Tensor Field

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 FieldGeometric Introduction, The Vector Bundle Explanation, Notation, The Cāˆž(M) Module Explanation, Applications, Tensor Calculus, Twisting By A Line Bundle, The Flat Case, Cocycles and Chain Rules

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

Tensor Derivative (continuum Mechanics) - Divergence of A 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 nāˆ’1 ...
Scalar-vector-tensor Decomposition
... In cosmological perturbation theory, the scalar-vector-tensor decomposition is a decomposition of the most general linearized perturbations of the ... 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 ... 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 ... The other vector bundles of tensors have comparable cocycles, which come from applying functorial properties of tensor constructions to the chain rule itself this is why they also are intrinsic (read, 'nat ... 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 ...
Saint-Venant's Compatibility Condition - Rank 2 Tensor Fields
... 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 ... are finite dimensional spaces of symmetric tensors with vanishing Saint-Venant's tensor that are not the symmetric derivative of a vector field ... lemma can be understood more clearly using the operator, where is a symmetric tensor field ...

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