In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of a physical space, in which case the field quantity acquires a unit of measurement. Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in material object, 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 magnitude and a direction, like velocity), a tensor field is a generalization of a scalar field and a 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. A tensor field, in common usage, is often referred to in the shorter form "tensor". For example, the Riemann curvature tensor refers a tensor field, as it associates a tensor to each point of a Riemannian manifold, a topological space.
Intrinsic approach
Geometric introduction
A scalar field is an assignment (mathematically a function) of a real or complex number to every point of a manifold. One example is the temperature field. The temperature at a point does not change even if the coordinate system changes. 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. Again, the length and direction of these vectors is coordinate-independent. Now consider more complicated fields. For example, if the manifold is Riemannian, then it has a metric field g {\displaystyle g} , such that given any two vectors v , w {\displaystyle v,w} at point x {\displaystyle x} , their inner product is g x ( v , w ) {\displaystyle g_{x}(v,w)} . The field g {\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. An intrinsic definition should specify tensor fields in a coordinate-independent way: independently of latitude and longitude, or whatever particular "cartographic projection" we are using to introduce numerical coordinates. One way to achieve this is using the concept of tensor bundles.
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. (There are vector bundles that are not tensor bundles: the Möbius band for instance.) Explicitly, a tensor bundle is the fiber bundle
V ⊗ ⋯ ⊗ V ⊗ V ∗ ⊗ ⋯ ⊗ V ∗ = V ⊗ p ⊗ ( V ∗ ) ⊗ q {\displaystyle V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*}=V^{\otimes p}\otimes (V^{*})^{\otimes q}}
where V is the tangent bundle of M and V∗ is the cotangent bundle of M, and its fibers are
V x ⊗ ⋯ ⊗ V x ⊗ V x ∗ ⊗ ⋯ ⊗ V x ∗ = V x ⊗ p ⊗ ( V x ∗ ) ⊗ q {\displaystyle V_{x}\otimes \cdots \otimes V_{x}\otimes V_{x}^{*}\otimes \cdots \otimes V_{x}^{*}=V_{x}^{\otimes p}\otimes (V_{x}^{*})^{\otimes q}}
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