What about quantities that are not secondrank covariant tensors. I know the author as a research scholar who has worked with me for several years. These two topics will be then combined to write down the formalism for special. 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. Eiffel also got this wrong, making argument types covariant instead of contravariant. Covariance and contravariance in scala kamil korzekwa. To make it more concrete, lets look at which operations in java are defined in terms of the subtype relation.
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. These and other pictorial examples of visualizing contravariant and covariant vectors are discussed in am. In typescript, argument types are bivariant both covariant and contravariant, which is unsound although now in typescript 2. One is called the contravariant vector or just the vector, and the other one is called the covariant vector or dual vector or onevector. By explicitly showing this in your derivation, there are a lot of things that i learned. In general, a covariant type parameter can be used as the return type of a delegate, and contravariant type parameters can be used as parameter types. With the notion of contravariant and covariant components of a vector, we make nonorthogonal basis to behave like orthonormal basis. Covariance and contravariance of vectors wikipedia. But it is also customary to call the collection of coefficients a i a vector more specifically, a contravariant vector, while we have already called. Im reading a quora answer on an intuitive explanation of covariantcontravariant components of vectors. In java, arrays are covariant this was still rather abstract. Abstract vector is a physical quantity and it does not depend on any coordinate system. Overriding method becomes variant with respect to return type. Under a rescaling of contravariant coordinates by a factor of k, covariant vectors scale by k.
The following code demonstrates the difference between assignment compatibility, covariance, and. If the e constitute an orthonormal basis, where g, then the two sets of components covariant and contravariant are numerically coincident. Turning to your question about changing frames of reference changing covariant to contravariant, this is not the case. Obviously, our space coordinates and time variable join to form a contravariant four vector x. 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. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. But in many physical situations we have to choose an nonorthogonal basis or oblique coordinate system.
One is called the contravariant vector or just the vector, and the other one is called the covariant vector or dual vector or. This is the main purpose of this article to explore the notion of contravariant and covariant vectors in di. A generic type parameter that is not marked covariant or contravariant is referred to as invariant. Now that we have contravariant vectors and covariant vectors, we can do something very interesting and combine them. If s is subtype of t then lists is is a subtype of listt. Vector is a physical quantity and it does not depend on any coordinate system. The valence or type of a tensor gives the number of covariant and contravariant component indices. Covariance and contravariance are collectively referred to as variance.
Chapter 10 vectors and tensors georgia institute of. Covariance and contravariance in generics microsoft docs. Im reading a quora answer on an intuitive explanation of covariant contravariant components of vectors. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. In cartesian space they must be an orthogonal norm preserving matrix. We now examine the dual or covariant vector counterpart to the contravariant vector. Covariant and contravariant approaches to topology article pdf available in international journal of mathematics and mathematical sciences 204 january 1997 with 35 reads how we measure reads. The gradient g is an example of a covariant tensor, and the differential position d dx is an example of a contravariant tensor. Change of basis, reciprocal basis vectors, covariant and. Covariance and contravariance of vectors semantic scholar. The difference between these two kinds of tensors is how they transform under a continuous change of coordinates. In equation 29 components of covariant vector dual covary with change of coordinate basis and hence the name covariant vector. Mar 31, 2020 the additivity of the corrections is necessary if the result of a covariant derivative is to be a tensor, since tensors are additive creatures.
Oct 20, 2016 covariant differentiation for a contravariant vector consider a vector v v. There are lots of articles, all over the place about the distinction between covariant vectors and contravariant vectors after struggling through many of them, i think im starting to get the idea. This is obtained from the basic contravariant field strength tensor by the substitutions. Covariant differentiation for a contravariant vector consider a vector v v. Physics expressing, exhibiting, or relating to covariant theory. Nazrul islams book entitled tensors and their applications. In general the upper indices can be lowered and the lower indices can be made to be upper indices, using the metric tensors. 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. Vectors, metric and the connection 1 contravariant and. Notice how the contravariant basis vector g is not differentiated. 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. Covariance and contravariance computer science wikipedia. What we choose to frame physics in terms of is in some sense a choice.
The additivity of the corrections is necessary if the result of a covariant derivative is to be a tensor, since tensors are additive creatures. This is explained in programming in scala odersky2008. What is the difference between contravariant and covariant tensors, and why do they transform differently under a change of coordinates. Feb 17, 2010 vector is a physical quantity and it does not depend on any coordinate system. Covariant definition of covariant by the free dictionary. This post is a short introdcution into covariant and contravariant coordinates in terms of linear algebra. First of, i have read many explanations on so and blogs about covariance and contravariance and a big thanks goes out to eric lippert for producing such a great series on covariance and contravariance however i have a more specific question that i am trying to get my head around a little bit. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. Using the product rule of derivation, the rate of change of the components v. 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. Net framework 4, variant type parameters are restricted to generic interface and generic.
Because we can convert covariant to contravariant objects with the metric there are many ways to frame a given set of physical laws. Therefore, the levicivita permutation symbols pseudotensor components can be used in expression of the contravariant basis. The following code demonstrates the difference between assignment compatibility, covariance, and contravariance. Covariance preserves assignment compatibility and contravariance reverses it. To make it more concrete, lets look at which operations in java are.
That is, is string a subtype of object, object a subtype of string, or is neither a subtype of the other. 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. Vectors, metric and the connection 1 contravariant and covariant vectors 1. A brief summary of facts about variance in the common language runtime. In ndimensional space a tensor of rank n has nn components. Tensors covariant differential and riemann tensor coursera. General relativitycontravariant and covariant indices. The important thing to observe here is that a is a fixed object that. Unfortunately, terminology is confusing and inconsistent.
Statistics varying with another variable quantity in a manner that leaves a. 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 components that are both covariant and contravariant. In a metric space, the contravariant and covariant tensors can be transformed to each other using the metric tensors g mn and g mn.
This book is a humble step of efforts made by him to prove him to be a dedicated and striving teacher who has worked relentlessly in this field. In this video, i describe the meaning of contravariant and covariant vector components. Let us exemplify covectors and their covariant behavior by introducing func. Softdrink is the common supertype of both tonicwater and cola, so the method addall above returns instance of vendingmachine of type softdrink covariant and contravariant type parameter as mutable field type. If we have a coordinate system with straight coordinate axes, the geometric explanation given is that a vectors covariant components in such a system will be perpendicular projections on the axes, whereas its contravariant components will be parallel projections.
Pdf covariant and contravariant approaches to topology. What are the physical differences between contravariant. We have a contravariant vector that describes the direction and speed at which we are going. We wish to write the velocity vector at some time t 0.
The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern. As always in physics, this notion and the way to talk about it is obscured and the connection to a proper mathematical treatment is left out. Mar 23, 2019 in this video, i describe the meaning of contravariant and covariant vector components. A is a fixed object usually called a vector that may be described in various coordinate systems. Intuitively, by a parallel vector field, we mean a vector field with the property that the vectors at different points are parallel. What are the physical differences between contravariant and. The terms covariant and contravariant were introduced by james joseph sylvester in 1851 in the context of associated algebraic forms theory.
Change of basis, reciprocal basis vectors, covariant and contravariant components of a vector and metric tensor. Appendix a relations between covariant and contravariant bases. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc. We have covariant vector that describes the rate and direction at which the temperature changes. Vectors, contravariant and covariant penn engineering. For an interface, covariant type parameters can be used as the return types of the interfaces methods, and contravariant type parameters can be used as the parameter types of the interfaces methods. We will discuss these three variances in detail in coming sections. 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. As mentioned in a previous video, tensors are invariant under coordinate transformations. Tk mn rp is a mixed tensor with contravariant rank 4 and covariant rank 2. In a nonorthogonal basis they will be di erent, and we must take care never to add contravariant components to covariant ones. Kolkers answer gives the gory detail, but heres a quick and dirty version. In general direction vector like velocity vector is contravariant vector and dual vector like gradient e. Covariant and contravariant vectors can be interconverted with use of the metric tensor, given by9.
Covariant and contravariant type parameter as mutable field type. Its worth is proportional to the density of noodles. Mp denoting the special 3ifyou,oreader,dohavesuchanunderstanding,contactmeimmediately,becauseisureasheckdont. 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. The basis for these names will be explained in the next section, but at this stage it is just a name used to distinguish two types of vector. Also this implies that there is the covariant four vector.
If i have covariant, but multiplying by this, i obtain contravariant vector. The expansion in orthonormal basis is mathematically simple. It gives me great pleasure to write the foreword to dr. So a covariant vector transforms in such a way that the transformation matrix is the transpose of the inverse of the lt transformation matrix for a contravariant vector. For a given nonorthogonal basis a, b, c, we can always construct an. Then the prototypical example of a contravariant vector is a.