In mathematics, physics, and engineering, 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 and engineering. As a tensor is a generalization of a scalar (a pure number representing a value, like length) and a vector (a geometrical arrow in space), 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.

Many mathematical structures informally called 'tensors' are actually 'tensor fields'. An example is the Riemann curvature tensor.

Read more about Tensor Field: Geometric Introduction, The Vector Bundle Explanation, Notation, The *C*ā(*M*) Module Explanation, Applications, Tensor Calculus, Twisting By A Line Bundle, The Flat Case, Cocycles and Chain Rules

### Other articles related to "tensor field, tensor, tensors, field, fields, tensor fields":

**Tensor Field**s

... condition takes the form of the vanishing of the Saint-Venant's

**tensor**defined by The result that, on a simply connected domain W=0 implies that strain is the ... connected domains there are finite dimensional spaces of symmetric

**tensors**with vanishing Saint-Venant's

**tensor**that are not the symmetric derivative of a vector

**field**... and Poincare's lemma can be understood more clearly using the operator, where is a symmetric

**tensor field**...

**Tensor Field**

... The divergence of a

**tensor field**is defined using the recursive relation where c is an arbitrary constant vector and v is a vector

**field**... If is a

**tensor field**of order n > 1 then the divergence of the

**field**is a

**tensor**of order nā1 ...

... In cosmological perturbation theory, the scalar-vector-

**tensor**decomposition is a decomposition of the most general linearized perturbations of the Friedmann-Lemaitre-Robertson-Walker metric into components according to ... into four scalars, two divergence-free spatial vector

**fields**(that is, with a spatial index running from 1 to 3), and a traceless, symmetric spatial

**tensor field**with vanishing doubly and singly longitudinal ... The vector and

**tensor fields**each have two independent components, so this decomposition encodes all ten degrees of freedom in the general metric perturbation ...

**Tensor Field**- Cocycles and Chain Rules

... As an advanced explanation of the

**tensor**concept, one can interpret the chain rule in the multivariable case, as applied to coordinate changes, also as the requirement for self-consistent concepts of ... The other vector bundles of

**tensors**have comparable cocycles, which come from applying functorial properties of

**tensor**constructions to the chain rule ... What is usually spoken of as the 'classical' approach to

**tensors**tries to read this backwards ā and is therefore a heuristic, post hoc approach rather than truly a foundational one ...

### Famous quotes containing the word field:

“The *field* of doom bears death as its harvest.”

—Aeschylus (525–456 B.C.)