To examine the transformation properties of a covariant tensor, first. The definition of contravariant tensor of rankis a simple generalization. Covariance and contravariance of vectors 3 for some invertible n. If w is a covariant vector, and v a contravariant vector, then vw is defined as wv. Covariant definition of covariant by the free dictionary. Covariance and contravariance computer science wikipedia. Vectors and covectors in special relativity jim napolitano march 12, 2010. 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. I read the english and german wikipedia article, and some other blog posts and articles from ibm. Covariant components are labeled with lower indices. 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. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. In ndimensional space a tensor of rank n has nn components.
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. Jul 20, 2017 tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Tensors, contravariant and covariant components of a vector. 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. Covariance preserves assignment compatibility and contravariance reverses it. 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.
Jun 15, 2011 similarly, contravariant vectors can be thought of as scalarvalued linear functions of one covariant vector each, with the following definition. Physics expressing, exhibiting, or relating to covariant theory. Kolkers answer gives the gory detail, but heres a quick and dirty version. Net framework 4, variant type parameters are restricted to generic interface and generic. More compactly, the q are the components of a tensor of type 2. Tensors can, in general, have rank greater than 2, and.
Contravariant and covariant components of a vector in a curvinilinear coordinate system. 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. Covariant and contravariant 4vector in special relativity. In this video, i describe the meaning of contravariant and covariant vector components.
A gentle introduction to tensors washington university in. 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. Covariance and contravariance of vectors semantic scholar. The gradient g is an example of a covariant tensor, and the differential position d dx is an example of a contravariant tensor. 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. General relativitycontravariant and covariant indices. Understanding tensoroperations, covariance, contravariance. Contravariant tensor article about contravariant tensor. One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor. Vectors, metric and the connection 1 contravariant and. Covariance and contravariance of vectors wikipedia.
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. When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. 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. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc.
Mar 23, 2019 in this video, i describe the meaning of contravariant and covariant vector components. I, too, have a concern about the clarity of the difference between covariant and contravariant transformations. Calling convention for overridden methods are covariant with out parameters and return values, and contravariant with normal parameters with the mode in. For instance, a rank2 contravariant tensor has the transformation property. However, sather supported both covariance and contravariance.
A brief summary of facts about variance in the common language runtime. 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. In general, these transformation properties differ from those of a contravariant tensor. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. We are now ready to get serious about electrodynamics. Suppose we are given a system of equations u 1 u 1 x, y, z u 2 u 2 x, y, z. Requirements 1 the derivative of a tensor must be a tensor. Now that we have talked about tensors, we need to figure out how to classify them. Understanding tensoroperations, covariance, contravariance, in the context of special relativity. Covariant derivative 22 mar 2012riemannchristoffel curvature tensor. An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0. All samples ive seen so far was just casting some object into. Chapter 10 vectors and tensors georgia institute of.
We have developed a group of coordinate transformations that preserves. This collection of notes attempts to demystify some of the vagaries of tensor. This procedure naturally generalizes to multilinear maps with different numbers of covariant and contravariant arguments and also shows where the transforms like a tensor bit comes from. Covariance and contravariance of vectors wikimili, the free. Or, in indices, a tangent vector is by definition a vector that is a linear combination of the coordinate partials. A tensor with only covariant indices explanation of covariant tensor. Introduction to tensors contravariant and covariant vectors. Covariance and contravariance in generics microsoft docs. The velocity vector in equation 3 corresponds to neither the covariant nor contravari. The following code demonstrates the difference between assignment compatibility, covariance, and contravariance. 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. Vectors, metric and the connection 1 contravariant and covariant vectors 1. The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation.
A generic type parameter that is not marked covariant or contravariant is referred to as invariant. Covariant tensor article about covariant tensor by the free. As mentioned in a previous video, tensors are invariant under coordinate transformations. 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. Contravariant tensor article about contravariant tensor by. For example, for a tensor of contravariant rank 2 and covariant rank 1. The difference between these two kinds of tensors is how they transform under a continuous change of coordinates. Today, i read some articles about covariance, contravariance and invariance in java. Tell a friend about us, add a link to this page, or visit the webmasters page for free fun content. An informal discussion on tensor calculus chris tiee abstract.
We wish to write the velocity vector at some time t 0. The discussion of rotations should come before covariance and contravariance. Statistics varying with another variable quantity in a manner that leaves a. Covariance and contravariance of vectors wikimili, the free encyclopedia wikimili, the free encyclopedia. Abstract vector is a physical quantity and it does not depend on any coordinate system. A tensor contraction occurs when one of a tensors free covariant indices. In cartesian space they must be an orthogonal norm preserving matrix. The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern.
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