Statistics varying with another variable quantity in a manner that leaves a specified relationship unchanged. 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 duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern. Tensor which depends on two directions one normal to the area and other that of the force on it. These tensorrelated topics anything related to which, shall henceforth be. Converting contravariant or covariant tensors to mixed. Another characteristic of a tensor is the dimension of the tensor, which is the count of each index. Tensor algebras covariant and contravariant tensors let be a vector space of dimension with dual space. Contravariance says you are prepared to handle a more general argument. So, so if we have two tensors, metric tensor and inverse metric tensor, to every contravariant vector, with the use of the metric tensor we can define. This video looks at the ideas of contravariant, covariant and physical components and how to convert the first two component types into physical component values. Covariant components are labeled with lower indices.
The covariant derivative of the r component in the r direction is the regular derivative. Although the assertion covariance and contravariance refer to how coordinates change under a change of basis seems about right, the next few sentences try to say roughly that vectors in v have contravariant components and vectors in v have covariant components. Converting contravariant or covariant tensors to mixed tensors. General relativitycontravariant and covariant indices. A di erent metric will, in general, identify an f 2v with a completely di erent ef 2v. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. What are the physical differences between contravariant and. Wikipedia covariance and contravariance of vectors tensor. It specifies precisely the method direction of projection used to derive the components by projecting the magnitude of the tensor quantity onto the coordinate system being used as the basis of the tensor. More compactly, the q are the components of a tensor of type 2. For example, if we have a matrix consisting of 3 rows, with 4 elements in each row.
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. A tensor of rank p,q is a multilinear map from a collection of vectors and dual vectors to some field. The same applies to covariant tensors of rank n or mixed tensor of rank n. If a monomial is a product of elements of with elements of, then the tensor is said to. 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. Indeed, a vector is a tensor of rank one, and a scalar is a tensor of rank zero. Contravariant, covariant and physical components of. What demonstrates that the vector is coordinate invariant or basis independent is the fact that covariant and contravariant objects must transform in opposite ways. From this we can deduce the 4 tensor form for the electromagnetic field. 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 dimension of contravariant vector is the inverse of the covariant vector and hence we expect the behaviour of contravariant vector and covariant vector under coordinate. Covariant and contravariant were always one of those complicated concepts that i was running away from. Contravariant is a mathematical term with a precise definition in tensor analysis. Covariant formulation of the tensor algebra of nonlinear elasticity.
An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0. The expansion in orthonormal basis is mathematically simple. Vectors, metric and the connection 1 contravariant and. The covariant derivative of the r component in the q direction is the regular derivative plus another term. Covariant definition of covariant by the free dictionary. This post is intended to give a developerfocused explanation of the terms without diving into the category theory behind them too much. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. Explain how the components of a general covariant tensor may be converted into those of the equivalent contravariant tensor, and vice versa. This blog post is also part of the fp complete haskell syllabus and part of our haskell training. In ndimensional space a tensor of rank n has nn components. It is an idea that seems to originate in physics but in fact it is a more a mathematical concept. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc. Covariant tensor article about covariant tensor by the.
Posted by michael snoyman 09 november, 2016 typeclasses such as bifunctor are often expressed in terms of whether they are covariant or contravariant. Tensors, contravariant and covariant components of a vector. When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. Covariant formulation of the tensor algebra of nonlinear. From this we can deduce the 4tensor form for the electromagnetic field. Explain the difference between a covariant tensor and a contravariant tensor, using the metric tensor as an example. Then the prototypical example of a contravariant vector is a. In physics we normally choose, so we will use this from now on where there are pmany terms and qmany terms. Covariant tensor article about covariant tensor by the free. The downside to this perspective is that you are really missing out on the rather rich mathematical intuition behind what a covariant or contravariant tensor really is and all the language of differential forms and tangent bundles and all that stuff. Having in mind the analogy to coordinate differentials and derivative of a scalar, take a look at this picture, which i think will help to make it clearer. The phrase contravariant tensor of order r is also used to denote a contravariant tensor field of order over a smooth manifold. Since you asked for an intuitive way to understand covariance and contravariance, i think this will do. This is almost completely unintelligible, since the space which contains the.
With the notion of contravariant and covariant components of a vector, we make nonorthogonal basis to behave like orthonormal basis. So, rather than saying contravariant vectors and covariant vectors are both covariant, we can. This metric observable tensor, in real observations where the observer accompanies his references, is the same that the analogous built general covariant tensor h. Chapter 10 vectors and tensors georgia institute of. Physics expressing, exhibiting, or relating to covariant theory.
Writing vector or tensor equations in generalized coordinate systems is a process familiar to students in classical mechanics. The g term accounts for the change in the coordinates. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering. Entangled states and quantum causality threshold in the general theory of relativity. 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. Contravariant, covariant and physical components of tensors. Covariance available in java is useful, to say that one returns a more specific result in the subtype. Wikipedia covariance and contravariance of vectors free download as pdf file. Ill have a crack at answering some of your questions. The full tensor algebra of is the subalgebra of the tensor algebra generated by monomials such that each belongs either to or to. Such a field assigns to each an element of, the fold tensor product of the tangent space to at.
Planetphysicscovariance and contravariance wikiversity. For more information, please see the wikipedia page on covariance and contravariance. Vectors, metric and the connection 1 contravariant and covariant vectors 1. Since whether tensor components are contravariant or covariant, how they are mixed, and the order of operations all impact the results it. But in many physical situations we have to choose an nonorthogonal basis or oblique coordinate system. Contravariant tensors of any rank are derived from the derivative of the coordinate axes covariant tensors of any rank are derived from the gradient which are also related to how the tensors components transform between different coordinate systems. Covariance and contravariance of vectors wikipedia. Here i describe it in an easy way and hope to help other people to free their soul too. Contravariance is a fundamental concept or property within tensor theory and applies to tensors of all ranks over all manifolds. Covariance and contravariance of vectors 3 for some invertible n. Tensors covariant differential and riemann tensor coursera. Lorentz covariance of maxwell equations scalar and vector potentials, and gauge invariance.
Any attempt to remove it to derive a stress tensor will destroy covariance. So, rather than saying contravariant vectors and covariant vectors are both covariant, we can say vectors and oneforms are both lorentz invariant. For the contravariant components it is customary to use superscripts for the indices such as j and k. Covariant and contravariant tensors let be a vector space of dimension with dual space. Kolkers answer gives the gory detail, but heres a quick and dirty version. Tensors this will be a brief summary of what we have already covered as it applies to tensors, plus a little about. What are the physical differences between contravariant. Pdf a guide on tensors is proposed for undergraduate students in physics or engineering that. The contravariant components of a tensor, for example, are obtained by resolving the tensor into components in terms of the socalled coordinate basis vectors. We may play this game in the euclidean space en with its \dot inner product. If n 2 quantities in a coordinate system are related to n 2 other quantities in another coordinate system by the transformation equations or, by our conventions, they are called components of a contravariant tensor of the second rank or of rank two. The terms contravariant and covariant apply based on how they are derived. The contravariant component of any arbitrary vector a is ai with superscript index and covariant component is a i with subscript index are taken to be understood.
Tensor sparse coding for region covariances 727 since the constrain t 7 is nonconvex, a conv ex relaxation of this constrain t involves minimizing the 1 norm of x instead of the 0 pseudonorm. Useful expressions for the contravariant coordinates of ov are, using the sine rule, 4 covariant coordinates the covariant components of a vector are defined by the transformation 5. So, if we have a metric tensor, and we have its inverse tensor with upper indices, where this is a kronecker symbol, which means that it is 1 if mu equals 2r for nz and zero otherwise. Tensors can, in general, have rank greater than 2, and often do. Feb 17, 2010 vector is a physical quantity and it does not depend on any coordinate system.
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. S 3d, 4thorder covariance tensor 3 x3 x3 x3 s 3d, equivalent 2ndorder covariance tensor 6 x6 x t s1 x quadratic function of elements of x, i. Since the space parts form the time component of a four vector, e must be the timespace part of a tensor of rank two. For instance, a covariant vector is a rankone covariant tensor and thus it is also referred to as a oneform. 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. But one day i decided to learn it, and finally, free my soul.
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. Introduction to tensors contravariant and covariant vectors rotation in 2space. Tensors, contravariant and covariant components of a. Pdf covariant energymomentum conservation in general. We wish to write the velocity vector at some time t 0. What lets me transform between different coordinate systems is some matrix we typically refer to it as the jacobian, for the general case. A covariant tensor of rank two is called a twoform, and so on.
The tensor itself is independent of the coordinate system we are using. Differenceconvert between covariantcontravariant tensors. Examples of vectors with contravariant components include the. For every contravariant index there is an a transformation and for every covariant index there is a b. 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. 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. Spectral decomposition of a 4thorder covariance tensor. Electromagnetic theory ii contents special relativity. Jun 24, 2017 this video looks at the ideas of contravariant, covariant and physical components and how to convert the first two component types into physical component values. In cartesian space they must be an orthogonal norm preserving matrix. Vector is a physical quantity and it does not depend on any coordinate system. From wikipedia, the complicated definition the reason for my. Covariance and contravariance of vectors semantic scholar.