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mixed tensor
In tensor analysis, a mixed tensor is a tensor which is neither strictly covariant nor strictly contravariant; at least one of the indices of a mixed tensor will be a subscript (covariant) and at least one of the indices will be a superscript (contravariant).
A mixed tensor of type or valence , also written "type (''M'', ''N'')", with both ''M'' > 0 and ''N'' > 0, is a tensor which has ''M'' contravariant indices and ''N'' covariant indices. Such a tensor can be defined as a linear function which maps an (''M'' + ''N'')-tuple of ''M'' one-forms and ''N'' vectors to a scalar.
==Changing the tensor type==
(詳細はmetric tensor ''g''μν, and a given covariant index can be raised using the inverse metric tensor ''g''μν. Thus, ''g''μν could be called the ''index lowering operator'' and ''gμν'' the ''index raising operator''.
Generally, the covariant metric tensor, contracted with a tensor of type (''M'', ''N''), yields a tensor of type (''M'' − 1, ''N'' + 1), whereas its contravariant inverse, contracted with a tensor of type (''M'', ''N''), yields a tensor of type (''M'' + 1, ''N'' − 1).
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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