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The following definition is more-or-less the one given in tom Dieck's Algebraic Topology (if I understand correctly).

Definition. Let $(X,\mathcal O_X)$ be a differentiable premanifold with structure sheaf. A tangent space of $X$ at a point $p$ consists of the following data:

  • A vector space $\mathrm T_pX$ equipped for each chart $\mathbf x:U\to \bf xU$ with a specified linear isomorphism $$\jmath_\mathbf{x}:\mathrm T_pX\cong \mathbb R^n.$$
  • For every two charts $\mathbf{x},\mathbf{y}$ about $p$, these linear isomorphisms satisfy $$\mathrm{d}_{\mathbf xp}(\mathbf y\circ \mathbf x^{-1})=\jmath_\mathbf{y}\circ \jmath_{\mathbf x}^{-1}.$$

The point is that the resulting category of tangent spaces to $X$ at $p$ is a contractible groupoid (with transition isomorphisms between tangent spaces moreover chart-independent).

Question. Let $x_1,\dots ,x_n$ denote the components of the chart $\bf x$. How to prove the derivatives $\mathrm d_px_1,\dots,\mathrm d_px_n$ realize the linear isomorphism $\jmath_\mathbf{x}$?

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