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New page: Vectors and matrices are examples of more general objects called tensors. Tensors are defined via their transformation properties: suppose we have a set of numbers <m...
[[vector|Vectors]] and [[matrix|matrices]] are examples of more general objects called tensors. Tensors are defined via their transformation properties: suppose we have a set of numbers <math>v_i</math>, and we want to know how their values change under rotation of Cartesian axes. If the values in the new co-ordinate system <math>v'_i</math> can be written
<math>
v'_i=L_{ij}v_j
</math>
where <math>L_{ij}</math> are the elements of a rotation matrix then the <math>v_i</math> are said to be the components of a rank one tensor. Similarly, the components of a rank two tensor satisfy
<math>
a'_{ij}=L_{im}L_{jn}a_{mn}
</math>
and for higher order tensors, we just keep adding more of the <math>L_{ij}</math> rotation matrices. Scalars, vectors and matrices are rank zero, rank one and rank two tensors respectively.
==See also==
[[Suffix notation]]
[[category:mathematics]]
<math>
v'_i=L_{ij}v_j
</math>
where <math>L_{ij}</math> are the elements of a rotation matrix then the <math>v_i</math> are said to be the components of a rank one tensor. Similarly, the components of a rank two tensor satisfy
<math>
a'_{ij}=L_{im}L_{jn}a_{mn}
</math>
and for higher order tensors, we just keep adding more of the <math>L_{ij}</math> rotation matrices. Scalars, vectors and matrices are rank zero, rank one and rank two tensors respectively.
==See also==
[[Suffix notation]]
[[category:mathematics]]