Covariant definition of covariant by the free dictionary. I, too, have a concern about the clarity of the difference between covariant and contravariant transformations. Vectors and covectors in special relativity jim napolitano march 12, 2010. Covariance and contravariance of vectors 5 then the frame f is related to the frame f by the inverse of the jacobian matrix of the coordinate transition. Thus contravariant vectors often called simply vectors are the simplest kind of contravariant tensor. 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.
Covariance and contravariance are collectively referred to as variance. This collection of notes attempts to demystify some of the vagaries of tensor. Kolkers answer gives the gory detail, but heres a quick and dirty version. The discussion of rotations should come before covariance and contravariance. Abstract vector is a physical quantity and it does not depend on any coordinate system. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. 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. We are now ready to get serious about electrodynamics. Covariance and contravariance in generics microsoft docs. Introduction to tensors contravariant and covariant vectors. The definition of contravariant tensor of rankis a simple generalization. The difference between these two kinds of tensors is how they transform under a continuous change of coordinates.
Covariance and contravariance of vectors wikipedia. Covariance and contravariance describes how a tensor transforms with respect to dilation, not a rotation or other lorentz transformation. Covariance and contravariance computer science wikipedia. Inasmuch as these notes may help others, the reader is free to use, distribute, and modify. Contravariant tensor article about contravariant tensor by. In this video, i describe the meaning of contravariant and covariant vector components. Covariance and contravariance of vectors wikimili, the free encyclopedia wikimili, the free encyclopedia. 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. A tensor with only covariant indices explanation of covariant tensor. Now that we have talked about tensors, we need to figure out how to classify them. Jul 20, 2017 tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Contravariant tensor article about contravariant tensor. If w is a covariant vector, and v a contravariant vector, then vw is defined as wv. Covariant and contravariant 4vector in special relativity.
An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. Covariance and contravariance of vectors 3 for some invertible n. In coordinatefree language, this means that an orientation on m can be specified by. Or, in indices, a tangent vector is by definition a vector that is a linear combination of the coordinate partials. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc. For instance, a rank2 contravariant tensor has the transformation property. More compactly, the q are the components of a tensor of type 2. Requirements 1 the derivative of a tensor must be a tensor. Chapter 10 vectors and tensors georgia institute of. When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. Understanding tensoroperations, covariance, contravariance. 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. 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 or quadratic forms should be understood as being elements of these spaces.
Covariance and contravariance of vectors wikimili, the free. The second is a brief introduction to coordinate free geometry, which necessitates a discussion of contravariant and covariant vectors. Tensors, contravariant and covariant components of a vector. What are the physical differences between contravariant and. However, sather supported both covariance and contravariance. The following code demonstrates the difference between assignment compatibility, covariance, and contravariance. Net framework 4, variant type parameters are restricted to generic interface and generic. In order to change a contravariant vector into a oneform or viseversa so that we can compute the inner product, we use the metric tensor as a lowering or raising operator. Suppose we are given a system of equations u 1 u 1 x, y, z u 2 u 2 x, y, z. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Physics expressing, exhibiting, or relating to covariant theory. General relativitycontravariant and covariant indices.
For other uses of covariant or contravariant, see covariance and contravariance in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis from one coordinate system to another. In cartesian space they must be an orthogonal norm preserving matrix. Jun 15, 2011 similarly, contravariant vectors can be thought of as scalarvalued linear functions of one covariant vector each, with the following definition. A tensor contraction occurs when one of a tensors free covariant indices. Covariance preserves assignment compatibility and contravariance reverses it. Covariant derivative 22 mar 2012riemannchristoffel curvature tensor. A guide on tensors is proposed for undergraduate students in physics or engineering that ties directly to vector calculus in orthonormal coordinate systems. The gradient g is an example of a covariant tensor, and the differential position d dx is an example of a contravariant tensor. In general, these transformation properties differ from those of a contravariant tensor. Calling convention for overridden methods are covariant with out parameters and return values, and contravariant with normal parameters with the mode in.
Vectors, metric and the connection 1 contravariant and. This collection of notes attempts to demystify some of the vagaries of tensor analysis, to emphasize the connections between the old way of doing things and the new. Contravariant and covariant components of a vector in a curvinilinear coordinate system. Vectors, metric and the connection 1 contravariant and covariant vectors 1. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern. This material is covered thoroughly in chapters one and seven in our textbook classical electrodynamics, 2nd ed. Covariant and contravariant tensors physics forums. In ndimensional space a tensor of rank n has nn 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 the minkowski metric tensor in the special relativity case. Covariant tensor article about covariant tensor by the free.
Tell a friend about us, add a link to this page, or visit the webmasters page for free fun content. Statistics varying with another variable quantity in a manner that leaves a. A gentle introduction to tensors washington university in. The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. An informal discussion on tensor calculus chris tiee abstract.
The velocity vector in equation 3 corresponds to neither the covariant nor contravari. Today, i read some articles about covariance, contravariance and invariance in java. We have developed a beautiful, geometric system for describing the coordinates in terms of which electrodynamics must be formulated for the speed of light to be an invariant. All samples ive seen so far was just casting some object into. Covariant components are labeled with lower indices. One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor. We wish to write the velocity vector at some time t 0.
As mentioned in a previous video, tensors are invariant under coordinate transformations. For example, for a tensor of contravariant rank 2 and covariant rank 1. To examine the transformation properties of a covariant tensor, first. Vectors and covectors in special relativity jim napolitano march 12, 2010 these notes are meant to accompany the course electromagnetic theory for the spring 2010 term at rpi. Understanding tensoroperations, covariance, contravariance, in the context of special relativity. We have developed a group of coordinate transformations that preserves. A generic type parameter that is not marked covariant or contravariant is referred to as invariant. I read the english and german wikipedia article, and some other blog posts and articles from ibm. Tensors can, in general, have rank greater than 2, and.
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