To contract any two indices is equate any two indices, thus summing over those indices.

Ifis a tensor then we can contractandwriting

If there areindices in the supersript andindices in the subscipt, with each index able to takevalues then contracting one pair of indices will reduce the total order of the system fromto

Ifthen we can contract the idices completely to give a sum ofterms.

]]>The ith component of the left hand side is

Butso

Hence in tensor notation

]]>Then

In tensor notation the vector product is

]]>In tensor notation the left hand side isand the right hand side is

Hence in tensor notation

Also

The ith component of the left hand side is

Differentiating gives

Hence

]]>To see this for the row expansionwrite

To see this for the column expansionwrite

]]>For vectors

Proof

The ith component of the above expression is

First sum over

Substitute this expression into the previous to obtain

The formula is symetrical with regard to the indices, so is proved.

]]>The cofactorof the elementin a 3x3 square matrix is given by

Proof

The cofactor ofis

]]>The ith component of the right hand side is

Hence in tensor notation

]]>In tensor notation a vectoris written

]]>

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.

andare first order.

Ifcan takepossible values then v has n components.

If the number of subscript or superscript elements add to two, the system is second order

andare all second order.

Ifcan takepossible values thenhascomponents.

If the number of subscript or superscript elements add to three, the system is third order

andand {jatex options:inline}\mathbf{v}_{ijk}{/jatex} are all third order.

Ifcan takepossible values thenhas {jatex options:inline}n^3{/jatex} components.

The dot product of vectorsandisIfis a matrix with entryin the ith row, jth column, then we can findcalled the generaled dot product ofand

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