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# tensor

## Definitions

### from The American Heritage® Dictionary of the English Language, 5th Edition.

• noun Anatomy A muscle that stretches or tightens a body part.
• noun Mathematics A set of quantities that obey certain transformation laws relating the bases in one generalized coordinate system to those of another and involving partial derivative sums. Vectors are simple tensors.

### from The Century Dictionary.

• noun In anatomy, one of several muscles which tighten a part, or make it tense, or put it upon the stretch: differing from an extensor in not changing the relative position or direction of the axis of the part: opposed to laxator.
• noun In mathematics, the modulus of a quaternion; the ratio in which it stretches the length of a vector.
• In anatomy, noting certain muscles whose function is to render fasciæ or other structures tense.

### from the GNU version of the Collaborative International Dictionary of English.

• noun (Anat.) A muscle that stretches a part, or renders it tense.
• noun (Geom.) The ratio of one vector to another in length, no regard being had to the direction of the two vectors; -- so called because considered as a stretching factor in changing one vector into another. See Versor.

• noun A muscle that stretches a part, or renders it tense.
• adjective Of or relating to tensors
• verb To compute the tensor product of two tensors.

• noun any of several muscles that cause an attached structure to become tense or firm
• noun a generalization of the concept of a vector

## Etymologies

### from The American Heritage® Dictionary of the English Language, 4th Edition

[New Latin tēnsor, from Latin tēnsus, past participle of tendere, to stretch; see tense.]

## Examples

• For those familiar with tensors, it should be clear that the metric tensor is actually a tensor field (a tensor is assigned to each point of our mathematical space).

• For those familiar with tensors, it should be clear that the metric tensor is actually a tensor field (a tensor is assigned to each point of our mathematical space).

• EDIT: A note from The Unapologetic Mathematician that I should add: the metric tensor is a bilinear function of two vectors at a given point, while the line element is a quadratic function of a single vector.

• A metric tensor is a function defined on a manifold (a vector space) that takes in two tangent vectors and produces a scalar quantity.

• As in general relativity the metric tensor is written as gij = ηij + hij …

• A metric tensor is NOT the same object as a metric, it is NOT the same object as its matrix representation, and it is NOT the same object as its associated line element.

• EDIT: A note from The Unapologetic Mathematician that I should add: the metric tensor is a bilinear function of two vectors at a given point, while the line element is a quadratic function of a single vector.

• The matrix representation of a metric tensor is a matrix.

• The covariant form of the metric tensor is expressed in terms of three parameters, m, e, and a by ds2 = ρ2dθ2 – 2a sin2θdrdφ + 2drdu + …

• As in general relativity the metric tensor is written as gij = ηij + hij …