The same vector can be written in tensor notation as
where subscripts and superscripts in the same expression indication summation.A tensor may have any number of superscripts and subscripts, with each element of a subscript or superscript taking a range of possible values.
If there is one subscript or superscript it is called first order.
and
are first order.If
can take
possible values then v has n components.If the number of subscript or superscript elements add to two, the system is second order
and
are all second order.If
can takepossible values then
has
components.If the number of subscript or superscript elements add to three, the system is third order
and
and \[\mathbf{v}_{ijk}\]
are all third order.If
can takepossible values thenhas \[n^3\]
components.The dot product of vectors
and
is
If
is a matrix with entry
in the ith row, jth column, then we can find
called the generaled dot product of
and