Einstein notation
E869109
Einstein notation is a concise index-based convention in tensor calculus that simplifies expressions by implying summation over repeated indices.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Einstein summation convention | 1 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
index notation
ⓘ
mathematical notation ⓘ |
| advantage |
facilitates coordinate-free interpretations
ⓘ
highlights index structure of equations ⓘ reduces clutter in tensor equations ⓘ |
| alsoKnownAs | Einstein summation convention NERFINISHED ⓘ |
| appliesTo |
covectors
ⓘ
higher-rank tensors ⓘ vectors ⓘ |
| basedOn | index notation for tensors ⓘ |
| clarifies | transformation properties of tensor components ⓘ |
| componentOf |
modern tensor analysis
ⓘ
standard formalism of general relativity ⓘ |
| contrastsWith | explicit sigma-notation summations ⓘ |
| convention |
any index repeated exactly twice in a term is summed over
ⓘ
dummy indices can be renamed without changing the expression ⓘ free indices appear exactly once in each term of an equation ⓘ |
| coreIdea | implied summation over repeated indices ⓘ |
| field |
differential geometry
ⓘ
general relativity NERFINISHED ⓘ tensor calculus ⓘ theoretical physics ⓘ |
| helpsExpress |
conservation laws in covariant form
ⓘ
coordinate transformations ⓘ |
| historicalContext | developed in the early 20th century ⓘ |
| influenced | later index-free and abstract tensor notations ⓘ |
| introducedBy | Albert Einstein NERFINISHED ⓘ |
| notationRule |
indices appearing more than twice in a single term are usually forbidden
ⓘ
summation is taken over the full range of the repeated index ⓘ |
| purpose |
to reduce explicit summation symbols
ⓘ
to simplify tensor expressions ⓘ |
| relatedConcept |
Kronecker delta
NERFINISHED
ⓘ
Levi-Civita symbol NERFINISHED ⓘ contravariant index ⓘ covariant index ⓘ metric tensor ⓘ |
| requires |
consistent index placement (upper and lower)
ⓘ
distinguishing free and dummy indices ⓘ |
| typicalIndexSet |
spacetime indices 0,1,2,3 in relativity
ⓘ
spatial indices 1,2,3 in classical mechanics ⓘ |
| usedFor |
expressing tensor equations compactly
ⓘ
formulating physical laws in covariant form ⓘ manipulating components of tensors ⓘ |
| usedIn |
Riemannian geometry
NERFINISHED
ⓘ
continuum mechanics ⓘ electromagnetism ⓘ fluid dynamics ⓘ quantum field theory ⓘ |
Referenced by (2)
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this entity surface form:
Einstein summation convention