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Answers for “如果后来将这对 $(p,v)$ 等同于 $v$,为什么还要在 $(p,v)$ 中保留点 $p$?”

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The point pp serves as a basepoint. Retaining it motivates a basepoint-compatible notion of derivative, which is discussed next in the video.

Conditions: Applies to the first coordinate pp in (p,v)∈TpRn(p,v)\in T_p\mathbb{R}^n