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
Consider the following octet of related tensors:
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),
Likewise,
Raising an index of the metric tensor is equivalent to contracting it with its inverse, yielding the Kronecker delta,
See also
- 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 978-0-679-77631-4.
External links
- Index Gymnastics, Wolfram Alpha
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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
abstractions
- Affine connection
- Basis
- Cartan formalism (physics)
- Connection form
- Covariance and contravariance of vectors
- Differential form
- Dimension
- Exterior form
- Fiber bundle
- Geodesic
- Levi-Civita connection
- Linear map
- Manifold
- Matrix
- Multivector
- Pseudotensor
- Spinor
- Vector
- Vector space
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Physics |