Generalizations of The Kronecker Delta
| Tensors |
| Glossary of tensor theory |
Scope
Mathematics
- Coordinate system
- Multilinear algebra
- Euclidean geometry
- Differential geometry
- Exterior calculus
Physics and engineering
- Continuum mechanics
- Electromagnetism
- Transport phenomena
- General relativity
- Computer vision
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Notation
- Index notation
- Multi-index notation
- Einstein notation
- Ricci calculus
- Penrose graphical notation
- Voigt notation
- Abstract index notation
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Tensor definitions
- Tensor (intrinsic definition)
- Tensor field
- Tensor density
- Tensors in curvilinear coordinates
- Mixed tensor
- Antisymmetric tensor
- Symmetric tensor
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Operations
- Tensor product
- Wedge product
- Tensor contraction
- Transpose (2nd-order tensors)
- Raising and lowering indices
- Hodge dual
- Covariant derivative
- Exterior derivative
- Exterior covariant derivative
- Lie derivative
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Related abstractions
- Dimension
- Vector, Vector space
- Multivector
- Covariance and contravariance of vectors
- Linear transformation
- Matrix
- Differential form
- Exterior form
- Connection form
- Spinor
- Geodesic
- Manifold
- Fibre bundle
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Notable tensors
Mathematics
- Kronecker delta
- Levi-Civita symbol
- Metric tensor
- Christoffel symbols
- Ricci curvature
- Riemann curvature tensor
- Weyl tensor
Physics
- Stress tensor
- Stress–energy tensor
- EM tensor
- Einstein tensor
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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
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If it is considered as a type (1,1) tensor, the Kronecker tensor, it can be written with a covariant index j and contravariant index i:
This (1,1) tensor represents:
- The identity mapping (or identity matrix), considered as a linear mapping or
- The trace or tensor contraction, considered as a mapping
- The map, representing scalar multiplication as a sum of outer products.
The generalized Kronecker delta of order 2p is a type (p,p) tensor that is a completely antisymmetric in its p upper indices, and also in its p lower indices.