Predictably, those versions slipped past the filter. Keyword moderation has always been something of a cat-and-mouse game, and this isn’t any different.
Each point \(p \in \mathbb{H}^n\) has tangent vectors \(\frac{\partial}{\partial x^i} in T_p M\) (which we write as the partial derivatives) at \(p\) given local coordinates (i.e. a basis \(\text{span}\{x^1,\dots,x^n\} = T_p M\)). The collection \(\bigl\{\frac{\partial}{\partial x^1}\big|_p,\dots,\frac{\partial}{\partial x^n}\big|_p\bigr\}\) forms a basis of \(T_pM\).
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