Not all relationships in nature are linear, but most are differentiable and so may be locally approximated with sums of multilinear maps. Similarly, mathematicians find there are some tensor relations which are more conveniently derived in a co-ordinate notation). (Of course, physicists and engineers are among the first to recognise that vectors and tensors have a physical significance as entities, which goes beyond the (often arbitrary) co-ordinate system in which their components are enumerated. Covariant vectors, for instance, can also be described as one-forms, or as the elements of the dual space to the contravariant vectors. The usual mathematics way, which involves defining certain vector spaces and not fixing any coordinate systems until bases are introduced when needed.The usual physics way of defining tensors, in terms of objects whose components transform according to certain rules, introducing the ideas of covariant or contravariant transformations.There are two ways of approaching the definition of tensors: To understand tensor fields, you need to first understand the basic idea of tensors. Note that the word "tensor" is often used as a shorthand for tensor field, which is a tensor value defined at every point in a manifold. But tensors are used also within other fields such as continuum mechanics, for example the strain tensor, (see linear elasticity). General Relativity is formulated completely in the language of tensors, which Einstein had learned from Levi-Civita himself with great difficulty. The tensor calculus achieved broader acceptance with the introduction of Einstein's theory of general relativity, around 1915. The notation was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential geometry, and made accessible to many mathematicians by the publication of Tullio Levi-Civita's classic text The Absolute Differential Calculus in 1900 (in Italian translations followed). The word was used in its current meaning by Woldemar Voigt in 1899.
The word "tensor" was first introduced by William Rowan Hamilton in 1846, but he used the word for what is now called modulus.
This article attempts to provide a non-technical introduction to the idea of tensors, and to provide an introduction to the articles which describe different, complementary treatments of the theory of tensors in detail. The abstract theory of tensors is a branch of linear algebra, now called multilinear algebra. In particular, tensors behave in specific ways under coordinate transformations. While tensors can be represented by multi-dimensional arrays of components, the point of having a tensor theory is to explain further implications of saying that a quantity is a tensor, beyond that specifying it requires a number of indexed components. In the field of diffusion tensor imaging, for instance, a tensor quantity that expresses the differential permeability of organs to water in varying directions is used to produce scans, for example of the brain. Tensors are of importance in physics and engineering. Tensors may be written down in terms of coordinate systems, as arrays of scalars, but are defined so as to be independent of any chosen frame of reference. The tensor concept includes the ideas of scalar, vector and linear operator. In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized 'quantity'.
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