As mentioned in a previous video, tensors are invariant under coordinate transformations. One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor. If we use the symbol a to denote a vector, and a b to denote the inner product between. We use this repeatedly in the algebra in sections below. 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. See also covariant tensor, fourvector, lorentz tensor, metric tensor, mixed tensor, tensor. Converting contravariant or covariant tensors to mixed tensors.
If your covariant derivative took in 1forms as the directional argument instead of vectors, it would not represent a connection, because there is no way to canonically tie together curves and 1forms without a tool like a metric tensor or a symplectic form. This quantity, rbj, is often called a christoffel symbol of the first kind, while rkj. An informal discussion on tensor calculus chris tiee abstract. Vectors, metric and the connection 1 contravariant and covariant vectors 1. Net framework 4, variant type parameters are restricted to generic interface and generic. Such a field assigns to each an element of, the fold tensor product of the tangent space to at. Covariance and contravariance of vectors wikipedia. General relativitycontravariant and covariant indices. Aug 03, 2006 essentially, there is no difference between the covariant and contravariant forms of the metric in the sense that they both measure things. Tensors can, in general, have rank greater than 2, and. Converting contravariant or covariant tensors to mixed.
Tensor analysis maksim levental phd student from somewhere. Furthermore a contravariant functor refers to the dual category, which is exactly the opposite of how physicists use it. Now lets consider a vector x whose contravariant components relative to the x axes of figure 2 are x 1, x 2, and lets multiply this by the covariant metric tensor as follows. So the compiler just outright refuses to compile your code. 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. Introduction to tensors contravariant and covariant vectors rotation in 2space. Covariant and contravariant tensors physics forums. Now what bothers me is this lowering and rising index stuff where one can transform a contravariant vector to a covariant one and viceversa by multiplying by the minkowski metric tensor in the special relativity case. Since our zoo can handle all animals, it can certainly handle fish, so lets try to add some fish to our zoo. Finally we are in a position to see how covariant and contravariant vectors differ in this metric. Covariance and contravariance are collectively referred to as variance. Covariant and contravariant indices can be used simultaneously in a mixed tensor. In cartesian space they must be an orthogonal norm preserving matrix. The terms contravariant and covariant apply based on how they are derived.
In this video, i describe the meaning of contravariant and covariant vector components. 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. 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. Mar 26, 2014 contravaraint metric tensor contravariant basis contravariant components contravariant metric tensor. A brief summary of facts about variance in the common language runtime. For 2dimensional polar coordinates, the metric is s 2. A di erent metric will, in general, identify an f 2v with a completely di erent ef 2v. If n 2 quantities in a coordinate system are related to n 2 other quantities in another coordinate system by the transformation equations or, by our conventions, they are called components of a contravariant tensor of the second rank or of rank two. Elements of tensor calculus 695 in brief given a veetor spaee e and its dual spaee e, t4e not ion of tensor arises when we study the multilinear forms on the spaee f. Covariant and contravariant components of a vector with a metric in a vector space v over a field k with a bilinear form g. The contravariant components of a tensor, for example, are obtained by resolving the tensor into components in terms of the socalled coordinate basis vectors.
Mar 23, 2019 in this video, i describe the meaning of contravariant and covariant vector components. For a deeper understanding, knowledge of manifolds and some pointset topology is required. The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. Hence displacement is a contravariant quantity and the gradient is a covariant quantity. Now that we have talked about tensors, we need to figure out how to classify them. We wish to write the velocity vector at some time t 0. Physicists use it this way, the terms contravariant and covariant in mathematics are defined for functors and not for tensors. In general direction vector like velocity vector is contravariant vector and dual vector like gradient e.
What different between covariant metric tensor and. A gentle introduction to tensors washington university in. An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0. 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. If both are covariant, or both are contravariant, one or the other must be raised or lowered by contracting it with the metric tensor before contracting the overall pair. For example, for a tensor of contravariant rank 2 and covariant rank 1. Proceeding a small step further, tensor theory requires background in multivariate calculus. Tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update. 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.
The phrase contravariant tensor of order r is also used to denote a contravariant tensor field of order over a smooth manifold. 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. In ndimensional space a tensor of rank n has nn components. In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface or higher dimensional differentiable manifold and produces a real number scalar gv, w in a way that generalizes many of the familiar properties of the dot product of vectors in euclidean space. For every contravariant index there is an a transformation and for every covariant index there is a b. Contravaraint metric tensor contravariant basis contravariant components contravariant metric tensor. Tensors, contravariant and covariant components of a vector.
Remember that summation is implied over the repeated index u, whereas the index v appears only once in any given product so this expression applies for any value of v. The tensor itself is independent of the coordinate system we are using. Aug 06, 2019 one part are ordinary vector spaces contravariant, the other dual vector spaces covariant. The dimension of contravariant vector is the inverse of the covariant vector and hence we expect the behaviour of contravariant vector and covariant vector under coordinate. More compactly, the q are the components of a tensor of type 2. This procedure naturally generalizes to multilinear maps with different numbers of covariant and contravariant arguments and also shows where the transforms like a tensor bit comes from. Vectors, metric and the connection 1 contravariant and. Metric tensor metrics minimal surface normal derivative normal velocity. Higher order tensors are in principle handled similarly, but they may be expressed with mixed coordinate types i. In equation 29 components of covariant vector dual covary with change of coordinate basis and hence the name covariant vector. 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 contravariant component of any arbitrary vector a is ai with superscript index and covariant component is a i with subscript index are taken to be understood. We may play this game in the euclidean space en with its \dot inner product.
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. The following code demonstrates the difference between assignment compatibility, covariance, and contravariance. Covariance preserves assignment compatibility and contravariance reverses it. Covariance and contravariance computer science wikipedia.
Second, tensor theory, at the most elementary level, requires only linear algebra and some calculus as prerequisites. Covariant vectors are written as rows 1 of components with subscripts. A generic type parameter that is not marked covariant or contravariant is referred to as invariant. Covariance and contravariance in generics microsoft docs. Chapter 10 vectors and tensors georgia institute of. This is done with the metric tensor g which turns a covector into a vector by the operation. If you have a metric texgtex on a manifold then it is usually regarded as being a map which takes two vectors into a real number. This collection of notes attempts to demystify some of the vagaries of tensor analysis, to emphasize the connections between the old way of doing things and the new. Sirs, as a physicist studying general relativity and quantum field theory, i feel like i have my head wrapped around upstairsdownstairs notation fairly well. The metric tensor is g ij and is most easily understood when represented by a square matrix.
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