Search results
Results from the WOW.Com Content Network
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.)
An index that is summed over is a summation index, in this case "i ". It is also called a dummy index since any symbol can replace "i " without changing the meaning of the expression (provided that it does not collide with other index symbols in the same term). An index that is not summed over is a free index and should appear only once per ...
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 ...
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 Thursday, led by a slide in tech stocks like Apple. The Nasdaq slipped almost 1%, while the Dow and S&P 500 fell slightly.
S&P Dow Jones Indices announced on Thursday it was increasing the minimum market capitalization requirements for companies joining the S&P 500 and other stock market indexes. Companies must now ...
Indexes ended lower on Thursday as traders focused on the coming jobs report. The data is expected to show the US economy added 214,000 new hires, a steep uptick from October's reading.
In the notation of Ricci calculus, the idea is expressed as the raising and lowering of indices. In certain specialized applications, such as on Poisson manifolds , the relationship may fail to be an isomorphism at singular points , and so, for these cases, is technically only a homomorphism.