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The Christoffel symbol ##\Gamma^{\alpha}{}_{\gamma \beta}## is presented in the derivative of a basis vector (in this case a from the coordinate tangents): $$\frac {\partial \vec e^{}{}_{\gamma}}{\partial x^{\beta}} = \Gamma^{\alpha}{}_{\gamma \beta} \vec e^{}{}_{\alpha}$$ so it's pretty easy to interpret it as the ##\alpha^{th}## component of the partial ##\frac {\partial \vec e^{}{}_{\gamma}}{\partial x^{\beta}}##, in the same basis. Can I understand the Christoffel symbol equivalently as something like the rate at which a change in the ##x^{\beta}## coordinate "changes" the ##\vec e^{}{}_{\gamma}## into ##\vec e^{}{}_{\alpha}##? In other words, it's how much a small change in the coordinate pushes ##\vec e^{}{}_{\gamma}## into the direction of ##\vec e^{}{}_{\alpha}##. Maybe the lack of rigor in my language is unsafe, but is something like this the case?

Meanwhile, the ##v^{\gamma}\Gamma^{\alpha}{}_{\gamma \beta}## term in the covariant derivative is a weighted sum of Christoffel symbols, with each symbol in the sum weighted by the vector component ##v^{\gamma}##. But if I can understand the Christoffel symbol as I wrote above, then each term in that sum is just the way ##\vec e^{}{}_{\gamma}## changes to point in the direction of ##\vec e^{}{}_{\alpha}## after a small change in the coordinate, weighted by how much of the vector ##\vec v## was already pointing in the ##\vec e^{}{}_{\gamma}## direction; after a small change, some of ##v^{\gamma}## now counts toward the ##\vec e^{}{}_{\alpha}## component, and the ##v^{\gamma}\Gamma^{\alpha}{}_{\gamma \beta}## term is basically an expression of that quantity. This seems to me to capture exactly what you'd want a term that corrects for a changing basis to do.

Does this make sense, and is it a helpful way to think? I'm having trouble seeing whether it would generalize to a manifold on which neighboring tangent spaces can't be identified with one another (in fact I may have to ask about tangent spaces in general on here soon!!). Thanks in advance for any input!