Most of the material presented in this chapter is taken from jackson, chap. Thus, the entries of covariant and contravariant vectors change according to different rules as well. 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. X 2 j is covariant in the first variable x 1 and contravariant in the second one x j 2. Similar to vectors, the covariant, contravariant and mixed components of a second order tensor are related to one another via the metric tensor, which raises or lowers the corresponding indices.
For other uses of covariant or contravariant, see covariance and contravariance disambiguation. Vectors vi are often called contravariant vectors, and dual vectors wi are called covariant vectors. We define three main types of tensors covariant tensors, contravariant tensors, and mixed tensors based on their behavior and properties with respect to general coordinate transformations. Hence g is a covariant symmetric tensor of rank two. A gentle introduction to tensors washington university in. Note that the rows of s appear as superscripts and the columns appear as. Since this gives us two choices for each transformation coefficient. The difference between these two kinds of tensors is how they transform under a continuous change of coordinates. We need to replace the matrix elements u ij in that equation by partial derivatives of the kinds occurring in eqs. I know the author as a research scholar who has worked with me for several years. Contravariant tensors are indicated with raised indices, i. 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.
Covariant differentiation we wish to organize physical properties and mathematical operations into covariant tensors. This in turn means that the difference between the covariant and contravariant indexes vanishes for the cartesian tensors. Appendices deal with the geometry of npipeds, tensor densities, tensor expansions, and covariant derivatives. Chapter 1 tensors in rectilinear coordinates in two dimensions. But in any theory where covariant and contravariant tensors differed, e and j would not be. Equations that equate a scalar to a scalar, or v to v, or t to t are lc. Akl is a contravariant tensor, a kl is a covariant tensor, and ak l is a mixed tensor. 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.
If we use the symbol a to denote a vector, and a b to denote the inner product between. Physicists use it this way, the terms contravariant and covariant in mathematics are defined for functors and not for tensors. The index symmetries have some important implications for ra bcd. For example, a tensor with contravariant rank 2 and covariant rank 1, written as a 2,1 tensor. Contravariant tensor and covariant tensor in hindi youtube. The lower index the covariant part represents one of the two basis vectors in the same flat 2dimensional tangent plane. Tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. In figure 5, a point has contravariant coordinates. We do so by generalizing the cartesian tensor transformation rule, eq. The terms covariant and contravariant were introduced by james joseph sylvester in 1851 in the context of associated algebraic forms theory. Also, the contravariant covariant forms of the metric tensor are expressed as the dot product of a pair of contravariant covariant basis vectors. The key difference in the transformation is where is sum is located. In that spirit we begin our discussion of rank 1 tensors.
We have now produced an invariant inner product for contravariant vectors by using a covariant vector w. This is the main purpose of this article to explore the notion of contravariant and covariant vectors in di. 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. Using the product rule of derivation, the rate of change of the components v. Pdf covariance and contravariance of vectors semantic scholar. If a position vector r defines a point s in covariant vector. For example, for a tensor of contravariant rank 2 and covariant rank 1. It gives me great pleasure to write the foreword to dr. With the notion of contravariant and covariant components of a vector, we make nonorthogonal basis to behave like orthonormal basis. For a given nonorthogonal basis a, b, c, we can always construct an. An introduction to vectors and tensors from a computational.
In particular, if ti are components of arbitrary variance that is, some of the components. In equation 29 components of covariant vector dual covary with change of coordinate basis and hence the name covariant vector. We show that once orthonormality is relaxed, a dual basis, together with the contravariant and covariant components, naturally emerges. Contravariant vector an overview sciencedirect topics. The term contravariant seems to be falling out of fashion, so these objects are often called just vectors. Tensors, contravariant and covariant components of a vector. Einstein relatively easy introduction to covariant. 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.
From one covariant set and one contravariant set we can always form an invariant x i aib i invariant, 1. Covariant and contravariant tensors physics forums. 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. Contravariant tensor an overview sciencedirect topics. For example, a tensor with contravariant rank 2 and covariant rank 1, written as a 2,1 tensor, transforms. A ranktwo contravariant tensor is a tensor with 16 elements and the lorentz operator is applied twice to transform to a new frame. Contravariant tensor and covariant tensor in hindi. Chapter 1 vectors and tensors university of oxford. Tensor techniques in physics learning development institute. But which covariant index in principle ra acd 6 ra bad 6 r a bca. The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern.
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. Contravariant tensors 2 any quantity that transforms in this way is called a contravariant tensor of rank 1, or, for short, a contravariant vector. To put it succinctly, tensors are geometrical objects over vector spaces, whose coordinates obey certain laws of transformation under change of basis. Differenceconvert between covariantcontravariant tensors. To turn a contravariant tensor into a covariant tensor index lowering, use the metric tensor to write 7 covariant and. Starting with this chapter, we will be using gaussian units for the maxwell equations and other related mathematical expressions. Tensors of rank greater than two are easily defined. Second, tensor theory, at the most elementary level, requires only. Pdf covariance and contravariance of vectors semantic. In tensor analysis, a mixed tensor is a tensor which is neither strictly covariant nor strictly contravariant. In general, these transformation properties differ from those of a contravariant tensor.
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. The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. Tensors consider an ndimensional space with arbitrary geometry and an object with components in the coord system and in the coord system. In this case, the upper index the contravariant part is one of two vector components in a flat 2dimensional tangent plane.
Nazrul islams book entitled tensors and their applications. We have a ij g img jna mn g jna n i g ima m j aij gimgjna mn g imaj m g. Covariant tensors are a type of tensor with differing transformation properties, denoted. A covariant tensor, denoted with a lowered index e. Complete contraction of a vector or a tensor forms a scalar. The terms contravariant and covariant apply based on how they are derived. 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. A contravariant vector is a contravariant tensor of rank 1. A vector v red represented by tangent basis vectors yellow, left. Let xi be coordinates of a point in x coordinate system and xi be coordinates of a same point in ycoordinate system. Oct 24, 2020 covariant differentiation for a contravariant vector.
On a generalization of the index notation for absolute. Covariance and contravariance of vectors wikipedia. The gradient g is an example of a covariant tensor, and the differential position d dx is an example of a contravariant tensor. X 1 x 2 can be interpreted as 1 covariant 1 contravariant tensor.
Tensors 235 sometimes, slightly abusing the language, people talk about covariant contravariant variables or. The above tensor t is a 1 covariant, 1 contravariant object, or a rank 2 tensor of type 1, 1 on 2. The normal vectors are called contravariant vectors, because they transform contrary to the basis vector columns. We can find the components of a vector a in the covariant basis by taking the dot product with. Transformation of the contravariant and covariant components. Column vectors that represent data vectors that describe the position of something. Furthermore a contravariant functor refers to the dual category, which is exactly the opposite of how physicists use it.
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. 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 multilinear algebra and tensor analysis, covariance and contravariance describe how the. First, tensors appear everywhere in physics, including classical mechanics, relativistic mechanics, electrodynamics, particle physics, and more. Transformation laws for cartesian coordinates and tensor components 11. In general direction vector like velocity vector is contravariant vector and dual vector like gradient e. Aug 06, 2019 one part are ordinary vector spaces contravariant, the other dual vector spaces covariant. The definitions for contravariant and covariant tensors are inevitably defined at the. 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. If both r and s are nonzero then its called a mixed tensor. 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. Tensors are said to be of the same kind when they have the same number and order and type of. With covariant and contravariant vectors defined, we are now ready to extend our analysis to tensors of arbitrary rank.
For orthonormal systems we are used to the fact that when a ake k, then unique components can be obtained via a dot product4. The covariant summation runs over the superscripts, whereas the contravariant summations runs over the subscripts as we did above. Thus, contravariant vectors are elements of v and covariant. To examine the transformation properties of a covariant tensor, first consider the gradient. In this chapter we introduce the concept of tensors using the system notation and definition of metric spaces. The same applies to covariant tensors of rank n or mixed tensor of rank n. 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. Vectors are simple and wellknown examples of tensors, but there is much more to tensor theory than vectors. In this section, the concept of contravariant and covariant vectors is extended to tensors. Dalarsson, in tensors, relativity, and cosmology second edition, 2015 abstract. The tangent vector tu from elementary calculus to a parametric curve.
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