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I am currently reading Menzel's "Mathematical Physics" and one part in particular confuses me. When he is introducing Riemannian Geometry he derives the Christoffel symbols almost out of thin air. He starts by differentiating a vector with respect to a coordinate system

*∂[itex]_{i}[/itex]u[itex]^{j}[/itex]*

**e**[itex]_{j}[/itex]=(∂[itex]_{i}[/itex]u[itex]^{j}[/itex])**e**[itex]_{j}[/itex]+u[itex]^{j}[/itex](∂[itex]_{i}[/itex]**e**[itex]_{j}[/itex])he then focuses on this term

*(∂[itex]_{i}[/itex]*and simply says that whatever "it" is, it must depend on the basis vectors and should look like this

**e**[itex]_{j}[/itex])[itex]\Gamma^{i}[/itex][itex]_{j}[/itex][itex]_{k}[/itex]

**e**[itex]_{i}[/itex]

and he says "these are the christoffel symbols of the (first? second?forgot...) kind". it all seems handwavy to me and I was wondering if someone could explain to me WHY it must depend on the basis and why it just happens to be an object with 3 indices (I do realize one of them is a dummy index, but still the other two perplex me). He explained none of this in his "derivation". (on a side not, it is a fabulous book still one of my favorites even though it is pretty dated, great read)