Pdf covariance and contravariance of vectors semantic scholar. Similarly, the set of numbers f with downstairs indices is called the components of a covariant vector. People commonly talk about contravariant and covariant vectors and tensors, when they really mean contravariant and covariant components. Otherwise, if the tensor has both covariant and contravariant. In figure 5, a point has contravariant coordinates. This is the historical origin of the names covariant and contravariant. The same applies to covariant tensors of rank n or mixed tensor of rank n. Contravariant tensor and covariant tensor in hindi.
Tensors may be multiplied by real numbers, and two tensors of the same degree may be added, because each tk. 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. Hence g is a covariant symmetric tensor of rank two. Covariant and contravariant vectors request pdf researchgate. We can find the components of a vector a in the covariant basis by taking the dot product with. What about quantities that are not secondrank covariant tensors. 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. In tensor analysis, a mixed tensor is a tensor which is neither strictly covariant nor strictly contravariant. Aug 23, 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 gradient g is an example of a covariant tensor, and the differential position d dx is an example of a contravariant tensor. In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities. The second is a brief introduction to coordinatefree geometry, which necessitates a discussion of contravariant and covariant vectors. Contravariant tensors of any rank are derived from the derivative of the coordinate axes covariant tensors of any rank are derived from the gradient which are also related to how the tensor s components transform between different coordinate systems.
For other uses of covariant or contravariant, see covariance and contravariance disambiguation. Tensors are said to be of the same kind when they have the same number and order and type of. It is equally rare that two repeated covariant or contravariant indices should be summed. Moreover, there is a multiplication, known as tensor product such that. Change of basis, reciprocal basis vectors, covariant and contravariant components of a vector and metric tensor. In general, these transformation properties differ from those of a contravariant tensor. Einstein relatively easy introduction to covariant. The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern. The term contravariant seems to be falling out of fashion, so these objects are often called just vectors. Tensor techniques in physics learning development institute. Also, the contravariant covariant forms of the metric tensor are expressed as the dot product of a pair of contravariant covariant basis vectors. Unfortunately, terminology is confusing and inconsistent. From one covariant set and one contravariant set we can always form an invariant x i aib i invariant, 1. Nazrul islams book entitled tensors and their applications.
In this section, the concept of contravariant and covariant vectors is extended to tensors. Contravariant tensor an overview sciencedirect topics. Sometimes the natural basis vectors are called covariant since their indices are downstairs and the dual basis vectors contravariant since their indices are upstairs. Tensors, contravariant and covariant components of a vector. Contravariant tensor and covariant tensor in hindi youtube. Covariant tensors are a type of tensor with differing transformation properties, denoted. 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. Tensors in rectilinear coordinates in two dimensions figure 5 let us consider how the oblique contravariant coordinates of a point transform on going to a new pair of oblique axes through the same origin. A covariant tensor, denoted with a lowered index e. The difference between these two kinds of tensors is how they transform under a continuous change of coordinates. Let xi be coordinates of a point in x coordinate system and xi be coordinates of a same point in ycoordinate system.
Most of the material presented in this chapter is taken from jackson, chap. I know the author as a research scholar who has worked with me for several years. A ranktwo contravariant tensor is a tensor with 16 elements and the lorentz operator is applied twice to transform to a new frame. This in turn means that the difference between the covariant and contravariant indexes vanishes for the cartesian tensors. 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 example, for a tensor of contravariant rank 2 and covariant rank 1. Transformation of contravariant covariant components. The key difference in the transformation is where is sum is located. Covariant differentiation we wish to organize physical properties and mathematical operations into covariant tensors. 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. For example, a tensor with contravariant rank 2 and covariant rank 1, written as a 2,1 tensor, transforms. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. Starting with this chapter, we will be using gaussian units for the maxwell equations and other related mathematical expressions. The definitions for contravariant and covariant tensors are inevitably defined at the. Chapter 1 tensors in rectilinear coordinates in two dimensions. Aug 06, 2019 one part are ordinary vector spaces contravariant, the other dual vector spaces covariant. Physicists use it this way, the terms contravariant and covariant in mathematics are defined for functors and not for tensors.
In equation 29 components of covariant vector dual covary with change of coordinate basis and hence the name covariant vector. Transformation of the covariant components of a lorentz tensor is defined as. The following code demonstrates the difference between assignment compatibility, covariance, and contravariance. The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. Furthermore a contravariant functor refers to the dual category, which is exactly the opposite of how physicists use it. The components of two tensors of any ranks and covariant contravariant characters can be multiplied, component by component, to make an object with all the indices of both factors. In general direction vector like velocity vector is contravariant vector and dual vector like gradient e. Contravariant tensors 2 any quantity that transforms in this way is called a contravariant tensor of rank 1, or, for short, a contravariant vector. Covariant and contravariant tensors physics forums. On a generalization of the index notation for absolute. After a while, you get used to this sort of nonsense. A covariant tensor is obtained by lowering the contravariant indexes of the tensor with the metric tensor.
For a given nonorthogonal basis a, b, c, we can always construct an. To turn a contravariant tensor into a covariant tensor index lowering, use the metric tensor to write 7 covariant and. Transformation of the contravariant and covariant components. A vector v red represented by tangent basis vectors. Thus, the entries of covariant and contravariant vectors change according to different rules as well. The following code demonstrates the difference between assignment compatibility, covariance, and. The normal vectors are called contravariant vectors, because they transform contrary to the basis vector columns. The new quantity, termed the direct product of the two tensors, can be shown to be a tensor whose rank is the sum of the ranks of the factors, and with covariant. Thus, contravariant vectors are elements of v and covariant. In that spirit we begin our discussion of rank 1 tensors. In particular, if ti are components of arbitrary variance that is, some of the components.
Using the product rule of derivation, the rate of change of the components v. To examine the transformation properties of a covariant tensor, first consider the gradient. If a certain free index is covariant contravariant in one term, it must be covariant contravariant in all terms. Tensors of rank greater than two are easily defined. 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. The terms contravariant and covariant apply based on how they are derived. A mixed tensor of type or valence, also written type m, n, with both m 0 and n 0, is a tensor which has m contravariant indices and n covariant indices. A basic, somewhat simplified 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. It gives me great pleasure to write the foreword to dr. Under a rescaling of contravariant coordinates by a factor of k, covariant vectors scale by k.
The tangent vector tu from elementary calculus to a parametric curve. Covariance preserves assignment compatibility and contravariance reverses it. Otherwise, if the tensor has both covariant and contravariant indices, it is said to be mixed. Thus, the entries of covariant and contravariant vectors change according to. With this convention a contravariant vector, with contravariant components, is written in terms of the covariant basis. The second is a brief introduction to coordinatefree geometry, which neces sitates a discussion of contravariant and covariant. Pdf covariance and contravariance of vectors semantic. The vector space or linear space, mve4 space, or just space of all k contravariant, l covariant tensors tensors of valence. Request pdf covariant and contravariant vectors vector is a physical quantity and it does not depend on any coordinate system.
That is you must contract the indexes of the tensor with the metric like. Oct 24, 2020 covariant differentiation for a contravariant vector. Akl is a contravariant tensor, a kl is a covariant tensor, and ak l is a mixed tensor. All the 2mnm index vectors in j m thus specify all the possible covariant, contravariant, and mixed components of a given mth order tensor defined on an ndimensional coordinate space. With the notion of contravariant and covariant components of a vector, we make nonorthogonal basis to behave like orthonormal basis. A given velocity vector for example has whatever direction and magnitude it has, independent of the coordinate system we use to express it. Tensors consider an ndimensional space with arbitrary geometry and an object with components in the coord system and in the coord system.
This is the main purpose of this article to explore the notion of contravariant and covariant vectors in di. Covariance and contravariance of vectors wikipedia. Vector tensor with rank covariant or contravariant 1. The main reason for the differentiation of covariant and contravariant vari ables is that under the change of bases, their coordinates change according to different rules. We have now produced an invariant inner product for contravariant vectors by using a covariant vector w. For example, a tensor with contravariant rank 2 and covariant rank 1, written as a 2,1 tensor. If a position vector r defines a point s in contravariant and covariant tensor pdf. Change of basis, reciprocal basis vectors, covariant and. A vector v red represented by tangent basis vectors yellow, left. If q 0 p 0, then all indices are covariant contravariant and the tensor is said to be covariant contravariant. This question points out a problem with the terminology.
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