Covariant formulation of the tensor algebra of nonlinear. We have made no restrictions upon our choice of basis vectors e. We wish to write the velocity vector at some time t 0. One part are ordinary vector spaces contravariant, the other dual vector spaces covariant. In other words, outside, the electric potential generated by the surface charge distribution, combined with that generated by the surface dipole distribution, completely cancels out the electric potential and, hence, the electric field produced by the charges distributed within. 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. This table shows important examples of tensors on vector spaces and tensor fields on manifolds.
Physics 221a fall 2019 appendixe introduction to tensor. Univariant definition of univariant by merriamwebster. In this video, i describe the meaning of contravariant and covariant vector components. If i have covariant, but multiplying by this, i obtain contravariant vector. Tensors are objects in multilinear algebra that can have aspects of both covariance and contravariance. Covariance and contravariance are collectively referred to as variance. However, sather supported both covariance and contravariance. In a metric space, the contravariant and covariant tensors can be transformed to each other using the metric tensors g mn and g mn.
Covariance and contravariance of vectors wikipedia. A contravariant tensor is a tensor having specific transformation properties c. Intuitive way to understand covariance and contravariance in. Vectors, metric and the connection 1 contravariant and. The expansion in orthonormal basis is mathematically simple. 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.
Covariant tensor article about covariant tensor by the. I know the author as a research scholar who has worked with me for several years. To examine the transformation properties of a contravariant tensor, first consider a tensor of rank 1 a vector. Kolecki national aeronautics and space administration glenn research center cleveland, ohio 445 tensor analysis is the type of subject that can make even the best of students shudder. With the notion of contravariant and covariant components of a vector, we. The valence or type of a tensor gives the number of covariant and contravariant component indices. A contravariant tensor is a tensor having specific transformation properties cf. Since you asked for an intuitive way to understand covariance and contravariance, i think this will do. Furthermore a contravariant functor refers to the dual category, which is exactly the opposite of how physicists use it. Tensors can, in general, have rank greater than 2, and. One important characteristic is the rank of a tensor, which is the number of indicies needed to specify the tensor. Finally we are in a position to see how covariant and contravariant vectors differ in this metric. This is the second volume of a twovolume work on vectors and tensors. Covariant definition of covariant by the free dictionary.
A brief summary of facts about variance in the common language runtime. General relativitycontravariant and covariant indices. Nevertheless it is convenient to use the notation of upper and lower indices in the sums in. An ordinary matrix is a rank 2 tensor, a vector is a rank 1 tensor, and a scalar is rank 0. In cartesian space they must be an orthogonal norm preserving matrix. Contravariant is a mathematical term with a precise definition in tensor analysis.
Intuitive way to understand covariance and contravariance. The \rank of the tensor is the number of separate indices it has. 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. Vectors and covectors in special relativity jim napolitano march 12, 2010 these notes are meant to accompany the course electromagnetic theory for the spring 2010 term at rpi. The velocity vector in equation 3 corresponds to neither the covariant nor contravari. This means that they have components that are both covariant and contravariant. We use this repeatedly in the algebra in sections below. In quantum computing and condensed matter, there are an important class of states called tensor network states, which have amplitudes that are obtained by contracting tensors. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle see affine connection. Abstract vector is a physical quantity and it does not depend on any coordinate system. On the other hand, the energymomentum tensor of a scalar field is more naturally written as a covariant tensor 4.
A generic type parameter that is not marked covariant or contravariant is referred to as invariant. Chapter 10 vectors and tensors in this chapter we explain how a vector space v gives rise to a family of associated tensor spaces, and how mathematical objects such as linear maps or quadratic forms should be understood as being elements of these spaces. Earlier versions of the itensor package in maxima used a notation that sometimes led to incorrect index ordering. Entangled states and quantum causality threshold in the general theory of relativity. The contravariant tensor conjugate to gij is denoted by gij satisfy 5. A covariant tensor, denoted with a lowered index e. An introduction to tensors for students of physics and. An introduction to tensors for students of physics and engineering joseph c. Contravariant tensor article about contravariant tensor by. The gradient g is an example of a covariant tensor, and the differential position d dx is an example of a contravariant tensor. The electromagnetic field tensor let us now investigate whether we can write the components of the electric and magnetic fields as the components of some proper 4 tensor. If, the covariant tensor is called a covariant vector.
Examples of how to use contravariant in a sentence from the cambridge dictionary labs. We are now ready to get serious about electrodynamics. The presentation is based on how various quantities transform under coordinate transformations, and is fairly standard. 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. Such a field assigns to each an element of, the fold tensor product of the tangent space to at. This question regards making sympys geometric algebra module use both covariant and contravariant vector forms to make the output much more compact. The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. Also, the contravariant covariant forms of the metric tensor are expressed as the dot product of a pair of contravariant covariant basis vectors. Khaled abdelwaged of student hanan hassan makallawi 2 covariant and contra variant vectors. Covariant formulation of the tensor algebra of nonlinear elasticity article in international journal of nonlinear mechanics 472 march 2012 with 115 reads how we measure reads.
Planetphysicscovariance and contravariance wikiversity. We have developed a beautiful, geometric system for describing the coordinates in terms of which electrodynamics must be formulated for the speed of light to be an invariant. Chapter 10 vectors and tensors georgia institute of. Expressions for lengths, areas and volumes of objects in the vector space can then be given in terms of tensors with covariant and contravariant indices.
The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern differential geometry uses more sophisticated indexfree methods to represent tensors. The difference between these two kinds of tensors is how they transform under a continuous change of coordinates. In that spirit we begin our discussion of rank 1 tensors. It specifies precisely the method direction of projection used to derive the components by projecting the magnitude of the tensor quantity onto the coordinate system being used as the basis of the tensor.
Covariance and contravariance in generics microsoft docs. 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. Now, if this energyforce 4vector equation is to be covariant so its transformed form is still a 4vector then the right hand sides must form a 4vector too. In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. The elements of this tensor can be substituted directly into the eigenvalue problem for general coordinate systems written in covariant notation. Tensors are fundamental geometrical objects just like vectors, in fact they are vectors. This is obtained from the basic contravariant field strength tensor by the substitutions. 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. Volume 1 is concerned with the algebra of vectors and tensors, while this volume is concerned with the geometrical aspects of vectors and tensors. This volume begins with a discussion of euclidean manifolds.
In multilinear algebra and tensor analysis, covariance and contravariance describe how the. Covariant and contravariant tensors physics forums. Discusses covariance and contravariance in generic and nongeneric delegates and provides a list of variant generic delegates in the. An informal discussion on tensor calculus chris tiee abstract. Covariant tensor article about covariant tensor by the free. The terms covariant and contravariant were introduced by james joseph sylvester in 1851 in the context of associated algebraic forms theory. Introduction to tensor calculus for general relativity mit. This collection of notes attempts to demystify some of the vagaries of tensor. Vectors, metric and the connection 1 contravariant and covariant vectors 1.
Under simple expansions and contractions of the coordinates, the reciprocity is. Tensors covariant differential and riemann tensor coursera. Feb 17, 2010 vector is a physical quantity and it does not depend on any coordinate system. In tensor analysis, a covariant vector varies more or less reciprocally to a corresponding contravariant vector. Covariance and contravariance computer science wikipedia. There are neither covariant or contravariant tensors for our case. For example, for a tensor of contravariant rank 2 and covariant rank 1. To make the lorentz transformations look nice and simple, we should instead express them in terms of 4vectors, where. Tell a friend about us, add a link to this page, or visit the webmasters page for free fun content.
If we use the symbol a to denote a vector, and a b to denote the inner product between. Introduction to the essentials of tensor calculus 6 the invariant measure of volume is easily constructed as. A scalar function that depends entirely on the principal invariants of a tensor is objective, i. Examples of how to use covariant in a sentence from the cambridge dictionary labs. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. Tensor which depends on two directions one normal to the area and other that of the force on it.
Dec 08, 2014 this course will eventually continue on patreon at textbook. Physics 221a fall 2019 appendixe introduction to tensor analysis. This is because they are written in the language of 3vectors. The tensor components have co and contravariant indices. Physicists use it this way, the terms contravariant and covariant in mathematics are defined for functors and not for tensors. First, tensors appear everywhere in physics, including classi. How can we identify the components of the magnetic field, which is a pseudovector, with any of the components of a proper4 tensor. Net framework 4, variant type parameters are restricted to generic interface and generic. I understand that, in curvilinear coordinates, one can define a covariant basis and a contravariant basis. Converting contravariant or covariant tensors to mixed tensors. I wanted to know of book suggestions that can help me overcome my fear of indices.
One especially important secondrank tensor is the metric tensor, which well talk about now. A gentle introduction to tensors boaz porat department of electrical engineering technion israel institute of technology. Transformation of the covariant components of a lorentz tensor is defined as. The differences of zircon titanium thermometry, iuhft value and the covariant relation of rbsr and ba among the two stages of magmatism show that tsona leucogranties originated from more than one melting reactions, they had experienced the change from dehydration melting to fluid fluxed melting, and the reason respond to this transformation. 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. Univariant definition is having one degree of freedom used of a physicalchemical system. It seems to me that any vector can be decomposed in either of those basis, thus one can have covariant components and contravariant components of the same vector, depending on the chosen basis.
This course will eventually continue on patreon at textbook. Transformation of contravariant covariant components. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering. This metric observable tensor, in real observations where the observer accompanies his references, is the same that the analogous built general covariant tensor h. In this section, the concept of contravariant and covariant vectors is extended to tensors. The article considers how this applies to some common type constructors.
To examine the transformation properties of a covariant tensor, first consider the gradient. Contravariant vector an overview sciencedirect topics. Now that we have talked about tensors, we need to figure out how to classify them. This property is commonly used in formulating closedform expressions for the strain energy density, or helmholtz free energy, of a nonlinear material possessing isotropic symmetry. Introduction these notes contain an introduction to tensor analysis as it is commonly used in physics, but mostly limited to the needs of this course. It gives me great pleasure to write the foreword to dr. Notice how the contravariant basis vector g is not differentiated. In this chapter we introduce a new kind of vector covector, one that will be es sential for the rest of this booklet. Conversion of mixed tensors into mixed tensors and into covariant or contravariant ones. In tensor analysis, a mixed tensor is a tensor which is neither strictly covariant nor strictly contravariant. In general the upper indices can be lowered and the lower indices can be made to be upper indices, using the metric tensors.
It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. A mixed type tensor transforms covariantly in its covariant indices and contravariantly in its contravariant indices, e. The transformation that describes the new basis vectors as a linear combination of the old basis vectors is defined as a covariant transformation. But in many physical situations we have to choose an nonorthogonal basis or oblique coordinate system. Note that in chapter ii it was shown that the last two expressions have the. The phrase contravariant tensor of order r is also used to denote a contravariant tensor field of order over a smooth manifold. The metric tensor represents a matrix with scalar elements or and is a tensor object which is used to raise or lower the index on another tensor object by an operation called contraction, thus allowing a covariant tensor to be converted to a contravariant tensor, and vice versa. Covariant and contravariant vectors tensor euclidean vector. As mentioned in a previous video, tensors are invariant under coordinate transformations. This means it is not necessary to transform the expression such as must be done for a tensor in the box on page iii1.
A covariant vector is specifically a vector which transforms with the basis vectors, a contravariant vector on the other hand is a vector that transforms against the basis vectors. In general, these transformation properties differ from those of a contravariant tensor. 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 corresponding covariant vector. Jul 20, 2017 tensors of rank 1, 2, and 3 visualized with covariant and contravariant components. This material is covered thoroughly in chapters one and seven in our textbook classical electrodynamics, 2nd ed. Although tensors are applied in a very broad range of physics and math. In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis. In ndimensional space a tensor of rank n has nn components. Second, tensor theory, at the most elementary level, requires only. Calling convention for overridden methods are covariant with out parameters and return values, and contravariant with normal parameters with the mode in.
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