Since you asked for an intuitive way to understand covariance and contravariance, i think this will do. Mathematically covariance is a measure of how much things vary together, and contravariance is the opposite of covariance. For more information, please see the wikipedia page on covariance and contravariance. Physics stack exchange is a question and answer site for active researchers, academics and students of physics. A gentle introduction to tensors washington university in. Covariance and contravariance are properties of a mapping function that associates one member of a set with another. This article shows you why they are important concepts for software developers and how you can leverage variance when you design your code. Physicists use it this way, the terms contravariant and covariant in mathematics are defined for functors and not for tensors. When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually.
So that we can get ready for a discussion of special relativity, we need to take up a distinction. A di erent metric will, in general, identify an f 2v with a completely di erent ef 2v. As mentioned in a previous video, tensors are invariant under. Is it foolish to distinguish between covariant and. Chapter 10 vectors and tensors georgia institute of. In this video, i describe the meaning of contravariant and covariant vector components.
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. The tensor product of the covariant and contravariant permutation pseudo tensors is. The difference between these two kinds of tensors is how they transform under a continuous change of coordinates. As adjectives the difference between covariant and invariant is that covariant is category theory of a functor which. To put it succinctly, tensors are geometrical objects over vector spaces, whose coordinates obey certain laws of transformation under change of basis. The discussion of rotations should come before covariance and contravariance. So, rather than saying contravariant vectors and covariant vectors are both covariant, we can say vectors and oneforms are both lorentz invariant. 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. Course notes tensor calculus and differential geometry. An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0. Kolkers answer gives the gory detail, but heres a quick and dirty version. 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 position vector. 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. Tensors, contravariant and covariant components of a. Covariance and contravariance computer science wikipedia. 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. I get that the contravariant has an upper index and the covariant has a lower index, but i dont really have an intuitive feel for how they are different.
Mar 23, 2019 in this video, i describe the meaning of contravariant and covariant vector components. Defining the christoffel connection by rjk xbxjxt the covariant differential can be written and the generalization of the expression 2. One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor. Of course, if you search the web, the descriptions, the mathematics, and the applications of these terms can seem bewildering. Tensors of rank greater than two are easily defined. Now that we have a notion for the difference between the transformation of a vector and the. The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern. In contextcomputingprogramminglangen terms the difference between covariant and invariant is that covariant is computingprogramming using or relating to covariance while invariant is computingprogramming neither covariant nor contravariant. 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. Jul 20, 2017 tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. 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 tensor s components transform between different coordinate systems. It is necessary in order to really understand what they are made of and why they.
What are the physical differences between contravariant and. 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. One part are ordinary vector spaces contravariant, the other dual vector spaces covariant. 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. On a generalization of the index notation for absolute. Now that we have talked about tensors, we need to figure out how to classify them. The contravariant tensor conjugate to gij is denoted by gij satisfy 5. We may play this game in the euclidean space en with its \dot inner product. Relationship between covariantcontravariant basis 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.
The problem here is that physicists often need to use differential geometry for example, for relativity long before they have seen a proper course on differential geometry. 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. Could someone explain, in as much detail as they are willing to give, how the contravariant tensors can be transformed into covariant tensors. As adjectives the difference between covariant and invariant is that covariant is category theory of a functor which preserves composition while invariant is not varying.
The key to the trransformation is having a metric around, i. If you are using cartesian coordinates with a euclidean metric, its difficult to notice the difference between contravariant and covariant mathematically, there is a. There are as many explanations, examples, and mathematical descriptions for covariance and contravariance as you care to explore, but all of this information adds to the confusion. Introduction to tensors contravariant and covariant vectors rotation in 2space. On an elementary level, if a is an m x n matrix then it defines a function r n r m while its transpose matrix a t is n x m and defines a function in the opposite direction r m r n. 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 tensor analysis, a covariant vector varies more or less reciprocally to a corresponding contravariant vector. If we use the symbol a to denote a vector, and a b to denote the inner product between. With the notion of contravariant and covariant components of a vector, we. Ill start with an imprecise and somewhat misleading illustration to motivate the difference between covariance and contravariance. When first learning this material i could apply the formulae but was very confused by what the concepts intuitively meant, and could not discern the difference between the esoteric paths termed covariant and contravariant. Covariance and contravariance in generics microsoft docs. However, in a more thorough and indepth treatment of mechanics, it is. Furthermore a contravariant functor refers to the dual category, which is exactly the opposite of how physicists use it.
Covariance and contravariance of vectors 3 for some invertible n. Prerequisites from linear algebra linear algebra forms the skeleton of tensor calculus and differential geometry. Doubts on covariant and contravariant vectors and on double tensors. Appendix a relations between covariant and contravariant bases. Covariant and contravariant tensors physics forums. The principle of general covariance we want to understand how the laws of physics, beyond those governing freelyfalling particles described by the geodesic equation, adapt to the curvature of spacetime. What are the physical differences between contravariant. Covariance and contravariance describes how a tensor transforms with respect to dilation, not a rotation or other lorentz transformation. Covariance preserves assignment compatibility and contravariance reverses it. The terms contravariant and covariant apply based on how they are derived. Tensors, contravariant and covariant components of a vector. Covariance and contravariance of vectors wikipedia.
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 following code demonstrates the difference between assignment compatibility, covariance, and contravariance. Variance is a topic that comes up fairly often in type systems and can be a bit confusing the first time you hear it. It is an idea that seems to originate in physics but in fact it is a more a mathematical concept. 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. The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern differential geometry uses more sophisticated indexfree methods to represent tensors. Understanding the difference between co and contra variant vectors. The gradient g is an example of a covariant tensor, and the differential position d dx is an example of a contravariant tensor. A contravariant vector is a tangent vector and a covariant vector is technically a linear functional on the collection of tangent vectors a covector. A brief summary of facts about variance in the common language runtime. More compactly, the q are the components of a tensor of type 2. Learning tensors for the first time, but im kind of stuck on the difference between covariant and contravariant this is for a relativity course, if that matters. Covariance and contravariance are collectively referred to as variance.
Net framework 4, variant type parameters are restricted to generic interface and generic. So there is actually no difference between a contravariant and covariant vector, there is a difference between a contravariant and covariant basis as is shown in arxiv. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Planetphysicscovariance and contravariance wikiversity. A disadvan tage of the notation is that, in the study of general properties of tensors, equations such as the transformation relations, the expressions for covariant derivatives, or the integrability condi. Covariant and contravariant bases are dual to one another and are physics nomenclature for constructs that arise in differential geometry.
Vectors are simple and wellknown examples of tensors, but there is much more to tensor theory than vectors. What is the difference between contravariant and covariant tensors, and why do they transform differently under a change of coordinates. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc. As mentioned in a previous video, tensors are invariant under coordinate transformations. In equation 29 components of covariant vector dual covary with change of coordinate basis and hence the name covariant vector. The question is what is the difference between covariance and contravariance. In that spirit we begin our discussion of rank 1 tensors.
I, too, have a concern about the clarity of the difference between covariant and contravariant transformations. A generic type parameter that is not marked covariant or contravariant is referred to as invariant. This extends naturally to the components of higher order tensors. General relativitycontravariant and covariant indices. 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.
Ultimately the distinction between covariance and contravariance comes from the distinction between a vector space and its dual space. These topics are usually encountered in fundamental mathematics courses. An inner product is between elements of the same vector space and not between two vector spaces, it is not how it is defined. 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.
I understand that contravariant tensors are like tangents and covariant are as normal vectors to the surface. In general direction vector like velocity vector is contravariant vector and dual vector like gradient e. A contravariant tensor in other words a vector, transform oppositely contra to the way basis vectors transform, while a covariant tensor or dual vector. Then the prototypical example of a contravariant vector is a. This blog post is also part of the fp complete haskell syllabus and part of our haskell training. A covariant relationship is indicated since the arrows travel in the same direction. Indeed, a vector is a tensor of rank one, and a scalar is a tensor of. This post is intended to give a developerfocused explanation of the terms without diving into the category theory behind them too much. The relationship between v and v is one of mutual duality, and to. Tensors can, in general, have rank greater than 2, and. Here, we abstain from giving an exact definition of even and odd permutations. 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. This means, an interface instance that has method with more derived return types than originally specified covariance or that has methods with less derived parameter types contravariance.
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