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∞(M) Module Explanation, Applications, Tensor Calculus, Twisting By A Line Bundle, The Flat Case, Cocycles and Chain Rules
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