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However, the dimension of the space is largely irrelevant in most tensor equations with the notable exception of the contracted Kronecker delta. Tensors are generalizations of scalars that have no indices , vectors that have exactly one index , and matrices that have exactly two indices to an arbitrary number of indices. Tensors provide a natural and concise mathematical framework for formulating and solving problems in areas of physics such as elasticity, fluid mechanics, and general relativity. The notation for a tensor is similar to that of a matrix i. In addition, a tensor with rank may be of mixed type , consisting of so-called "contravariant" upper indices and "covariant" lower indices.

Note that the positions of the slots in which contravariant and covariant indices are placed are significant so, for example, is distinct from.

Vector and Tensor Analysis with Applications

While the distinction between covariant and contravariant indices must be made for general tensors, the two are equivalent for tensors in three-dimensional Euclidean space , and such tensors are known as Cartesian tensors. Objects that transform like zeroth-rank tensors are called scalars , those that transform like first-rank tensors are called vectors , and those that transform like second-rank tensors are called matrices.


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In tensor notation, a vector would be written , where , Tensors may be operated on by other tensors such as metric tensors , the permutation tensor , or the Kronecker delta or by tensor operators such as the covariant derivative. The manipulation of tensor indices to produce identities or to simplify expressions is known as index gymnastics , which includes index lowering and index raising as special cases.

These can be achieved through multiplication by a so-called metric tensor , , , etc.

Tensor notation can provide a very concise way of writing vector and more general identities. For example, in tensor notation, the dot product is simply written. Similarly, the cross product can be concisely written as. Contravariant second-rank tensors are objects which transform as. Covariant second-rank tensors are objects which transform as.

If two tensors and have the same rank and the same covariant and contravariant indices, then they can be added in the obvious way,. The generalization of the dot product applied to tensors is called tensor contraction , and consists of setting two unlike indices equal to each other and then summing using the Einstein summation convention. Various types of derivatives can be taken of tensors, the most common being the comma derivative and covariant derivative. If the components of any tensor of any tensor rank vanish in one particular coordinate system, they vanish in all coordinate systems.

A transformation of the variables of a tensor changes the tensor into another whose components are linear homogeneous functions of the components of the original tensor. A tensor space of type can be described as a vector space tensor product between copies of vector fields and copies of the dual vector fields, i.

Tensor Calculus 4b: Position Vector, Covariant Basis, Covariant Metric Tensor, Contravariant Basis

For example,. Tensors of type form a vector space. This description generalized to any tensor type, and an invertible linear map induces a map , where is the dual vector space and the Jacobian , defined by. This definition can be extended similarly to other tensor products of and.

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When there is a change of coordinates , then tensors transform similarly, with the Jacobian of the linear transformation. Portions of this entry contributed by Todd Rowland. Abraham, R. Manifolds, Tensor Analysis, and Applications, 2nd ed. New York: Springer-Verlag, Akivis, M. An Introduction to Linear Algebra and Tensors. New York: Dover, Arfken, G.

Orlando, FL: Academic Press, pp. Aris, R. Bishop, R. Tensor Analysis on Manifolds. Borisenko, A.

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Vector and Tensor Analysis with Applications. Bott, R. Hence, this textbook addresses graduate students as well as scientists working in this field and in particular dealing with multi-physical problems. In each chapter numerous exercises are included, allowing for self-study and intense practice. Solutions to the exercises are also provided.

Springer Professional. Back to the search result list. Table of Contents Frontmatter Chapter 1. So far we have considered solely real vectors and real vector spaces.

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For the purposes of this chapter an introduction of complex vectors is, however, necessary. Indeed, in the following we will see that the existence of a solution of an eigenvalue problem even for real second-order tensors can be guaranteed only within a complex vector space. Fourth-order tensors play an important role in continuum mechanics where they appear as elasticity and compliance tensors.


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In this section we define fourth-order tensors and learn some basic operations with them. In the previous chapter we discussed isotropic and anisotropic tensor functions and their general representations. Of particular interest in continuum mechanics are isotropic tensor-valued functions of one arbitrary not necessarily symmetric tensor.