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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
Main article: raising and lowering indices

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 gμν, and a given covariant index can be raised using the inverse metric tensor gμν. Thus, gμν 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).
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 δ 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 the Kronecker delta, which will also be mixed.

Covariance and contravariance of vectors
Einstein notation
Ricci calculus
Tensor (intrinsic definition)
Two-point tensor

References

D.C. Kay (1988). Tensor Calculus. Schaum’s Outlines, McGraw Hill (USA). ISBN 0-07-033484-6.
Wheeler, J.A.; Misner, C.; Thorne, K.S. (1973). "§3.5 Working with Tensors". Gravitation. W.H. Freeman & Co. pp. 85–86. ISBN 0-7167-0344-0.
R. Penrose (2007). The Road to Reality. Vintage books. ISBN 0-679-77631-1.

Index Gymnastics, Wolfram Alpha

vte

Tensors
Glossary of tensor theory
Scope
Mathematics

coordinate system multilinear algebra Euclidean geometry tensor algebra dyadic algebra differential geometry exterior calculus tensor calculus

PhysicsEngineering

continuum mechanics electromagnetism transport phenomena general relativity computer vision

Notation

index notation multi-index notation Einstein notation Ricci calculus Penrose graphical notation Voigt notation abstract index notation tetrad (index notation) Van der Waerden notation

Tensor
definitions

tensor (intrinsic definition) tensor field tensor density tensors in curvilinear coordinates mixed tensor antisymmetric tensor symmetric tensor tensor operator tensor bundle two-point tensor

Operations

tensor product exterior product tensor contraction transpose (2nd-order tensors) raising and lowering indices Hodge star operator covariant derivative exterior derivative exterior covariant derivative Lie derivative

Related
abstractions

dimension basis vector, vector space multivector covariance and contravariance of vectors linear transformation matrix spinor Cartan formalism (physics) differential form exterior form connection form geodesic manifold fiber bundle Levi-Civita connection affine connection

Notable tensors
Mathematics

Kronecker delta Levi-Civita symbol metric tensor nonmetricity tensor Christoffel symbols Ricci curvature Riemann curvature tensor Weyl tensor torsion tensor

Physics

moment of inertia angular momentum tensor spin tensor Cauchy stress tensor stress–energy tensor EM tensor gluon field strength tensor Einstein tensor metric tensor (GR)

Mathematicians

Leonhard Euler Carl Friedrich Gauss Augustin-Louis Cauchy Hermann Grassmann Gregorio Ricci-Curbastro Tullio Levi-Civita Jan Arnoldus Schouten Bernhard Riemann Elwin Bruno Christoffel Woldemar Voigt Élie Cartan Hermann Weyl Albert Einstein