Covariance and contravariant tensor pdf file

This post is intended to give a developerfocused explanation of the terms without diving into the category theory behind them too much. An informal discussion on tensor calculus chris tiee abstract. Tensors can, in general, have rank greater than 2, and. The differences of zircon titanium thermometry, iuhft value and the covariant relation of rbsr and ba among the two stages of magmatism show that tsona leucogranties originated from more than one melting reactions, they had experienced the change from dehydration melting to fluid fluxed melting, and the reason respond to this transformation may be the start of the ew extension in southern. Demonstrates the concepts of covariance and contravariance in the java type system gist. Covariance and contravariance in scala knoldus blogs. Vectors are neither covariant nor contravariant, but they have covariant components and contravariant components. Explain the difference between a covariant tensor and a contravariant tensor, using the metric tensor as an example. Lorentz covariance of maxwell equations scalar and vector potentials, and gauge invariance relativistic motion of charged particles action principle for electromagnetism. Intuitive way to understand covariance and contravariance.

Vectors, metric and the connection 1 contravariant and. For what its worth i cleaned up a document i wrote a while ago which tries to give a slightly lesssucky overview of covariance and contravariance without getting tangled in the whole differential geometry of it. Abstract vector is a physical quantity and it does not depend on any 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. Contravariant adaptation on the manifold of invertible. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc. The covariant gradient the usual gradient is converted to the contravariant or natural gradient using the inverse of a riemannian metric tensor which is derived for the manifold of invertible. In mathematics and theoretical physics, covariance and contravariance are concepts used in many areas, generalizing in a sense invariance, i. This collection of notes attempts to demystify some of the vagaries of tensor.

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. Tensors, contravariant and covariant components of a vector. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. For instance, a rank2 contravariant tensor has the transformation property. Is there a fundamental difference in the way contravariant and covariant components can be used, or could one simply interchange them. But in many physical situations we have to choose an nonorthogonal basis or oblique coordinate system.

Pattern matching pca pdf pdfbox pdf plugin pem file performace performance. Intuitive way to understand covariance and contravariance in. 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. Explain how the components of a general covariant tensor may be converted into those of the equivalent contravariant tensor, and vice versa.

Differenceconvert between covariantcontravariant tensors. This material is covered thoroughly in chapters one and seven in our textbook classical electrodynamics, 2nd ed. Vectors are covariant, but the components of vectors are contravariant. Covariant tensor article about covariant tensor by the. Before moving to details of covariance and contravariance, below image is a good example to start with. 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. When the manifold is equipped with a metric, covariant and contravariant indices become very closely related to oneanother. 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. The tensor product of the covariant and contravariant permutation pseudo tensors is a sixthorder tensor. Suppose we are given a system of equations u 1 u 1 x, y, z u 2 u 2 x, y, z. Understanding tensoroperations, covariance, contravariance, in the context of special relativity. If you want to express this newtonian fluid constitutive equation in terms of actual tensor notation, the partial derivatives on the right hand side need to be replaced by covariant derivatives, and the kronecker delta needs to be replaced by the appropriately indexed representation of the metric tensor. The expansion in orthonormal basis is mathematically simple.

Twenty years ago, in an article titled \ covariance and contravariance. A change of scale on the reference axes corresponds to a change of units in the problem. Contravariant indices can be gotten by contracting with the matrix inverse of the metric tensor. Today, i read some articles about covariance, contravariance and invariance in java. Hence acceleration is also a contravariant tensor of rank two. We wish to write the velocity vector at some time t 0.

In ndimensional space a tensor of rank n has nn components. Covariance of the dirac equation 3 which is called the feynmanslash. If we use the symbol a to denote a vector, and a b to denote the inner product between. An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0. Vector is a physical quantity and it does not depend on any coordinate system. 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 in physics, a basis is sometimes thought of as a set of reference axes. Contravariant and covariant components of a vector in a curvinilinear coordinate system. Covariance and contravariance of vectors wikipedia. Indeed, a vector is a tensor of rank one, and a scalar is a tensor of rank zero.

Electromagnetic theory ii contents special relativity. As mentioned in a previous video, tensors are invariant. Introduction to tensors contravariant and covariant vectors rotation in 2space. Chapter 10 vectors and tensors georgia institute of. Jul 20, 2017 tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Planetphysicscovariance and contravariance wikiversity. In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities. Dec 08, 2014 this course will continue on patreon at textbook. Vectors, metric and the connection 1 contravariant and covariant vectors.

I think i may have some pretty big misunderstandings here so thanks in advance for any answersclarity. A mapping which preserves but reverses a relation is called a contravariant mapping. Now that we have talked about tensors, we need to figure out how to classify them. The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation.

A device which can compare two animals can also compare two tigers, but a device which can compare two tigers cannot necessarily compare any two animals. Note that in chapter ii it was shown that the last two expressions have the. Tensor which depends on two directions one normal to the area and other that of the force on it. Covariant definition of covariant by the free dictionary. 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. 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. Requirements 1 the derivative of a tensor must be a tensor. For more information, please see the wikipedia page on covariance and contravariance. Tensors, contravariant and covariant components of a. Covariance and contravariance in generics microsoft docs.

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. Entangled states and quantum causality threshold in the general theory of relativity. We are now ready to get serious about electrodynamics. First of all, remember that the reason of having covariant or contravariant tensors is because you want to represent the same thing in a different coordinate system.

Contravariant indices can be turned into covariant indices by contracting with the metric tensor. Introduction and examples modeling mean structure modeling covariance structure mean and covariance models for tensor valued data peter ho statistics, biostatistics and the csss. The velocity vector in equation 3 corresponds to neither the covariant nor contravari. Introduction and examples modeling mean structure modeling covariance structure. Examples of vectors with contravariant components include the position of an. Appendix a relations between covariant and contravariant bases. The discussion of rotations should come before covariance and contravariance. Since you asked for an intuitive way to understand covariance and contravariance, i think this will do. Similarly, the set of numbers f with downstairs indices is called the components of a covariant vector. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. The second is a brief introduction to coordinatefree geometry, which neces sitates a discussion of contravariant and covariant.

An interface or delegate type can have both covariant and contravariant type parameters. I read the english and german wikipedia article, and some other blog posts and articles from ibm. This means it is not necessary to transform the expression such as must be done for a tensor in the box on page iii1. More compactly, the q are the components of a tensor of type 2. Tensor notation tensor notation in three dimensions. We present here a brief summary of tensor notation in three dimensions simply to refresh the memory of the reader and provide a smooth introduction to the relativistic tensor notation which follows. The question is what is the difference between covariance and contravariance.

Understanding tensoroperations, covariance, contravariance. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering. The gradient of a scalar field is a covariant vector field. It is an idea that seems to originate in physics but in fact it is a more a mathematical concept. Secondly, if a tensor is a ultimately a map to the reals, where is that evident in this video. Covariant derivative 22 mar 2012riemannchristoffel curvature tensor. What are the physical differences between contravariant. Udayanandan associate professor department of physics nehru arts and science college, kanhangad 1. Demonstrates the concepts of covariance and contravariance. Kolkers answer gives the gory detail, but heres a quick and dirty version. 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.

In mathematical terms, they occur in a foundational way in linear algebra and multilinear algebra, differential geometry and other branches of geometry, category theory and. General relativitycontravariant and covariant indices. A polar vector in 3 dimensions 3d is an object which, given a coordinate frame, can be. Then the prototypical example of a contravariant vector is a. The definition of contravariant tensor of rankis a simple generalization. Covariant tensor article about covariant tensor by the free. 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. This blog post is also part of the fp complete haskell syllabus and part of our haskell training.

One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor. 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. Second, tensor theory, at the most elementary level, requires only. We have developed a group of coordinate transformations that preserves. With the notion of contravariant and covariant components of a vector, we make nonorthogonal basis to behave like orthonormal basis. 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. Feb 17, 2010 vector is a physical quantity and it does not depend on any coordinate system. The terms contravariant and covariant apply based on how they are derived. Vectors and covectors in special relativity jim napolitano march 12, 2010. The foolish purpose of this post is to tackle the concept of tensor, while trying to keep it accessible to the widest audience. For instance, a rank2 contravariant tensor has the. Introduction and examples modeling mean structure modeling covariance structure mean and covariance models for tensorvalued data peter ho statistics, biostatistics and the csss university of washington. In this video, i describe the meaning of contravariant and covariant vector components.

When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. 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. Covariance and contravariance describes how a tensor transforms with respect to dilation, not a rotation or other lorentz transformation. I understand that, depending on how something transforms, its either covariant or contravariant and that that determines whether the index is written at the top or bottom. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. He does a nice job in the video showing that a tensor can be thought of as an array of numbers, but how is this formal definition coming into play.

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