The same applies to covariant tensors of rank n or mixed tensor of rank n. The components of a covariant vector transform like a gradient and obey the transformation law. This video is a nice explanation for people studying relativity, but in general i think that making a big deal about how vectors and tensors. A contravariant tensor has components written with raised indices, for example a 4velocity, whereas the components of a covariant tensor are written with lowered indices, for example a metric. Having in mind the analogy to coordinate differentials and derivative of a scalar, take a look at this picture, which i think will help to make it clearer. Introduction to tensors contravariant and covariant vectors rotation in 2space. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering. To examine the transformation properties of a covariant tensor, first. Covariance and contravariance of vectors wikipedia. The valence or type of a tensor gives the number of covariant and contravariant component indices. Tensors this will be a brief summary of what we have already covered as it applies to tensors, plus a little about. What are the physical differences between contravariant. 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.
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. When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. Now, if this energyforce 4vector equation is to be covariant so its transformed form is still a 4vector then the right hand sides must form a 4vector too. The terms covariant and contravariant were introduced by james joseph sylvester in 1851 in the. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. If we apply the same correction to the derivatives of other secondrank contravariant tensors, we will get nonzero results, and they will be the right nonzero results.
In ndimensional space a tensor of rank n has nn components. Transformations of contravariant and covariant tensor. 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. Introduction to tensor calculus for general relativity mit. Kolkers answer gives the gory detail, but heres a quick and dirty version. With the notion of contravariant and covariant components of a vector, we. Appendix a relations between covariant and contravariant bases. 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. More compactly, the q are the components of a tensor of type 2. However, the last two equations show that the distinction between contravariant and covariant tensor operators persists for. For other uses of covariant or contravariant, see covariance and contravariance disambiguation. Physics 221a fall 2019 appendixe introduction to tensor.
As mentioned in a previous video, tensors are invariant under coordinate transformations. Then the prototypical example of a contravariant vector is a. Contraction may be performed on any pair of covariant and contravariant indices. Vectors, metric and the connection 1 contravariant and. In that spirit we begin our discussion of rank 1 tensors. A covariant tensor, denoted with a lowered index e. Covariance and contravariance in generics microsoft docs. Physics expressing, exhibiting, or relating to covariant theory. This book has been presented in such a clear and easy way that the students will have no difficulty. Metrinilic property covariant metric tensor covariant tensor curl curvature normal. The tensor product of the covariant and contravariant permutation pseudo tensors is a sixthorder tensor. Statistics varying with another variable quantity in a manner that leaves a specified relationship unchanged. There are also tensors with mixed indices, for example the riemann tensor is often given as. We wish to write the velocity vector at some time t 0.
Physics 221a fall 2019 appendixe introduction to tensor analysis 1. In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Net framework 4, variant type parameters are restricted to generic interface and generic. Notice how the contravariant basis vector g is not differentiated. If, the covariant tensor is called a covariant vector. The covariant derivative is the derivative that under a general coordinate transformation transforms covariantly. In general, these transformation properties differ from those of a contravariant tensor to examine the transformation properties of a covariant tensor, first consider the gradient. The distinction between contravariant and covariant transformations was established in section 4.
This means that they have components that are both covariant and contravariant. Tensors covariant differential and riemann tensor coursera. The downside to this perspective is that you are really missing out on the rather rich mathematical intuition behind what a covariant or contravariant tensor really is and all the language of differential forms and tangent bundles and all that stuff. Vectors, metric and the connection 1 contravariant and covariant vectors 1. We now examine the dual or covariant vector counterpart to the contravariant vector. A generic type parameter that is not marked covariant or contravariant is referred to as invariant. This is obtained from the basic contravariant field strength tensor by the substitutions.
The presentation is based on how various quantities transform under coordinate transformations, and is fairly standard. Introduction these notes contain an introduction to tensor analysis as it is commonly used in physics, but mostly limited to the needs of this course. The contravariant components of a tensor, for example, are obtained by resolving the tensor into components in terms of the socalled coordinate basis vectors. Nazrul islams book entitled tensors and their applications. A brief summary of facts about variance in the common language runtime. Physics stack exchange is a question and answer site for active researchers, academics and students of physics. The third chapter extends tensor theory to spaces other than vector spaces. In general, these transformation properties differ from those of a contravariant tensor. 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.
The contravariant tensor conjugate to gij is denoted by gij satisfy 5. It gives me great pleasure to write the foreword to dr. Covariance and contravariance are collectively referred to as variance. This is the main purpose of this article to explore the notion of contravariant and covariant vectors in di. Vectors, contravariant and covariant penn engineering.
The normal vectors are called contravariant vectors, because they transform contrary to the basis vector columns. 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. The velocity vector in equation 3 corresponds to neither the covariant nor contravari. A gentle introduction to tensors boaz porat department of electrical engineering technion israel institute of technology. For a covariant transformation, we switch the prime and unprimed, but then due to the nature of the contravariant and covariant vectors, the indices also switches, i. The components of this tensor, which can be in covariant g. Covariance and contravariance computer science wikipedia. The gradient g is an example of a covariant tensor, and the differential position d dx is an example of a contravariant tensor. 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. Lets work in the three dimensions of classical space forget time, relativity, fourvectors etc. In this video, i describe the meaning of contravariant and covariant vector components.
Covariance and contravariance of vectors semantic scholar. I know the author as a research scholar who has worked with me for several years. This post is a short introdcution into covariant and contravariant coordinates in terms of linear algebra. For example, for a tensor of contravariant rank 2 and covariant rank 1. 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 rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. Covariant definition of covariant by the free dictionary. The tensor itself is independent of the coordinate system we are using. For a given nonorthogonal basis a, b, c, we can always construct an.
If i have covariant, but multiplying by this, i obtain contravariant vector. First, tensors appear everywhere in physics, including classi. The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern. The difference between these two kinds of tensors is how they transform under a continuous change of coordinates. Typically, when tensor mathematics is applied, the meaning of each index has been defined beforehand. The second is a brief introduction to coordinatefree geometry, which neces sitates a discussion of contravariant and covariant. Generally, the physical dimensions of the components and basis vectors of the covariant and contravariant forms of a tensor are di erent.
394 632 129 886 60 1347 475 1327 710 1178 35 1080 1400 785 365 197 1343 652 414 1128 477 1184 893 765 568 407 760 597 863 631 293 188 375