor of type 2. 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 the ordered pair, an element of is called a contravariant vector. Introduction to tensors contravariant and covariant vectors rotation in 2space. Discusses covariance and contravariance in generic and nongeneric delegates and provides a list of variant generic delegates in the. Applying covariant versus contravariant electromagnetic.
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