Contraction may be performed on any pair of covariant and contravariant indices. We wish to write the velocity vector at some time t 0. Introduction to tensor calculus for general relativity mit. Obviously, our space coordinates and time variable join to form a contravariant four vector x. The tensor product of the covariant and contravariant permutation pseudo tensors is a sixthorder tensor.
Covariant, contravariant, rank tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Let be finite dimensional, let be a basis of and let be the basis dual to it of. With the notion of contravariant and covariant components of a vector, we make nonorthogonal basis to behave like orthonormal basis. The transformation that describes the new basis vectors as a linear combination of the old basis vectors is defined as a covariant transformation.
Covariance and contravariance of vectors semantic scholar. Suppose we are given a system of equations u 1 u 1 x, y, z u 2 u 2 x, y, z. What is the difference between contravariant and covariant tensors, and why do they transform differently under a change of coordinates. Under a rescaling of contravariant coordinates by a factor of k, covariant vectors scale by k. Abstract vector is a physical quantity and it does not depend on any coordinate system. For example, for a tensor of contravariant rank 2 and covariant rank 1. 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 rules of quantum mechanics. The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. Contravariant and covariant components of a vector in a curvinilinear coordinate system. Then and the set of all tensors of the form, forms a basis for. The gradient g is an example of a covariant tensor, and the differential position d dx is an example of a contravariant tensor. In this section, the concept of contravariant and covariant vectors is extended to tensors.
It is intended to serve as a bridge from the point where most undergraduate students leave off in their studies of mathematics to the place where most texts on tensor analysis begin. If the e constitute an orthonormal basis, where g, then the two sets of components covariant and contravariant are numerically coincident. Mar 31, 2020 the additivity of the corrections is necessary if the result of a covariant derivative is to be a tensor, since tensors are additive creatures. The additivity of the corrections is necessary if the result of a covariant derivative is to be a tensor, since tensors are additive creatures. This book has been presented in such a clear and easy way that the students will have no difficulty. In general, these transformation properties differ from those of a contravariant tensor. The second is a brief introduction to coordinatefree geometry, which neces sitates a discussion of contravariant and covariant. The contravariant components of a vector are obtained by projecting onto the coordinate axes. A brief summary of facts about variance in the common language runtime. This means that they have components that are both covariant and contravariant. Multiplying a point by such a matrix will annihilate its covariant index, leaving a result that has a free contravariant index, making the result be a point. For other uses of covariant or contravariant, see covariance and contravariance disambiguation.
We know experimentally that charge is a lorentz scalar. We do so by generalizing the cartesiantensor transformation rule, eq. True or covariant derivatives of tensor felds give tensor. 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 velocity vector in equation 3 corresponds to neither the covariant nor contravari. The same applies to covariant tensors of rank n or mixed tensor of rank n. 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. Covariance and contravariance of vectors wikipedia. The principal mathematical entity considered in this volume is a field, which is defined on a domain in a euclidean manifold. Even if that sentence does not make sense right now, it will soon, so bear with me.
Jul 20, 2017 tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Tensors, contravariant and covariant components of a. Now that we have talked about tensors, we need to figure out how to classify them. Kindle file format harry lass vector tensor analysis. A gentle introduction to tensors washington university in. In ndimensional space a tensor of rank n has nn components. Covariance and contravariance are collectively referred to as variance. A transformation matrix has one contravariant and one covariant index. The g term accounts for the change in the coordinates. Vectors, contravariant and covariant penn engineering. With covariant and contravariant vectors defined, we are now ready to extend our analysis to tensors of arbitrary rank. Vectors, metric and the connection 1 contravariant and.
Tensors this will be a brief summary of what we have already covered as it applies to tensors, plus a little about. This covariant contravariant distinction doesnt seem to mean much right now because they are the same for. If i have covariant, but multiplying by this, i obtain contravariant vector. Net framework 4, variant type parameters are restricted to generic interface and generic. Also, the contravariant covariant forms of the metric tensor are expressed as the dot product of a pair of contravariant covariant basis vectors. 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. To put it succinctly, tensors are geometrical objects over vector spaces, whose coordinates obey certain laws of transformation under change of basis. The space is itself a vector space over with respect to the addition of covariant tensors of the same valency and multiplication of them by scalars. 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. So a covariant vector transforms in such a way that the transformation matrix is the transpose of the inverse of the lt transformation matrix for a contravariant vector. Vectors, metric and the connection 1 contravariant and covariant vectors 1. Let us exemplify covectors and their covariant behavior by introducing func.
Tensors can, in general, have rank greater than 2, and often do. Since this gives us two choices for each transformation coefficient. General relativitycontravariant and covariant indices. We have covariant vector that describes the rate and direction at which the temperature changes. The terms contravariant and covariant apply based on how they are derived. Then the elements of the fold tensor product are called contravariant tensors or contravariant tensors of valency or order. This covariant contravariant distinction doesnt seem to mean much right now because they are the same for this vector v. The terms covariant and contravariant were introduced by james joseph sylvester in 1851 in the context of associated algebraic forms theory.
By explicitly showing this in your derivation, there are a lot of things that i learned. As mentioned in a previous video, tensors are invariant under coordinate transformations. Contravariant vector an overview sciencedirect topics. More compactly, the q are the components of a tensor of type 2. To examine the transformation properties of a covariant tensor, first. If we have a coordinate system with straight coordinate axes, the geometric explanation given is that a vectors covariant components in such a system will be perpendicular projections on the axes, whereas its contravariant components will be parallel projections. We need to replace the matrix elements u ij in that equation by partial derivatives of the kinds occurring in eqs. 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. For this reason contravariant tensors are sometimes directly defined as multilinear functionals on. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. The inverse of a covariant transformation is a contravariant transformation. Magnetic permeability and material stress we have just seen that vectors can be multiplied by scalars to produce new vectors with the same sense or direction. Appendix a relations between covariant and contravariant bases. A basic explanation of the covariance and contravariance of vectors and of tensors too, since vectors are tensors of rank math1math is best done with the help of a geometric representation or illustration.
The valence or type of a tensor gives the number of covariant and contravariant component indices. In this video, i describe the meaning of contravariant and covariant vector components. Tensors covariant differential and riemann tensor coursera. Generally, the physical dimensions of the components and basis vectors of the covariant and contravariant forms of a tensor are di erent. Im reading a quora answer on an intuitive explanation of covariantcontravariant components of vectors. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering. An introduction to tensors for students of physics and. Dan fleisch briefly explains some vector and tensor concepts from a students guide to vectors and tensors. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. The covariant derivative of the r component in the q direction is the regular derivative plus another term.
Whenever a vector should be invariant under a change of basis, that is to say it should represent the same geometrical or physical object having the same magnitude and direction as before, its components must transform according to the contravariant rule. The covariant components are the result of a projection, so they are images of a linear form applied to the vector. Covariance and contravariance in generics microsoft docs. This volume begins with a discussion of euclidean manifolds. 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. An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0. We have a contravariant vector that describes the direction and speed at which we are going. More generally, let be a commutative ring with unit element and a unitary module over. The precise rule 11 12 is that the return types of all methods in the interface must be valid covariantly and all the method parameter types. 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 tensors contravariant and covariant vectors rotation in 2space. In general, we can specify a unit vector u, at any location we wish, to point in any direction we please.
What about quantities that are not secondrank covariant tensors. Chapter 10 vectors and tensors georgia institute of. In cartesian space they must be an orthogonal norm preserving matrix. One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor. The difference between these two kinds of tensors is how they transform under a continuous change of coordinates. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Also this implies that there is the covariant four vector. Mar 23, 2019 in this video, i describe the meaning of contravariant and covariant vector components. The covariant derivative of the r component in the r direction is the regular derivative. Now that we have contravariant vectors and covariant vectors, we can do something very interesting and combine them. In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis. Tensors are objects in multilinear algebra that can have aspects of both covariance and contravariance. In a nonorthogonal basis they will be di erent, and we must take care never to add contravariant components to covariant ones.
Covariance and contravariance computer science wikipedia. In the lexicon of category theory, covariance and contravariance are properties of functors. Conventionally, indices identifying the basis vectors are placed as lower indices and so are all entities. 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. 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. A generic type parameter that is not marked covariant or contravariant is referred to as invariant. From this we can deduce the 4tensor form for the electromagnetic field. Just as any product of covariant and or contravariant vectors.
161 1316 1329 598 592 296 1102 573 151 105 614 1323 1294 1378 1523 379 589 259 856 1052 828 424 1329 354 995 766 1459 427 661