Introduction to tensors contravariant and covariant vectors rotation in 2space. Covariance and contravariance are collectively referred to as variance. Obviously, our space coordinates and time variable join to form a contravariant four vector x. The covariant derivative of the r component in the q direction is the regular derivative plus another term. Tensors covariant differential and riemann tensor coursera. Why do we say that a vector can have both covariant and contravariant components. The terms covariant and contravariant were introduced by james joseph sylvester in 1851 in the context of associated algebraic forms theory. We know experimentally that charge is a lorentz scalar. True or covariant derivatives of tensor felds give tensor.
Dan fleisch briefly explains some vector and tensor concepts from a students guide to vectors and tensors. 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. The gradient g is an example of a covariant tensor, and the differential position d dx is an example of a contravariant tensor. 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. From this we can deduce the 4tensor form for the electromagnetic field. A gentle introduction to tensors boaz porat department of electrical engineering technion israel institute of technology. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. Covariance and contravariance of vectors semantic scholar. 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. Let us exemplify covectors and their covariant behavior by introducing func. This means that they have components that are both covariant and contravariant. Contraction may be performed on any pair of covariant and contravariant 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. In ndimensional space a tensor of rank n has nn components. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. More compactly, the q are the components of a tensor of type 2. We need to replace the matrix elements u ij in that equation by partial derivatives of the kinds occurring in eqs. 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. 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. 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 we use the symbol a to denote a vector, and a b to denote the inner product between. Tensors are objects in multilinear algebra that can have aspects of both covariance and contravariance. The terms contravariant and covariant apply based on how they are derived.
Generally, the physical dimensions of the components and basis vectors of the covariant and contravariant forms of a tensor are di erent. 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. Also, the contravariant covariant forms of the metric tensor are expressed as the dot product of a pair of contravariant covariant basis vectors. The inverse of a covariant transformation is a contravariant transformation. An introduction to tensors for students of physics and. Conventionally, indices identifying the basis vectors are placed as lower indices and so are all entities. 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. Jul 20, 2017 tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. For example, for a tensor of contravariant rank 2 and covariant rank 1. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. Then and the set of all tensors of the form, forms a basis for.
The difference between these two kinds of tensors is how they transform under a continuous change of coordinates. If i have covariant, but multiplying by this, i obtain contravariant vector. A gentle introduction to tensors washington university in. The covariant derivative of the r component in the r direction is the regular derivative. 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. A generic type parameter that is not marked covariant or contravariant is referred to as invariant.
We have covariant vector that describes the rate and direction at which the temperature changes. The same applies to covariant tensors of rank n or mixed tensor of rank n. What about quantities that are not secondrank covariant tensors. A transformation matrix has one contravariant and one covariant index.
Net framework 4, variant type parameters are restricted to generic interface and generic. Now that we have talked about tensors, we need to figure out how to classify them. This book has been presented in such a clear and easy way that the students will have no difficulty. Kindle file format harry lass vector tensor analysis. This covariant contravariant distinction doesnt seem to mean much right now because they are the same for this vector v. 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.
Vectors, metric and the connection 1 contravariant and covariant vectors 1. The second is a brief introduction to coordinatefree geometry, which neces sitates a discussion of contravariant and covariant. Abstract vector is a physical quantity and it does not depend on any coordinate system. Covariance and contravariance of vectors wikipedia. In a nonorthogonal basis they will be di erent, and we must take care never to add contravariant components to covariant ones. The components of this tensor, which can be in covariant g. Vectors, contravariant and covariant penn engineering. For other uses of covariant or contravariant, see covariance and contravariance disambiguation. With the notion of contravariant and covariant components of a vector, we make nonorthogonal basis to behave like orthonormal basis. The covariant components are the result of a projection, so they are images of a linear form applied to the vector.
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. In general, these transformation properties differ from those of a contravariant tensor. A brief summary of facts about variance in the common language runtime. With covariant and contravariant vectors defined, we are now ready to extend our analysis to tensors of arbitrary rank. Then the elements of the fold tensor product are called contravariant tensors or contravariant tensors of valency or order. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. 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 this section, the concept of contravariant and covariant vectors is extended to tensors. One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor. Since this gives us two choices for each transformation coefficient. The principal mathematical entity considered in this volume is a field, which is defined on a domain in a euclidean manifold. Vectors, metric and the connection 1 contravariant and. The g term accounts for the change in the coordinates.
Appendix a relations between covariant and contravariant bases. Vectors are simple and wellknown examples of tensors, but there is much more to tensor theory than vectors. The tensor product of the covariant and contravariant permutation pseudo tensors is a sixthorder 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. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering. Also this implies that there is the covariant four vector. Even if that sentence does not make sense right now, it will soon, so bear with me. In general, we can specify a unit vector u, at any location we wish, to point in any direction we please. Introduction to tensor calculus for general relativity mit.
The contravariant components of a vector are obtained by projecting onto the coordinate axes. 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. Tensors can, in general, have rank greater than 2, and often do. The additivity of the corrections is necessary if the result of a covariant derivative is to be a tensor, since tensors are additive creatures. General relativitycontravariant and covariant indices. In cartesian space they must be an orthogonal norm preserving matrix. Under a rescaling of contravariant coordinates by a factor of k, covariant vectors scale by k. Chapter 10 vectors and tensors georgia institute of. Mar 23, 2019 in this video, i describe the meaning of contravariant and covariant vector components. If the e constitute an orthonormal basis, where g, then the two sets of components covariant and contravariant are numerically coincident. The valence or type of a tensor gives the number of covariant and contravariant component indices. As mentioned in a previous video, tensors are invariant under coordinate transformations.
Im reading a quora answer on an intuitive explanation of covariantcontravariant components of vectors. Now that we have contravariant vectors and covariant vectors, we can do something very interesting and combine them. Covariant, contravariant, rank tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. To put it succinctly, tensors are geometrical objects over vector spaces, whose coordinates obey certain laws of transformation under change of basis. We have a contravariant vector that describes the direction and speed at which we are going. The transformation that describes the new basis vectors as a linear combination of the old basis vectors is defined as a covariant transformation. This covariant contravariant distinction doesnt seem to mean much right now because they are the same for. In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis. Let be finite dimensional, let be a basis of and let be the basis dual to it of.
To examine the transformation properties of a covariant tensor, first. What is the difference between contravariant and covariant tensors, and why do they transform differently under a change of coordinates. By explicitly showing this in your derivation, there are a lot of things that i learned. 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. Covariance and contravariance in generics microsoft docs. An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0. The velocity vector in equation 3 corresponds to neither the covariant nor contravari. We do so by generalizing the cartesiantensor transformation rule, eq. Just as any product of covariant and or contravariant vectors. We wish to write the velocity vector at some time t 0. 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.
For this reason contravariant tensors are sometimes directly defined as multilinear functionals on. 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. Covariance and contravariance computer science wikipedia. 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.
This volume begins with a discussion of euclidean manifolds. Contravariant vector an overview sciencedirect topics. In the lexicon of category theory, covariance and contravariance are properties of functors. Contravariant and covariant components of a vector in a curvinilinear coordinate system. The vector space or linear space, mve4 space, or just space of all kcontravariant, covariant tensors tensors of valence k at the point p in a manifold m will be denoted tk mp, with tmp and t.
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