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 called "tensors" are tensor fields. For example, the Riemann curvature tensor is not a tensor, as the name implies, but a tensor field: It is named after Bernhard Riemann, and associates a tensor to each point of a Riemannian manifold, which is a topological space.

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.

The general idea of tensor field combines the requirement of richer geometry – for example, an ellipsoid varying from point to point, in the case of a metric tensor – with the idea that we don't want our notion to depend on the particular method of mapping the surface. It should exist independently of latitude and longitude, or whatever particular 'cartographic projection' we are using to introduce numerical coordinates.

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