This post is intended to give a developerfocused explanation of the terms without diving into the category theory behind them too much. What different between covariant metric tensor and. Jul 20, 2017 tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Invariance, covariance and contravariance explained before we go further, lets examine what invariant, covariant, and contravariant parameters and return types mean. Vectors, metric and the connection 1 contravariant and covariant vectors 1. For instance, a covariant vector is a rankone covariant tensor and thus it is also referred to as a oneform. From wikipedia, the complicated definition the reason for my. Spectral decomposition of a 4thorder covariance tensor. In quantum computing and condensed matter, there are an important class of states called tensor network states, which have amplitudes that are obtained by contracting tensors. This extends naturally to the components of higher order tensors. Now, the only question i have left is, what is gained by talking about contravariance and covariance, rather than just vector spaces and their dual spaces. 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. Pdf a guide on tensors is proposed for undergraduate students in physics or engineering that.
Or, in indices, a tangent vector is by definition a vector that is a linear combination of the coordinate partials. After spending some time looking at tensors, we can now expose the problem of how to differentiate a tensor. Jan 04, 2007 dear fellows, do any one have an idea of whether there must be a system tensor in order to be able to transform from the covariant form of a certain tensor to its contravariant one. Metrinilic property covariant metric tensor covariant tensor. For more information, please see the wikipedia page on covariance and contravariance. A covariant tensor of rank two is called a twoform, and so on. Vectors are simple and wellknown examples of tensors, but there is much more to tensor theory than vectors. A guide on tensors is proposed for undergraduate students in physics or engineering that ties directly to vector calculus in orthonormal coordinate systems. The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. Einstein relatively easy introduction to covariant. For example, for a tensor of contravariant rank 2 and covariant rank 1. This post is a short introdcution into covariant and contravariant coordinates in terms of linear algebra. In this video, i describe the meaning of contravariant and covariant vector components. 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.
In ndimensional space a tensor of rank n has nn components. I know the author as a research scholar who has worked with me for several years. Transformation of contravariant covariant components. Combine multiple words with dashes, and seperate tags with spaces. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering. Conversion of mixed tensors into mixed tensors and into covariant or contravariant ones. Harris, in mathematics for physical science and engineering, 2014. Tensors covariant differential and riemann tensor coursera. First we will explain the distinction between the covariant and contravariant components of vectors, thinking of vectorfields where a vector is defined at a. If you have a metric texgtex on a manifold then it is usually regarded as being a map which takes two vectors into a real number. Planetphysicscovariance and contravariance wikiversity.
It is an idea that seems to originate in physics but in fact it is a more a mathematical concept. Covariant and contravariant were always one of those complicated concepts that i was running away from. Tensors, contravariant and covariant components of a vector. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. Confusions about covariant and contravariant vectors. In cartesian space they must be an orthogonal norm preserving matrix. With the notion of contravariant and covariant components of a vector, we make nonorthogonal basis to behave like orthonormal basis. Tensors this will be a brief summary of what we have already covered as it applies to tensors, plus a little about. I know the author as a research scholar who has worked with me.
Sep 10, 2010 the title says it all, basically im trying to figure out what the difference is between the two tensors levicivita that are 3rd rank. Tensors this will be a brief summary of what we have already covered as it applies to tensors, plus a little about tensors in general. In this section, the concept of contravariant and covariant vectors is extended to tensors. To put it succinctly, tensors are geometrical objects over vector spaces, whose coordinates obey certain laws of transformation under change of basis. But one day i decided to learn it, and finally, free my soul. Kolkers answer gives the gory detail, but heres a quick and dirty version. So, rather than saying contravariant vectors and covariant vectors are both covariant, we can say vectors and oneforms are both lorentz invariant. In the video it is said that the contravariant vectors are the vectors that i get through adding together the basevectors each basevector multiplied by a certain amount.
In this context the covariant contravariant distinction is unimportant, because the way that the tensors transform under a change of basis just follows from the usual. Then the elements of the fold tensor product are called contravariant tensors or contravariant tensors of valency or order. A gentle introduction to tensors boaz porat department of electrical engineering technion israel institute of technology. Today, i read some articles about covariance, contravariance and invariance in java. Writing vector or tensor equations in generalized coordinate systems is a process familiar to students in classical mechanics. 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. Covariance, in probability theory and statistics, the measure of. Vectors, metric and the connection 1 contravariant and.
Tk mn rp is a mixed tensor with contravariant rank 4 and covariant rank 2. We wish to write the velocity vector at some time t 0. Covariance and contravariance of functors, in category theory. This blog post is also part of the fp complete haskell syllabus and part of our haskell training. Chapter 10 vectors and tensors georgia institute of. Then the prototypical example of a contravariant vector is a. Here i describe it in an easy way and hope to help other people to free their soul too. What is the difference between contravariant and covariant tensors, and why do they transform differently under a change of coordinates. They find a transformation matrix from the contravariant components of a vector, to the covariant components. Converting contravariant or covariant tensors to mixed tensors. Jun 24, 2017 this video looks at the ideas of contravariant, covariant and physical components and how to convert the first two component types into physical component values. The terms contravariant and covariant apply based on how they are derived.
A gentle introduction to tensors washington university in. I read the english and german wikipedia article, and some other blog posts and articles from ibm. 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. An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0. Essentially, there is no difference between the covariant and contravariant forms of the metric in the sense that they both measure things. Skip to main content this banner text can have markup. It gives me great pleasure to write the foreword to dr. More compactly, the q are the components of a tensor of type 2. Requirements 1 the derivative of a tensor must be a tensor. More generally, let be a commutative ring with unit element and a unitary module over. This is by construction, the notion of covariance and contravariance comes when we ask ourselves the question. For this reason contravariant tensors are sometimes directly defined as multilinear functionals on. And does covariance and contravariance of vectors have any relation to the contravariant functor mapping a vector space v to its dual v.
Covariance and contravariance of vectors semantic scholar. For a given coordinate system, there exist two uniquely determined sets of base vector systems one is the covariant and another is the contravariant base vector system. Covariance and contravariance of vectors, in mathematics and theoretical physics. Tensors are objects in multilinear algebra that can have aspects of both covariance and contravariance. Reciprocal base system covariant and contravariant vectors. Are contravariant and covariant vectors the same when the base is an orthonormalbasis.
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. In general the upper indices can be lowered and the lower indices can be. Nazrul islams book entitled tensors and their applications. Tensors of rank greater than two are easily defined. The same notion appears in quantum mechanics as ket and bra vectors and we. Feb 17, 2010 vector is a physical quantity and it does not depend on any coordinate system. Functions, and that of the greatest algebraical common measure pdf, philosophical. But in many physical situations we have to choose an nonorthogonal basis or oblique coordinate system. Covariant derivative and covariant tensor and contravariant tensor tags are words are used to describe and categorize your content. 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.
It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. Within the general scheme for the construction of tensors, a covariant vector is identified with a covariant tensor of valency 1. The difference between these two kinds of tensors is how they transform under a continuous change of coordinates. 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. Transformation of the covariant components of a lorentz tensor is defined as. This book has been presented in such a clear and easy way that the students will have no difficulty. So, the important thing to take away is that vectors are basis independent. 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 differentiation for a contravariant vector consider a vector v v.
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. Tensors, contravariant and covariant components of a. The velocity vector in equation 3 corresponds to neither the covariant nor contravari. Difference between covariant and contravariant levicivita. In mathematics and theoretical physics, covariance and contravariance are concepts used in many areas, generalizing in a sense invariance, i. Covariance and contravariance computer science, whether a type system preserves the ordering.
Wikipedia covariance and contravariance of vectors tensor. Spaces with contravariant and covariant affine connections. Contravariant, covariant and physical components of tensors. 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 transformation, a and its covariant counterpart, b, can be any linear transformation on the coordinates such as the general coordinate transformations we considered in chapter ii. Contravariant and covariant as transforms there is a lot more behind the concepts of contravariant and covariant tensors of any rank than the fact that their basis vectors are mutually orthogonal to each other. If we use the symbol a to denote a vector, and a b to denote the inner product between. General relativitycontravariant and covariant indices.
Abstract vector is a physical quantity and it does not depend on any coordinate system. How must components of vectorsdual vectors change under a change of coordinates, such that the vector remains invariant. Second, tensor theory, at the most elementary level, requires only. What are the physical differences between contravariant and. If i have covariant, but multiplying by this, i obtain contravariant vector. Wikipedia covariance and contravariance of vectors free download as pdf file. No part of this ebook may be reproduced in any form, by photostat, microfilm, xerography, or. 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. As mentioned in a previous video, tensors are invariant under coordinate transformations. This is a bit important to get rigid basics about tensors. Now that we have talked about tensors, we need to figure out how to classify them. Covariance and contravariance of vectors wikipedia. Contravariant vector an overview sciencedirect topics.
Difference between covariant and contravariant levicivita tensor. What are the physical differences between contravariant. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. Examples of vectors with contravariant components include the. An informal discussion on tensor calculus chris tiee abstract. 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. An element of the vector space dual to an dimensional vector space, that is, a linear functional linear form on. Cliquer ici pour le pdf en francais the foolish purpose of this post is to tackle the concept of tensor, while trying to keep it accessible to the widest audience possible. 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. One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor.
When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. This collection of notes attempts to demystify some of the vagaries of tensor. With covariant and contravariant vectors defined, we are now ready to extend our analysis to tensors of arbitrary rank. Tensors can, in general, have rank greater than 2, and.
In the ordered pair, an element of is called a contravariant vector. In a metric space, the contravariant and covariant tensors can be transformed to each other using the metric tensors g mn and g mn. Twenty years ago, in an article titled \ covariance and contravariance. The expansion in orthonormal basis is mathematically simple.