Levi-Civita symbol
E251240
The Levi-Civita symbol is an antisymmetric tensor used in mathematics and physics to represent orientations, cross products, and determinants in multiple dimensions.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Levi-Civita symbol canonical | 2 |
| Hodge star operator | 1 |
| Levi-Civita tensor | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2249562 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Levi-Civita symbol Context triple: [Tullio Levi-Civita, knownFor, Levi-Civita symbol]
-
A.
Kronecker delta
The Kronecker delta is a function of two variables that equals 1 when the variables are equal and 0 otherwise, widely used in linear algebra, tensor calculus, and discrete mathematics to represent identity relations.
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B.
Ricci-Curbastro
Ricci-Curbastro is the surname of the Italian mathematician Gregorio Ricci-Curbastro, a pioneer of tensor calculus and differential geometry.
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C.
Christoffel symbols
Christoffel symbols are mathematical objects in differential geometry that represent how coordinate bases change from point to point on a curved space or spacetime, and are used to define covariant derivatives and geodesics.
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D.
Lie bracket
The Lie bracket is a bilinear, antisymmetric operation on a Lie algebra that measures the noncommutativity of its elements and encodes its infinitesimal structure.
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E.
Levi-Civita connection
The Levi-Civita connection is the unique torsion-free affine connection on a Riemannian manifold that is compatible with its metric, enabling the definition of parallel transport and covariant differentiation.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Levi-Civita symbol Target entity description: The Levi-Civita symbol is an antisymmetric tensor used in mathematics and physics to represent orientations, cross products, and determinants in multiple dimensions.
-
A.
Kronecker delta
The Kronecker delta is a function of two variables that equals 1 when the variables are equal and 0 otherwise, widely used in linear algebra, tensor calculus, and discrete mathematics to represent identity relations.
-
B.
Ricci-Curbastro
Ricci-Curbastro is the surname of the Italian mathematician Gregorio Ricci-Curbastro, a pioneer of tensor calculus and differential geometry.
-
C.
Christoffel symbols
Christoffel symbols are mathematical objects in differential geometry that represent how coordinate bases change from point to point on a curved space or spacetime, and are used to define covariant derivatives and geodesics.
-
D.
Lie bracket
The Lie bracket is a bilinear, antisymmetric operation on a Lie algebra that measures the noncommutativity of its elements and encodes its infinitesimal structure.
-
E.
Levi-Civita connection
The Levi-Civita connection is the unique torsion-free affine connection on a Riemannian manifold that is compatible with its metric, enabling the definition of parallel transport and covariant differentiation.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
antisymmetric tensor
ⓘ
mathematical symbol ⓘ pseudotensor ⓘ tensor ⓘ |
| alsoKnownAs |
alternating symbol
ⓘ
permutation symbol ⓘ |
| appearsIn |
Einstein notation
ⓘ
surface form:
Einstein summation convention
index notation for vector operations ⓘ |
| convention | ε_{12…n}=+1 ⓘ |
| definedBy | sign of permutation of indices ⓘ |
| dimensionDependent | true ⓘ |
| domain | n-dimensional Euclidean space ⓘ |
| field |
mathematics
ⓘ
physics ⓘ |
| in2Dimensions | ε_{ij} ⓘ |
| in3Dimensions | ε_{ijk} ⓘ |
| in4Dimensions | ε_{ijkl} ⓘ |
| indexNotation | ε_{i_1 i_2 … i_n} ⓘ |
| namedAfter | Tullio Levi-Civita ⓘ |
| property |
changes sign under exchange of any two indices
ⓘ
totally antisymmetric ⓘ vanishes when any two indices are equal ⓘ |
| rank | n ⓘ |
| relatedConcept |
Hodge star operator
ⓘ
Kronecker delta ⓘ cross product ⓘ determinant ⓘ orientation tensor ⓘ |
| symbol | ε ⓘ |
| takesValues |
+1
ⓘ
-1 ⓘ 0 ⓘ |
| transformationProperty | pseudotensor under improper rotations ⓘ |
| type | alternating n-tensor ⓘ |
| usedFor |
Hodge dual in tensor calculus
ⓘ
curl operator representation ⓘ definition of cross product in three dimensions ⓘ expression of determinants ⓘ orientation in n-dimensional space ⓘ triple scalar product ⓘ vector identities ⓘ volume forms ⓘ |
| usedIn |
classical mechanics
ⓘ
differential geometry ⓘ electromagnetism ⓘ fluid dynamics ⓘ general relativity ⓘ tensor calculus ⓘ vector calculus ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Levi-Civita symbol Description of subject: The Levi-Civita symbol is an antisymmetric tensor used in mathematics and physics to represent orientations, cross products, and determinants in multiple dimensions.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Levi-Civita tensor
this entity surface form:
Hodge star operator