In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a pure number plus a direction, like velocity), a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space. If a tensor A is defined on a vector fields set X(M) over a module M, we call A a tensor field on M.^[1]

Many mathematical structures called "tensors" are also tensor fields. For example, the Riemann curvature tensor is a tensor field as it associates a tensor to each point of a Riemannian manifold, which is a topological space.

Definition

Let M be a manifold, for instance the Euclidean plane Rⁿ.

Definition. A tensor field of type (p, q) is a section

${\displaystyle T\ \in \ \Gamma (M,V^{\otimes p}\otimes (V^{*})^{\otimes q})}$

where V is a vector bundle on M, V^* is its dual and ⊗ is the tensor product of vector bundles.

Equivalently, it is a collection of elements T_x ∈ V_x^⊗p ⊗ (V_x^*)^⊗q for all points x ∈ M, arranging into a smooth map T : M → V^⊗p ⊗ (V^*)^⊗q. Elements T_x are called tensors.

Often we take V = TM to be the tangent bundle of M.

Geometric introduction

Intuitively, a vector field is best visualized as an "arrow" attached to each point of a region, with variable length and direction. One example of a vector field on a curved space is a weather map showing horizontal wind velocity at each point of the Earth's surface.

Now consider more complicated fields. For example, if the manifold is Riemannian, then it has a metric field ${\displaystyle g}$, such that given any two vectors ${\displaystyle v,w}$ at point ${\displaystyle x}$, their inner product is ${\displaystyle g_{x}(v,w)}$. The field ${\displaystyle g}$ could be given in matrix form, but it depends on a choice of coordinates. It could instead be given as an ellipsoid of radius 1 at each point, which is coordinate-free. Applied to the Earth's surface, this is Tissot's indicatrix.

In general, we want to specify tensor fields in a coordinate-independent way: It should exist independently of latitude and longitude, or whatever particular "cartographic projection" we are using to introduce numerical coordinates.

Via coordinate transitions

Following Schouten (1951) and McConnell (1957), the concept of a tensor relies on a concept of a reference frame (or coordinate system), which may be fixed (relative to some background reference frame), but in general may be allowed to vary within some class of transformations of these coordinate systems.^[2]

For example, coordinates belonging to the n-dimensional real coordinate space ${\displaystyle \mathbb {R} ^{n}}$ may be subjected to arbitrary affine transformations:

${\displaystyle x^{k}\mapsto A_{j}^{k}x^{j}+a^{k}}$

(with n-dimensional indices, summation implied). A covariant vector, or covector, is a system of functions ${\displaystyle v_{k}}$ that transforms under this affine transformation by the rule

${\displaystyle v_{k}\mapsto v_{i}A_{k}^{i}.}$

The list of Cartesian coordinate basis vectors ${\displaystyle \mathbf {e} _{k}}$ transforms as a covector, since under the affine transformation ${\displaystyle \mathbf {e} _{k}\mapsto A_{k}^{i}\mathbf {e} _{i}}$. A contravariant vector is a system of functions ${\displaystyle v^{k}}$ of the coordinates that, under such an affine transformation undergoes a transformation

${\displaystyle v^{k}\mapsto (A^{-1})_{j}^{k}v^{j}.}$

This is precisely the requirement needed to ensure that the quantity ${\displaystyle v^{k}\mathbf {e} _{k}}$ is an invariant object that does not depend on the coordinate system chosen. More generally, a tensor of valence (p,q) has p downstairs indices and q upstairs indices, with the transformation law being

${\displaystyle {T_{i_{1}\cdots i_{p}}}^{j_{1}\cdots j_{q}}\mapsto A_{i_{1}}^{i'_{1}}\cdots A_{i_{p}}^{i'_{p}}{T_{i'_{1}\cdots i'_{p}}}^{j'_{1}\cdots j'_{q}}(A^{-1})_{j'_{1}}^{j_{1}}\cdots (A^{-1})_{j'_{q}}^{j_{q}}.}$

The concept of a tensor field may be obtained by specializing the allowed coordinate transformations to be smooth (or differentiable, analytic, etc.). A covector field is a function ${\displaystyle v_{k}}$ of the coordinates that transforms by the Jacobian of the transition functions (in the given class). Likewise, a contravariant vector field ${\displaystyle v^{k}}$ transforms by the inverse Jacobian.

Tensor bundles

A tensor bundle is a fiber bundle where the fiber is a tensor product of any number of copies of the tangent space and/or cotangent space of the base space, which is a manifold. As such, the fiber is a vector space and the tensor bundle is a special kind of vector bundle.

The vector bundle is a natural idea of "vector space depending continuously (or smoothly) on parameters" – the parameters being the points of a manifold M. For example, a vector space of one dimension depending on an angle could look like a Möbius strip or alternatively like a cylinder. Given a vector bundle V over M, the corresponding field concept is called a section of the bundle: for m varying over M, a choice of vector

v_m in V_m,

where V_m is the vector space "at" m.

Since the tensor product concept is independent of any choice of basis, taking the tensor product of two vector bundles on M is routine. Starting with the tangent bundle (the bundle of tangent spaces) the whole apparatus explained at component-free treatment of tensors carries over in a routine way – again independently of coordinates, as mentioned in the introduction.

We therefore can give a definition of tensor field, namely as a section of some tensor bundle. (There are vector bundles that are not tensor bundles: the Möbius band for instance.) This is then guaranteed geometric content, since everything has been done in an intrinsic way. More precisely, a tensor field assigns to any given point of the manifold a tensor in the space

${\displaystyle V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*},}$

where V is the tangent space at that point and V^∗ is the cotangent space. See also tangent bundle and cotangent bundle.

Given two tensor bundles E → M and F → M, a linear map A: Γ(E) → Γ(F) from the space of sections of E to sections of F can be considered itself as a tensor section of ${\displaystyle \scriptstyle E^{*}\otimes F}$ if and only if it satisfies A(fs) = fA(s), for each section s in Γ(E) and each smooth function f on M. Thus a tensor section is not only a linear map on the vector space of sections, but a C^∞(M)-linear map on the module of sections. This property is used to check, for example, that even though the Lie derivative and covariant derivative are not tensors, the torsion and curvature tensors built from them are.

Notation

The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus, the tangent bundle TM = T(M) might sometimes be written as

${\displaystyle T_{0}^{1}(M)=T(M)=TM}$

to emphasize that the tangent bundle is the range space of the (1,0) tensor fields (i.e., vector fields) on the manifold M. This should not be confused with the very similar looking notation Zdroj:https://en.wikipedia.org?pojem=Tensor_field
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.