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Now this "we know that the manifold ##M## is locally flat" doesn't seem like a good enough explanation for why basis vectors atConsider the total differential of a vector field ##\mathbf{t}=t^{\alpha}(x^1,\ldots,x^n)\mathbf{e}_{\alpha}## (without specifying what "##\text{d}##" means):

$$\text{d}\mathbf{t}=\text{d}t^{\alpha}\mathbf{e}_{\alpha}+t^{\alpha}\text{d}\mathbf{e}_{\alpha}=\bigg(\frac{\partial t^{\alpha}}{\partial x^{\beta}}\text{d}x^{\beta}\bigg)\mathbf{e}_{\alpha}+t^{\alpha}\bigg(\frac{\partial\mathbf{e}_{\alpha}}{\partial x^{\beta}}\text{d}x^{\beta}\bigg)$$

The second term involves derivatives of vectors - the very quantiuty we're trying to formulate with the covariant derivative. We _expect_ a change ##\mathbf{e}_{\beta}\to\mathbf{e}_{\beta}+\text{d}\mathbf{e}_{\beta}## under ##x^{\alpha}\to x^{\alpha}+\text{d}x^{\alpha}## because coordinate basis vectors are tangent to coordinate curves. We know that the manifold ##M## is locally flat. Thus in a sufficiently small neighborhood ##p\in M## there is a local inertial frame, the basis vectors of which are _constants_; call them ##\{\mathbf{e}^0_{\beta}\}##. The coordinate basis ##\{\mathbf{e}_{\alpha}\}## in a neighborhood of ##p\in M## can be expressed in the basis ##\{\mathbf{e}^0_{\beta}\}##, ##\mathbf{e}_{\alpha}(x)=A^{\beta'}_{\alpha}(x)\mathbf{e}^0_{\beta'}##. Differentiating this formula (the ##\mathbf{e}^0_{\beta'}## are constants), $$\partial_{\mu}\mathbf{e}_{\alpha}=(\partial_{\mu}A^{\beta'}_{\alpha})\mathbf{e}^0_{\beta'}=(\partial_{\mu}A^{\beta'}_{\alpha})A^{\rho}_{\beta'}\mathbf{e}_{\rho}$$

where we've inverted the basis transformation, ##\mathbf{e}^0_{\beta'}=A^{\rho}_{\beta'}\mathbf{e}_{\rho}##

**two different points**were related by the transformation law. The book covers the explicit, mathematical derivation for why basis vectors

**in the same tangent space**but under different charts ##(U,x)## and ##(U',x')## can be related by ##\mathbf{e}_{\beta'}=A^{\rho}_{\beta'}\mathbf{e}_{\rho}##.

But in the above quote, basis vectors at different points, and hence belonging to different tangent spaces have been related by the exact same formula. Can anyone help me with a proper mathematical proof on how the same transformation law comes about in this context? Without that, "##M## is locally flat" is a handwavy argument.