It is an idea that seems to originate in physics but in fact it is a more a mathematical concept. This means that they have both covariant and contravariant components, or both vector and covector components. 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. Its worth noting that, since x u g uv x u, we have. Abstract vector is a physical quantity and it does not depend on any coordinate system. More generally, let be a commutative ring with unit element and a unitary module over. The terms contravariant and covariant apply based on how they are derived. Transformation laws and their relation to fancyschmancy tensors 17 2. Covariant and contravariant indices can be used simultaneously in a mixed. 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. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components.
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. This means that they have both covariant and contravariant components, or both vector and dual vector components. Most texts on upstairsdownstairs notation seem to either not address the signs of the entries of mixed tensors at all, or just address the sign convention for vectors and take mixed tensors as granted and proceed. Aug 06, 2019 one part are ordinary vector spaces contravariant, the other dual vector spaces covariant. Covariant and contravariant tensors physics forums. Although tensors are applied in a very broad range of physics and math ematics, this. 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. 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. Covariance and contravariance computer science wikipedia. Since whether tensor components are contravariant or covariant, how they are mixed, and the order of operations all impact the results it is imperative to track for correct application of methods. I just was reading a qft book where the author mixed up the transformation laws.
What is the physical meaning of a contravariant and a. If both r and s are nonzero then its called a mixed tensor. General relativitycontravariant and covariant indices. First we will explain the distinction between the covariant and contravariant components of vectors, thinking of vectorfields where a vector is defined at a. Covariance and contravariance fabulous adventures in coding. 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. I have already answered in detail a number of questions. We define three main types of tensorscovariant tensors, contravariant tensors, and mixed tensors based on their behavior and properties with respect to general coordinate transformations. One part are ordinary vector spaces contravariant, the other dual vector spaces covariant. Covariance and contravariance are collectively referred to as variance.
Vectors, metric and the connection 1 contravariant and. These coordinates are the same in all coordinate systems. This book has been presented in such a clear and easy way that the students will have no difficulty in understanding it. What demonstrates that the vector is coordinate invariant or basis independent is the fact that covariant and contravariant objects must transform in opposite ways. Tensors consider an ndimensional space with arbitrary geometry and an object with components in the coord system and in the coord system. When we derived the condition for a contravariant tensor, we considered. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. Note our care in 17 and 18 so that when we mix contravariant and covariant indices, it is.
Covariance and contravariance in generics microsoft docs. The velocity vector in equation 3 corresponds to neither the covariant nor contravari. Planetphysicscovariance and contravariance wikiversity. For example, the tensor, a mixed tensor of rank 5, contravariant of order 3 and covariant of order 2, is defined by the following relation note the pattern of the indices in the successive fractions. The distinction between covariance and contravariance is particularly important for computations with tensors, which often have mixed variance. The vector space or linear space, mve4 space, or just space of all kcontravariant, covariant tensors tensors of valence k at the point p in a manifold m will be denoted tk mp, with tmp and t. Covariance preserves assignment compatibility and contravariance reverses it. Ill have a crack at answering some of your questions. Tensors of rank greater than two are easily defined.
When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. So when stating that a quan tity is a tensor one should add with respect to name transformation. For this reason contravariant tensors are sometimes directly defined as multilinear functionals on. A brief summary of facts about variance in the common language runtime. For other uses of covariant or contravariant, see covariance and. Contravariant tensor covariant tensor mixed tensor. The covariant derivative of the r component in the r direction is the regular derivative.
The precise rule 11 12 is that the return types of all methods in the interface must be valid covariantly and all the method parameter types. Tk mn rp is a mixed tensor with contravariant rank 4 and covariant rank 2. Net framework 4, variant type parameters are restricted to generic interface and generic. Covariance generic with parameterized type decorated with out covariance is widely used with immutable collections i. Tensors can, in general, have rank greater than 2, and often do. Converting contravariant or covariant tensors to mixed tensors. Jul 21, 2017 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. Higher order tensors are in principle handled similarly, but they may be expressed with mixed coordinate types i. The metric tensor is g ij and is most easily understood when represented by a square matrix. However, upon deeper reflection that makes perfect sense. Now that we have talked about tensors, we need to figure out how to classify them. If we use the symbol a to denote a vector, and a b to denote the inner product between. Tensors this will be a brief summary of what we have already covered as it applies to tensors, plus a little about.
Covariance and contravariance of vectors wikimili, the best. What lets me transform between different coordinate systems is some matrix we typically refer to it as the jacobian. 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. In the lexicon of category theory, covariance and contravariance are properties of functors. In more modern terms, the transformation properties of the covariant indices of a tensor are given by a pullback. Vectors, metric and the connection 1 contravariant and covariant vectors 1. This blog post is also part of the fp complete haskell syllabus and part of our haskell training. A generic type parameter that is not marked covariant or contravariant is referred to as invariant. At this point people often wonder how we can talk about a vector being contravariant or covariant when the direction and magnitude of a vector which are its defining properties are actually invariant with. Without the use of tensors, a vector is an object that lives in a space with particular structures imposed upon it addition and scalar multiplication. Now, going back to our laymens picture of a vector, namely an arrow pointing from some defined origin, we begin to tackle your question.
The g term accounts for the change in the coordinates. Tensors are objects in multilinear algebra that can have aspects of both covariance and contravariance. Indeed, a vector is a tensor of rank one, and a scalar is a tensor of. Jul 29, 20 this illustrates a subtle and slightly unfortunate design choice in the method type inference algorithm, which of course was designed long before covariance and contravariance were added to the language. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. Covariant and contravariant indices can be used simultaneously in a mixed tensor see also covariant tensor, fourvector, lorentz tensor, metric tensor, mixed tensor, tensor.
Jul 20, 2017 tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. We can also have mixed tensors, which are covariant with respect to some of their indices and contravariant with respect to others. Tensors, contravariant and covariant components of a vector. Physicists use it this way, the terms contravariant and covariant in mathematics are defined for functors and not for tensors. A mixed tensor of type or valence, also written type m, n, with both m 0 and n 0, is a tensor which has m contravariant indices and n covariant indices. Tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update. For more information, please see the wikipedia page on covariance and contravariance. The contravariant basis vector gk of the curvilinear coordinate of uk at the point. That is, a parameter that was declared covariant must not occur in any contravariant positions where a position is contravariant if it occurs under an odd number of contravariant type constructors. Covariance and contravariance of vectors wikipedia. In the process of contraction one contravariant index and one covariant index of a mixed tensor are set equal and the repeated index is summed over, the result is a tensor of. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering. In this chapter we introduce the concept of tensors using the system notation and definition of metric spaces.
The covariant derivative of the r component in the q direction is the regular derivative plus another term. Linear functionals, volume forms, and rethinking integration 15 2. A basic explanation of the covariance and contravariance of vectors and of tensors too, since vectors are tensors of rank math1math is best done with the help of a geometric representation or illustration. Covariance and contravariance of vectors wikimili, the. The following code demonstrates the difference between assignment compatibility, covariance, and contravariance. The algebraic operation by which the rank of a mixed tensor is lowered by 2 is known as contraction.
Converting contravariant or covariant tensors to mixed. These operations are called raising and lowering of indices, because they convert x from a superscripted to a subscripted variable, or vice versa. Icomparable at first glance you might ask yourself why this is not a circular definition. Feb 06, 2018 this topic or question is basically all related to the transformation rules followed by the components of a vector which is a tensor of rank 1 or a tensor in different coordinate systems. Then the elements of the fold tensor product are called contravariant tensors or contravariant tensors of valency or order. This post is intended to give a developerfocused explanation of the terms without diving into the category theory behind them too much. Tensors can, in general, have rank greater than 2, and. A di erent metric will, in general, identify an f 2v with a completely di erent ef 2v.
This topic or question is basically all related to the transformation rules followed by the components of a vector which is a tensor of rank 1 or a tensor in different coordinate systems. To examine the transformation properties of a covariant tensor, first consider the. Im working on the electromagnetic stressenergy tensor and ive found this in a book by landaulifshitz. Appendix a relations between covariant and contravariant bases. 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. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. Chapter 10 vectors and tensors georgia institute of. Furthermore a contravariant functor refers to the dual category, which is exactly the opposite of how physicists use it. Tensors, contravariant and covariant components of a. The same applies to covariant tensors of rank n or mixed tensor of rank n.
The mixed tensors can be written in the unitary covariant bases using eqs. In ndimensional space a tensor of rank n has nn components. We may play this game in the euclidean space en with its \dot inner product. One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor. The contravariant components of a vector are obtained by projecting onto the coordinate axes. Variance is a topic that comes up fairly often in type systems and can be a bit confusing the first time you hear it. 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. We wish to write the velocity vector at some time t 0. Covariance and contravariance of vectors semantic scholar. 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. Some constructions of multilinear algebra are of mixed variance, which prevents them from being.
295 742 1144 1142 458 541 709 563 1033 140 1059 691 279 754 1199 1501 1463 223 444 707 1598 988 1457 437 1572 1026 1558 1398 1375 467 1317 1023 982 254 127 151 1324 1422