In equation 29 components of covariant vector dual covary with change of coordinate basis and hence the name covariant vector. If, the covariant tensor is called a covariant vector. In general the upper indices can be lowered and the lower indices can be made to be upper indices, using the metric tensors. Similarly, the set of numbers f with downstairs indices is called the components of a covariant vector. In a metric space, the contravariant and covariant tensors can be transformed to each other using the metric tensors g mn and g mn. 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.
Intuitive way to understand covariance and contravariance. Comparing the lefthand matrix with the previous expression for s 2 in terms of the covariant components, we see that. An informal discussion on tensor calculus chris tiee abstract. Contravariant components of metric tensor mathematics. Suppose we are given a system of equations u 1 u 1 x, y, z u 2 u 2 x, y, z. Chapter 10 vectors and tensors georgia institute of. The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. In 4dimensional spacetime, tensors like the riemann curvature tensor are of order 4 with 44 256 components. It is helpful to begin the study of tensors with vectors, tensors of order 1 with only four components. In cartesian space they must be an orthogonal norm preserving matrix. Understanding the role of the metric in linking the various forms of tensors1 and, more importantly, in di.
First we will explain the distinction between the covariant and contravariant components of vectors, thinking of vectorfields where a vector is defined at a point rather than as a. Thus, contravariant vectors are elements of v and covariant vectors are elements of v. The exponential change of finsler metric and relation. The resulting complications include differentiation of spatially varying base vectors, the metric tensor, dual base vectors, contravariant and covariant components, tensor. The terms contravariant and covariant apply based on how they are derived. This volume begins with a discussion of euclidean manifolds. This is the second volume of a twovolume work on vectors and tensors. The contravariant metric coefficients in the contravariant metric tensor m1. They are the doubly contravariant components of the metric tensor.
For this it suffices to take as the components of the contravariant tensor the values of the linear mapping at where are the basis elements in dual to, that is, and conversely. Change of basis, reciprocal basis vectors, covariant and. Since you asked for an intuitive way to understand covariance and contravariance, i think this will do. Zero tensor 0 has all its components zero in all coord systems. Well, a tensor is neither covariant nor contravariant, while it can be expressed by its covariant, contravariant, or mixed components. Covariant and contravariant tensors physics forums.
V k which may be referred to as the metric tensor, there is little distinction between covariant and contravariant vectors, because the bilinear form allows covectors to be identified with vectors. Vectors and tensors in curved space time asaf peer1 may 20, 2015 this part of the course is based on refs. Recall that any n linearly independent vectors in n form a basis in n. Introduction to tensor calculus for general relativity mit. Why do we say that a vector can have both covariant and contravariant components.
Thus gijis a rank2 contravariant tensor, and is the inverse of g ijwhich is a rank2 covariant tensor. I feel the way im editing videos is really inefficient. When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. One especially important secondrank tensor is the metric tensor. 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 tensors components transform between different coordinate systems. Tensors, contravariant and covariant components of a. The \rank of the tensor is the number of separate indices it has. Is there a purpose of using covariant or contravariant tensors other than convenience or ease in a particular coordinate system. Covariant and contravariant components of a vector with a metric in a vector space v over a field k with a bilinear form g. In ndimensional space a tensor of rank n has nn components. 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 relationship between v and v is one of mutual duality, and to. Imagine a particle moving along some path in the 2dimensional.
Volume 1 is concerned with the algebra of vectors and tensors, while this volume is concerned with the geometrical aspects of vectors and tensors. Covariant and contravariant components of a vector in. Covariant and contravariant components of a vector with a metric. The main thing to remember here is that the contravariant components of a vector are the number of times we take each vector of the basis. Covariance and contravariance of vectors wikipedia. Tk mn rp is a mixed tensor with contravariant rank 4 and covariant rank 2. For this reason contravariant tensors are sometimes directly defined as multilinear functionals on. Transformation laws for cartesian coordinates and tensor components. Contravariant and covariant components of a vector. The components of this tensor, which can be in covariant g ij or contravariant gij forms, are in general continuous variable functions of coordinates, i.
Tensors are objects on a vector space that does not change, as a whole, under a coordinate transformation. The conjugate metric, gkl, is the inverse to the metric tensor, and therefore. The contravariant and mixed metric tensors for flat spacetime are the same this follows by considering the coordinate transformation matrices that define co and contravariance. Such a transformation can be written using a matrix a, of which. The \stick is imbedded in the \lasagna and the inner product is equal to the number of noodles pierced by the stick. To put it succinctly, tensors are geometrical objects over vector spaces, whose coordinates obey certain laws of transformation under change of basis. Introduction using the equivalence principle, we have studied the trajectories of free test particles. We reserve the dot product notation for the metric and inverse metric tensors. Contravariant and covariant components of a vector in a curvinilinear coordinate system. First of all, remember that the reason of having covariant or contravariant tensors is because you want to represent the same thing in a different coordinate system. Vectors, metric and the connection 1 contravariant and. Also what is the meaning of mixed components used in physics, is there a physical significance in choosing one over. Tensors for beginners albert tarantola september 15, 2004 1 tensor notations the velocity of the wind at the top of eiffels tower, at a given moment, can be represented by a vector v with components, in some local, given, basis, vi i 1,2,3.
We wish to write the velocity vector at some time t. Another reason to learn curvilinear coordinates even if you never explicitly apply the knowledge to any practical problems is that you will develop a far deeper understanding of cartesian tensor analysis. I have 3 more videos planned for the noncalculus videos. The exponential change of finsler metric and relation between imbedding class numbers of their tangent riemannian spaces by b. We continue our discussion of gravitation and general relativity. The components of this tensor, which can be in covariant. 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. Tensors covariant differential and riemann tensor coursera. If two tensors of the same type have all their components equal in one coord system, then their components are equal in all coord systems. With the notion of contravariant and covariant components of a vector. On the nature of the cauchy stress tensor imechanica. Covariance and contravariance of vectors semantic scholar. Vectors are simple and wellknown examples of tensors, but there is much more to tensor theory than vectors.
This shows that all these mathematical quantities belong to a family in 3 dimensional world. So, if we have a metric tensor, and we have its inverse tensor with upper indices, where this is a kronecker symbol, which means that it is 1 if mu equals 2r for nz and zero otherwise. In general direction vector like velocity vector is contravariant vector and dual vector like gradient e. Pictorial representation of the inner product between a contravariant vector and a co variant vector. Covariant and contravariant components of a vector with a metric in a finitedimensional vector space v over a field k with a symmetric bilinear form g. Thus a metric tensor is a covariant symmetric tensor. 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.1068 285 190 859 163 836 224 704 1132 469 1355 509 514 1120 1360 1252 229 702 325 506 2 603 372 354 400 1400 1596 719 1039 604 1457 1085 890 263 833 1361 1000 1084 166 688 607 874 70 1158 326