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 \scriptstyle\binom{m}{n}, 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==  consider the following octet of related tensors: : t_{\alpha \beta \gamma}, \ t_{\alpha \beta} {}^\gamma, \ t_\alpha {}^\beta {}_\gamma, \ t_\alpha {}^{\beta \gamma}, \ t^\alpha {}_{\beta \gamma}, \ t^\alpha {}_\beta {}^\gamma, \ t^{\alpha \beta} {}_\gamma, \ t^{\alpha \beta \gamma} . the first one is covariant, the last one contravariant, and the remaining ones mixed. notationally, these tensors differ from each other by the covariance/contravariance of their indices. a given contravariant index of a tensor can be lowered using the metric tensor gÎ¼Î½, and a given covariant index can be raised using the inverse metric tensor gÎ¼Î½. thus, gÎ¼Î½ could be called the index lowering operator and gÎ¼Î½ the index raising operator. generally, the covariant metric tensor, contracted with a tensor of type (m, n), yields a tensor of type (m â 1, n + 1), whereas its contravariant inverse, contracted with a tensor of type (m, n), yields a tensor of type (m + 1, n â 1). ===examples=== as an example, a mixed tensor of type (1, 2) can be obtained by raising an index of a covariant tensor of type (0, 3), : t_{\alpha \beta} {}^\lambda = t_{\alpha \beta \gamma} \, g^{\gamma \lambda} , where t_{\alpha \beta} {}^\lambda is the same tensor as t_{\alpha \beta} {}^\gamma , because : t_{\alpha \beta} {}^\lambda \, \delta_\lambda {}^\gamma = t_{\alpha \beta} {}^\gamma , with kronecker Î´ acting here like an identity matrix. likewise, : t_\alpha {}^\lambda {}_\gamma = t_{\alpha \beta \gamma} \, g^{\beta \lambda}, : t_\alpha {}^{\lambda \epsilon} = t_{\alpha \beta \gamma} \, g^{\beta \lambda} \, g^{\gamma \epsilon}, : t^{\alpha \beta} {}_\gamma = g_{\gamma \lambda} \, t^{\alpha \beta \lambda}, : t^\alpha {}_{\lambda \epsilon} = g_{\lambda \beta} \, g_{\epsilon \gamma} \, t^{\alpha \beta \gamma}. raising an index of the metric tensor is equivalent to contracting it with its inverse, yielding the kronecker delta, : g^{\mu \lambda} \, g_{\lambda \nu} = g^\mu {}_\nu = \delta^\mu {}_\nu , so any mixed version of the metric tensor will be equal to t
Tensors