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