Tensor

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A tensor is a certain kind of geometrical entity which generalizes the concepts of scalar, vector and linear operator. Tensors are of importance in differential geometry, physics and engineering. Einstein's theory of general relativity is formulated completely in the language of tensors.

As a simple example, consider a ship in the water. We want to describe its response to an applied force. Force is a vector, and the ship will respond with an acceleration, which is also a vector. The acceleration will in general not be in the same direction as the force, because of the particular shape of the ship's body. However, it turns out that the relationship between force and acceleration is linear. Such a relationship is described by a tensor of type (1,1). In engineering, the stresses inside a rigid body or fluid are also described by a tensor; the word "tensor" is Latin for something that stretches, i.e. causes tension. If a particular surface element inside the material is singled out, the material on one side of the surface will apply a force on the other side. In general, this force will not be orthogonal to the surface, but it will depend on the orientation of the surface in a linear manner. This is described by a tensor of type (2,0), or more precisely by a tensor field of type (2,0) since the stresses may change from point to point.

Not all relationships in nature are linear, but most are differentiable and so may be approximated with sums of multilinear maps. Thus most quantities in the physical sciences can be expressed as tensors.

Formally, a tensor is defined as follows. Given any two real vector spaces V, W, their tensor product is a real vector space V Ä W together with a bilinear map Ä: V x W -> V Ä W. If {ei} and {fj} are bases for V and W, the {ei Ä fj} is a basis for V Ä W, and the dimension of V Ä W is given by the product of the dimensions of V and W. This tensor product can be generalized to more than just two vector spaces. A tensor on the vector space V is then defined to be an element of the tensor product V Ä V Ä ... Ä V Ä V* Ä V* Ä ... Ä V*, where V* is the dual space of V. If there are m copies of V and n copies of V* in our product, the tensor is said to be of type (m, n) and of contravariant rank m and covariant rank n. The tensors of rank zero are just the scalars R, those of contravariant rank 1 the vectors in V, and those of covariant rank 1 the one-forms in V* (for this reason the last two spaces are often called the contravariant and covariant vectors).

Note that the (1,1) tensors V Ä V* are isomorphic in a natural way to the space of linear transformations (i.e. matrices) from V to V. An inner product V x V -> R corresponds in a natural way to a (0,2) tensor in V* Ä V*, called the associated metric and usually denoted g.

In differential geometry, physics and engineering, we usually deal with tensor fields on differentiable manifolds. (The term "tensor" is sometimes used as a shorthand for "tensor field".) For instance, the curvature tensor is discussed in differential geometry and the stress-energy tensor is important in physics and engineering. Both of these are related by Einstein's theory of general relativity. In engineering, the underlying manifold will often be Euclidean 3-space. A tensor field assigns to any given point of the manifold a tensor in the space V Ä V Ä ... Ä V Ä V* Ä V* Ä ... Ä V*, where V is the tangent space at that point and V* is the cotangent space.

For any given coordinate system we have a basis {ei} for the tangent space V (note that this may vary from point-to-point if the manifold is not linear), and a corresponding dual basis {ei} for the cotangent space V* (see dual space). The difference between the raised and lowered indices is there to remind us of the way the components transform.

For example purposes, then, take a tensor A in the space V Ä V Ä V*. The components relative to our coordinate system can be written

A = Aijk (ei Ä ej Ä ek)

Here we used the Einstein notation, a convention useful when dealing with coordinate equations: when an index variable appears both raised and lowered on the same side of an equation, we are summing over all its possible values. In physics we often use the expression Aijk to represent the tensor, just as vectors are usually treated in terms of their components. This can be visualized as an n x n x n array of numbers. In a different coordinate system, say given to us as a basis {ei'}, the components will be different. If (xi'i) is our transformation matrix (note it is not a tensor, since it represents a change of basis rather than a geometrical entity) and if (yii') is its inverse, then our components vary per

Ai'j'k' = xi'i xj'j ykk' Aijk

In older texts this transformation rule often serves as the definition of a tensor. Formally, this means that tensors were introduced as specific representations of the group of all changes of coordinate systems.


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