Tensor

In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized 'quantity'. The tensor concept includes the ideas of scalars, vectors, and linear operators. Tensors may be written down in terms of coordinate systems, as arrays of scalars, but are defined so as to be independent of any chosen frame of reference. Tensors are of importance in physics and engineering. In the field of diffusion tensor imaging, for instance, a tensor quantity that expresses the differential permeability of organs to water in varying directions is used to produce scans of the brain. Perhaps the most important engineering examples are the stress tensor and strain tensor, which are both 2nd rank tensors, and are related in a general linear material by a fourth rank elasticity tensor.

Related Topics:
Mathematics - Geometrical - Scalar - Vectors - Linear operator - Coordinate system - Frame of reference - Physics - Engineering - Diffusion tensor imaging - Permeability - Brain - Stress tensor - Strain tensor - 2nd rank - Elasticity tensor

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

While tensors can be represented by multi-dimensional arrays of components, the point of having a tensor theory is to explain further implications of saying that a quantity is a tensor, beyond that specifying it requires a number of indexed components. In particular, tensors behave in specific ways under coordinate transformations. The abstract theory of tensors is a branch of linear algebra, now called multilinear algebra.

Related Topics:
Coordinate transformation - Linear algebra - Multilinear algebra

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

This article attempts to provide a non-technical introduction to the idea of tensors, and to provide an introduction to the articles which describe different, complementary treatments of the theory of tensors in detail.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~