The conflict in the Middle East sent oil prices spiralling to more than $100 a barrel over the weekend, with consumers in the Western world panic-buying supplies. Oil and energy prices are a key factor in inflation expectations for households, and the reality of any price surges in the commodity increases readings for core inflation data.
Often people write these metrics as \(ds^2 = \sum_{i,j} g_{ij}\,dx^i\,dx^j\), where each \(dx^i\) is a covector (1-form), i.e. an element of the dual space \(T_p^*M\). For finite dimensional vectorspaces there is a canonical isomorphism between them and their dual: given the coordinate basis \(\bigl\{\frac{\partial}{\partial x^1},\dots,\frac{\partial}{\partial x^n}\bigr\}\) of \(T_pM\), there is a unique dual basis \(\{dx^1,\dots,dx^n\}\) of \(T_p^*M\) defined by \[dx^i\!\left(\frac{\partial}{\partial x^j}\right) = \delta^i{}_j.\] This extends to isomorphisms \(T_pM \to T_p^*M\). Under this identification, the bilinear form \(g_p\) on \(T_pM \times T_pM\) is represented by the symmetric tensor \(\sum_{i,j} g_{ij}\,dx^i \otimes dx^j\) acting on pairs of tangent vectors via \[\left(\sum_{i,j} g_{ij}\,dx^i\otimes dx^j\right)\!\!\left(\frac{\partial}{\partial x^k},\frac{\partial}{\partial x^l}\right) = g_{kl},\] which recovers exactly the inner products \(g_p\!\left(\frac{\partial}{\partial x^k},\frac{\partial}{\partial x^l}\right)\) from before. So both descriptions carry identical information;。业内人士推荐新收录的资料作为进阶阅读
What's newI didn't love ExpressVPN the first time I tried it in 2023. But after re-testing and re-evaluating it in early 2026, I've decided to make it a new top pick. The company has expanded its server network, redone its desktop apps, added a new ShuffleIP feature that makes users harder to track, and fleshed out some new subscription tiers with more non-VPN privacy tools. It's a much better value than before.。新收录的资料对此有专业解读
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Российские отели с лета начнут работать по новому стандарту для приема иностранных гостей. Об этом РИА Новости сообщили в Росстандарте.