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. One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor. Linear functionals, volume forms, and rethinking integration 15 2. 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. Im working on the electromagnetic stressenergy tensor and ive found this in a book by landaulifshitz. Contravariant tensor covariant tensor mixed tensor. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering.
Vectors, metric and the connection 1 contravariant and. In ndimensional space a tensor of rank n has nn components. Tensors can, in general, have rank greater than 2, and. 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. This blog post is also part of the fp complete haskell syllabus and part of our haskell training. When we derived the condition for a contravariant tensor, we considered. A generic type parameter that is not marked covariant or contravariant is referred to as invariant. Covariance and contravariance computer science wikipedia. Tensors can, in general, have rank greater than 2, and often do. In more modern terms, the transformation properties of the covariant indices of a tensor are given by a pullback. 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 and contravariance are collectively referred to as variance. Covariance and contravariance of vectors wikimili, the best. To examine the transformation properties of a covariant tensor, first consider the. 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. Appendix a relations between covariant and contravariant bases. Although tensors are applied in a very broad range of physics and math ematics, this. The contravariant components of a vector are obtained by projecting onto the coordinate axes. One part are ordinary vector spaces contravariant, the other dual vector spaces covariant. 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. 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.
Indeed, a vector is a tensor of rank one, and a scalar is a tensor of. Jul 20, 2017 tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Chapter 10 vectors and tensors georgia institute of. The covariant derivative of the r component in the r direction is the regular derivative. What lets me transform between different coordinate systems is some matrix we typically refer to it as the jacobian. Converting contravariant or covariant tensors to mixed tensors. Vectors, metric and the connection 1 contravariant and covariant vectors 1.
Covariance and contravariance in generics microsoft docs. 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 g term accounts for the change in the coordinates. Physicists use it this way, the terms contravariant and covariant in mathematics are defined for functors and not for tensors. The same applies to covariant tensors of rank n or mixed tensor of rank n.
When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. Converting contravariant or covariant tensors to mixed. The covariant derivative of the r component in the q direction is the regular derivative plus another term. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. For more information, please see the wikipedia page on covariance and contravariance. In cartesian space they must be an orthogonal norm preserving matrix. More generally, let be a commutative ring with unit element and a unitary module over. Covariant and contravariant indices can be used simultaneously in a mixed. If we use the symbol a to denote a vector, and a b to denote the inner product between. The velocity vector in equation 3 corresponds to neither the covariant nor contravari. Tensors this will be a brief summary of what we have already covered as it applies to tensors, plus a little about. In the lexicon of category theory, covariance and contravariance are properties of functors.
Abstract vector is a physical quantity and it does not depend on any coordinate system. Now, going back to our laymens picture of a vector, namely an arrow pointing from some defined origin, we begin to tackle your question. So when stating that a quan tity is a tensor one should add with respect to name transformation. What is the physical meaning of a contravariant and a. We can also have mixed tensors, which are covariant with respect to some of their indices and contravariant with respect to others. Now that we have talked about tensors, we need to figure out how to classify them. Covariant and contravariant indices can be used simultaneously in a mixed tensor see also covariant tensor, fourvector, lorentz tensor, metric tensor, mixed tensor, tensor.
Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. This means that they have both covariant and contravariant components, or both vector and dual vector components. These coordinates are the same in all coordinate systems. It is an idea that seems to originate in physics but in fact it is a more a mathematical concept. We may play this game in the euclidean space en with its \dot inner product. The algebraic operation by which the rank of a mixed tensor is lowered by 2 is known as contraction. Some constructions of multilinear algebra are of mixed variance, which prevents them from being.
First we will explain the distinction between the covariant and contravariant components of vectors, thinking of vectorfields where a vector is defined at a. 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 mixed tensors can be written in the unitary covariant bases using eqs. Aug 06, 2019 one part are ordinary vector spaces contravariant, the other dual vector spaces covariant. Its worth noting that, since x u g uv x u, we have. A di erent metric will, in general, identify an f 2v with a completely di erent ef 2v.
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. An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0. Ill have a crack at answering some of your questions. 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.
Covariance generic with parameterized type decorated with out covariance is widely used with immutable collections i. The terms contravariant and covariant apply based on how they are derived. 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. 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. Higher order tensors are in principle handled similarly, but they may be expressed with mixed coordinate types i. Furthermore a contravariant functor refers to the dual category, which is exactly the opposite of how physicists use it. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. 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. The distinction between covariance and contravariance is particularly important for computations with tensors, which often have mixed variance. Icomparable at first glance you might ask yourself why this is not a circular definition. Note our care in 17 and 18 so that when we mix contravariant and covariant indices, it is.
A brief summary of facts about variance in the common language runtime. 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. Transformation laws and their relation to fancyschmancy tensors 17 2. What demonstrates that the vector is coordinate invariant or basis independent is the fact that covariant and contravariant objects must transform in opposite ways. 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. This book has been presented in such a clear and easy way that the students will have no difficulty in understanding it. Variance is a topic that comes up fairly often in type systems and can be a bit confusing the first time you hear it. Covariance and contravariance of vectors wikipedia. For other uses of covariant or contravariant, see covariance and. Tensors are objects in multilinear algebra that can have aspects of both covariance and contravariance. The contravariant basis vector gk of the curvilinear coordinate of uk at the point. 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 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.
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. 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. Tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update. I just was reading a qft book where the author mixed up the transformation laws. This means that they have both covariant and contravariant components, or both vector and covector components.
Then the elements of the fold tensor product are called contravariant tensors or contravariant tensors of valency or order. However, upon deeper reflection that makes perfect sense. For this reason contravariant tensors are sometimes directly defined as multilinear functionals on. In this chapter we introduce the concept of tensors using the system notation and definition of metric spaces.
If both r and s are nonzero then its called a mixed tensor. Covariant and contravariant tensors physics forums. 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. This post is intended to give a developerfocused explanation of the terms without diving into the category theory behind them too much. 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. We wish to write the velocity vector at some time t 0. 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. Covariance preserves assignment compatibility and contravariance reverses it. Tensors consider an ndimensional space with arbitrary geometry and an object with components in the coord system and in the coord system. Covariance and contravariance of vectors wikimili, the.
Tk mn rp is a mixed tensor with contravariant rank 4 and covariant rank 2. 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. Planetphysicscovariance and contravariance wikiversity. General relativitycontravariant and covariant indices. Net framework 4, variant type parameters are restricted to generic interface and generic. 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. The metric tensor is g ij and is most easily understood when represented by a square matrix. I have already answered in detail a number of questions. These operations are called raising and lowering of indices, because they convert x from a superscripted to a subscripted variable, or vice versa. Tensors, contravariant and covariant components of a vector. Covariance and contravariance of vectors semantic scholar. The following code demonstrates the difference between assignment compatibility, covariance, and contravariance. Tensors, contravariant and covariant components of a.