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