The difference between these two kinds of tensors is how they transform under a continuous change of coordinates. Abstract vector is a physical quantity and it does not depend on any coordinate system. This is almost completely unintelligible, since the space which contains the. The valence or type of a tensor gives the number of covariant and contravariant component indices. Under a rescaling of contravariant coordinates by a factor of k, covariant vectors scale by k. It gives me great pleasure to write the foreword to dr. The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern. 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 corresponding covariant 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. A generic type parameter that is not marked covariant or contravariant is referred to as invariant. Finally, we must write maxwells equations in covariant form. But in many physical situations we have to choose an nonorthogonal basis or oblique coordinate system. Therefore, the levicivita permutation symbols pseudotensor components can be used in expression of the contravariant basis. Is there a notion of a parallel field on a manifold. What we choose to frame physics in terms of is in some sense a choice.
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. Turning to your question about changing frames of reference changing covariant to contravariant, this is not the case. Covariant and contravariant vectors can be interconverted with use of the metric tensor, given by9. Its worth is proportional to the density of noodles. Eiffel also got this wrong, making argument types covariant instead of contravariant. For a given nonorthogonal basis a, b, c, we can always construct an. 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. Intuitively, by a parallel vector field, we mean a vector field with the property that the vectors at different points are parallel. Im reading a quora answer on an intuitive explanation of covariant contravariant components of vectors. To make it more concrete, lets look at which operations in java are defined in terms of the subtype relation. General relativitycontravariant and covariant indices. Physics expressing, exhibiting, or relating to covariant theory. 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. Covariance and contravariance of vectors semantic scholar.
Appendix a relations between covariant and contravariant bases. Let us exemplify covectors and their covariant behavior by introducing func. Unfortunately, terminology is confusing and inconsistent. In equation 29 components of covariant vector dual covary with change of coordinate basis and hence the name covariant vector. 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 kind of inheritance relationship between two parameterized types is known as covariant scala covariance syntax. By explicitly showing this in your derivation, there are a lot of things that i learned. In typescript, argument types are bivariant both covariant and contravariant, which is unsound although now in typescript 2. This is obtained from the basic contravariant field strength tensor by the substitutions. Statistics varying with another variable quantity in a manner that leaves a.
Also this implies that there is the covariant four vector. Mar 23, 2019 in this video, i describe the meaning of contravariant and covariant vector components. Now what bothers me is this lowering and rising index stuff where one can transform a contravariant vector to a covariant one and viceversa by multiplying by. 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. Covariance and contravariance in generics microsoft docs. Using the product rule of derivation, the rate of change of the components v. Overriding method becomes variant with respect to return type. The important thing to observe here is that a is a fixed object that. What is the difference between contravariant and covariant tensors, and why do they transform differently under a change of coordinates.
Covariance and contravariance computer science wikipedia. As mentioned in a previous video, tensors are invariant under coordinate transformations. Vectors, metric and the connection 1 contravariant and. What are the physical differences between contravariant. If the e constitute an orthonormal basis, where g, then the two sets of components covariant and contravariant are numerically coincident. 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. 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. 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. The additivity of the corrections is necessary if the result of a covariant derivative is to be a tensor, since tensors are additive creatures.
Covariant and contravariant type parameter as mutable field type. The following code demonstrates the difference between assignment compatibility, covariance, and contravariance. Change of basis, reciprocal basis vectors, covariant and. In this video, i describe the meaning of contravariant and covariant vector components.
In general the upper indices can be lowered and the lower indices can be made to be upper indices, using the metric tensors. 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. 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. It is an idea that seems to originate in physics but in fact it is a more a mathematical concept.
Change of basis, reciprocal basis vectors, covariant and contravariant components of a vector and metric tensor. What about quantities that are not secondrank covariant tensors. We will discuss these three variances in detail in coming sections. 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. Chapter 10 vectors and tensors georgia institute of. Tk mn rp is a mixed tensor with contravariant rank 4 and covariant rank 2. Because we can convert covariant to contravariant objects with the metric there are many ways to frame a given set of physical laws. Covariant definition of covariant by the free dictionary. Vectors, metric and the connection 1 contravariant and covariant vectors 1. The following code demonstrates the difference between assignment compatibility, covariance, and. If s is subtype of t then lists is is a subtype of listt. Im reading a quora answer on an intuitive explanation of covariantcontravariant components of vectors. We have covariant vector that describes the rate and direction at which the temperature changes. 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.
Pdf covariant and contravariant approaches to topology. This means that they have components that are both covariant and contravariant. 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. With the notion of contravariant and covariant components of a vector, we make nonorthogonal basis to behave like orthonormal basis. A is a fixed object usually called a vector that may be described in various coordinate systems. These two topics will be then combined to write down the formalism for special. One is called the contravariant vector or just the vector, and the other one is called the covariant vector or dual vector or. One is called the contravariant vector or just the vector, and the other one is called the covariant vector or dual vector or onevector. Vectors, contravariant and covariant penn engineering. We now examine the dual or covariant vector counterpart to the contravariant vector. 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.
This is explained in programming in scala odersky2008. In a nonorthogonal basis they will be di erent, and we must take care never to add contravariant components to covariant ones. To make it more concrete, lets look at which operations in java are. 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. Covariance and contravariance in scala kamil korzekwa. In ndimensional space a tensor of rank n has nn components.
This is the main purpose of this article to explore the notion of contravariant and covariant vectors in di. We have a contravariant vector that describes the direction and speed at which we are going. In java, arrays are covariant this was still rather abstract. That is, is string a subtype of object, object a subtype of string, or is neither a subtype of the other. Vector is a physical quantity and it does not depend on any coordinate system. The expansion in orthonormal basis is mathematically simple. Nazrul islams book entitled tensors and their applications. If i have covariant, but multiplying by this, i obtain contravariant vector. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc. 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.
Net framework 4, variant type parameters are restricted to generic interface and generic. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. 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. Tensors covariant differential and riemann tensor coursera. In a metric space, the contravariant and covariant tensors can be transformed to each other using the metric tensors g mn and g mn. 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. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. 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. Kolkers answer gives the gory detail, but heres a quick and dirty version. Covariant differentiation for a contravariant vector consider a vector v v.
Oct 20, 2016 covariant differentiation for a contravariant vector consider a vector v v. Mp denoting the special 3ifyou,oreader,dohavesuchanunderstanding,contactmeimmediately,becauseisureasheckdont. The gradient g is an example of a covariant tensor, and the differential position d dx is an example of a contravariant tensor. This post is a short introdcution into covariant and contravariant coordinates in terms of linear algebra. Now that we have contravariant vectors and covariant vectors, we can do something very interesting and combine them. Then the prototypical example of a contravariant vector is a.
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. Covariance and contravariance are collectively referred to as variance. In cartesian space they must be an orthogonal norm preserving matrix. A brief summary of facts about variance in the common language runtime. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. In general direction vector like velocity vector is contravariant vector and dual vector like gradient e. Feb 17, 2010 vector is a physical quantity and it does not depend on any coordinate system. These and other pictorial examples of visualizing contravariant and covariant vectors are discussed in am. 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. Covariance preserves assignment compatibility and contravariance reverses it.
Covariance and contravariance of vectors wikipedia. I know the author as a research scholar who has worked with me for several years. 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. We wish to write the velocity vector at some time t 0. 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. Notice how the contravariant basis vector g is not differentiated. Obviously, our space coordinates and time variable join to form a contravariant four vector x.
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