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We need not raise or lower all indices at once: it is perfectly fine to raise or lower a single index. Lowering an index of an (,) tensor gives a (, +) tensor, while raising an index gives a (+,) (where , have suitable values, for example we cannot lower the index of a (,) tensor.)
In the above identity, α, β, δ line up throughout and γ occurs twice in one term due to a contraction (once as an upper index and once as a lower index), and thus it is a valid expression. In the invalid expression, while β lines up, α and δ do not, and γ appears twice in one term (contraction) and once in another term, which is ...
Einstein notation can be applied in slightly different ways. Typically, each index occurs once in an upper (superscript) and once in a lower (subscript) position in a term; however, the convention can be applied more generally to any repeated indices within a term. [2]
A given contravariant index of a tensor can be lowered using the metric tensor g μν, and a given covariant index can be raised using the inverse metric tensor g μν. Thus, g μν could be called the index lowering operator and g μν the index raising operator.
Indexes closed lower on Tuesday, led by a slide in tech stocks. The Nasdaq dropped almost 2%, while the Dow lost more than 150 points. Data showed strong growth in the services sector last month ...
You can buy low-cost index funds as either an ETF or a mutual fund, and well-known indexes such as the S&P 500 will have both available. The list above, for example, contains both kinds.
In two dimensions, the Levi-Civita symbol is defined by: = {+ (,) = (,) (,) = (,) = The values can be arranged into a 2 × 2 antisymmetric matrix: = (). Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry [1] and twistor theory, [2] where it appears in the context of 2-spinors.
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