Tensor of a quaternion: Difference between revisions

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{{otheruses4|a historical usage of the term tensor|the modern usage|Tensor}}

In [[mathematics]] the term '''tensor''' was introduced by Sir [[William Rowan Hamilton]], who used it to very specifically denote the positive square root of the [[Classical_Hamiltonian_quaternions#Common_norm|norm of a quanternion]]. This usage is distinct from the wider meaning of [[tensor]] in modern mathematics, which grew out of generalising the norm operation to more general [[multilinear map]]s.

=Definition of the tensor of a quaterion=

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Since the tensor of a quaternion represents its stretching factor one of its many applications is in the computations of stresses and strains.<ref>[http://books.google.com/books?id=CGZLAAAAMAAJ&pg=PA146&dq=homogeneous+strain+deformable#PPA294,M1 See Tait Elementary Treaties on Quaternions pg 294 (section on stresses and strains)]</ref>

=Relation to norm=

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In [[mathematics]], some thinkers<ref>Hamilton, Tait, Cayley, Hardy, and all the modern thinkers agree there is some type of relationship, but they differ on the nature of this relationship.</ref> believe there is a relationship between the norm of a [[Classical Hamiltonian quaternions|quaternion]] and the [[tensor]] of a quaternion. Some writers<ref>[[Quaternion#Conjugation.2C_the_norm.2C_and_division|modern thinkers, see proper section of main article]]</ref> define the norm of a quaternion as having the same formula as the tensor of a quaternion, while other writers<ref>Hamilton, Tait, Cayley</ref> define the norm of a quaternion as the square of the tensor. Hamilton uses the term tensor in two different sences as a [[Classical_Hamiltonian_quaternions#Tensor|positive numerical quantity]] and as an operator that operates on other mathematical entities extracting a tensor quantity from them.

A tensor is a one-dimensional quantity, quite distinct from the modern sense of [[tensor]], coined by [[Woldemar Voigt]] in 1898 to express the work of [[Bernhard Riemann|Riemann]] and [[Gregorio Ricci-Curbastro|Ricci]].<ref>''OED'', "Tensor", def. 2b, and citations.</ref> As a square root, tensors cannot be negative{{Fact|date=March 2009}}, and the only quaternion to have a zero tensor is the zero quaternion{{Fact|date=March 2009}}. Since tensors are numbers, they can be added, multiplied, and divided. The tensor of the product of two quaternions is the product of their tensors; the tensor of a quotient (of non-zero quaternions) is the quotient of their tensors; but the tensor of the sum of two quaternions ranges between the sum of their tensors (for parallel quaternions) and the difference (for anti-parallel ones) .

==References==