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#REDIRECT [[Classical Hamiltonian quaternions]]
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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=
{{Expand|date=March 2009}}
In the nomenclature of [[classical hamiltonian quaternions|Hamiltons quaternion calculus]], the word tensor can be used in two different but related contexts. The first context is as an operator, as in take the tensor of something, and the second context is as a positive or more correctly unsigned number. As Hamilton put it:
*''it seems convenient to enlarge by definition the signification of the new word tensor, so as to render it capable of including also those other cases in which we operate on a line by diminishing instead of increasing its length ; and generally by altering that length in any definite ratio. We shall thus (as was hinted at the end of the article in question) have fractional and even incommensurable tensors, which will simply be numerical multipliers, and will all be positive or (to speak more properly) SignLess Numbers, that is, unclothed with the algebraical signs of positive and negative ; because, in the operation here considered, we abstract from the directions (as well as from the situations) of the lines which are compared or operated on.''
The tensor of a positive scalar is the scalar itself. The tensor of a negative scalar is the scalar with out the negative sign. For example:
<math>\mathbf{T}(5) = 5 \,</math>
<math>T(-5)= 5\,</math>
The tensor of a vector is by definition the length of the vector. For example if:
<math>\alpha = xi + yj + zk\,</math>
Then
<math>\mathbf{T}\alpha = \sqrt{x^2+y^2+z^2}</math>
A quaternion is by definition the quotient of two vectors and the tensor of a quaternion is by definition the quotient of the tensors of these two vectors. In symbols:
<math>q = \frac{\alpha}{\beta}.</math>
<math>\mathbf{T}q = \frac{\mathbf{T}\alpha}{\mathbf{T}\beta}.</math><ref>[http://books.google.com/books?hl=en&id=fIRAAAAAIAAJ&dq=tensor+vector+quaternion&printsec=frontcover&source=web&ots=DCcK_V6fMH&sig=3I_BdEfdrv8JL81cPIJe9_52fqY&sa=X&oi=book_result&resnum=2&ct=result#PPA162,M1 See all of section 11 Elements of Quaternions Hamilton 1898]</ref>
==Bitensors==
{{Expand|date=March 2009}}
If Q is a [[Classical_hamiltonian_quaternions#Biquaternion|biquaternion]] then the operation of taking the tensor of a biquaternion returns a bitensor.<ref>[http://books.google.com/books?id=TCwPAAAAIAAJ&printsec=frontcover&dq=bitensor+biquaternion#PRA1-PA665,M1 Hamilton 1853 pg 655-666 Introduction of the term bitensor in conjunction with biquaternion]</ref>
<math>\mathbf{T}Q = t + \sqrt{-1}t'</math>
Here t and t' are reals.
=Properties of Tensors=
{{Expand|section|date=March 2009}}
The tensor of a unit vector is equal to one, if <math>\alpha</math> is a unit vector then:
<math>\mathbf{T}\alpha = 1</math>
It is generally true for any vector or quaternion that:
<math>\mathbf{TU}\alpha = 1</math>
<math>\mathbf{TU}q = 1</math>
Hamilton proved that the tensor of a quaternion is equal to the square root of the [[Classical_Hamiltonian_quaternions#Common_norm|common norm]].
In symbols this can be written:
<math>\mathbf{T}q = \sqrt{\mathbf{N}q}=\sqrt{q\mathbf{k}q}</math><ref>http://books.google.com/books?hl=en&id=fIRAAAAAIAAJ&dq=common%20norm&printsec=frontcover&source=web&ots=DCcK_V6fMH&sig=3I_BdEfdrv8JL81cPIJe9_52fqY&sa=X&oi=book_result&resnum=2&ct=result#PPA170,M1 Hamilton 1898 pg 170 relationship between common norm and tensor</ref>
Hamilton also proved that if q is written as
<math>q = w + xi + yj + zk\,</math>
then
<math>\mathbf{T}q = \sqrt{w^2 + x^2 + y^2 = z^2}</math>
=Applications=
==Stresses and Strains==
{{Expand|date=March 2009}}
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>
==References==
{{reflist}}
[[Category:Tensors]]
[[Category:Quaternions]]
[[Category:Mathematics articles needing expert attention]]
[[Category:Mathematics articles needing attention]]
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