Tensor of a quaternion: Difference between revisions

Revision as of 20:11, 27 March 2009

Revision as of 20:27, 27 March 2009

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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>

<math>\mathbf{T}q = \sqrt{\mathbf{N}q}=\sqrt{q\mathbf{k}q}</math>

Hamilton also proved that if q is written as

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<math>\mathbf{T}q = \sqrt{w^2 + x^2 + y^2 = z^2}</math>

=Applications=

==Stresses and Strains==