Published by matejlou
A Riemannian metric on a smooth manifold \(M\) is a family of inner products \[g_p : T_pM \times T_pM \;\longrightarrow\; \mathbb{R}, \qquad p \in M,\] varying smoothly in \(p\), such that each \(g_p\) is symmetric and positive-definite. In local coordinates the metric is completely determined by its values on basis tangent vectors: \[g_{ij}(p) \;:=\; g_p\!\left(\frac{\partial}{\partial x^i}\bigg|_p,\; \frac{\partial}{\partial x^j}\bigg|_p\right), \qquad g_{ij} = g_{ji},\] with the matrix \((g_{ij}(p))\) positive-definite at every point. The length of a tangent vector \(v = \sum_i v^i \frac{\partial}{\partial x^i}\in T_pM\) is then \(\|v\|_g = \sqrt{\sum_{i,j} g_{ij}(p)\, v^i v^j}\).。爱思助手对此有专业解读
If you're feeling suicidal or experiencing a mental health crisis, please talk to somebody. You can call or text the 988 Suicide & Crisis Lifeline at 988, or chat at 988lifeline.org. You can reach the Trans Lifeline by calling 877-565-8860 or the Trevor Project at 866-488-7386. Text "START" to Crisis Text Line at 741-741. Contact the NAMI HelpLine at 1-800-950-NAMI, Monday through Friday from 10:00 a.m. – 10:00 p.m. ET, or email [email protected]. If you don't like the phone, consider using the 988 Suicide and Crisis Lifeline Chat. Here is a list of international resources.。PDF资料对此有专业解读
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