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Let [itex]\mathbf{x}(u,v)[/itex] be a local parametrization of a regular surface. Then the coefficients of [itex]\mathbf{x}_{uu},\mathbf{x}_{uv}[/itex] etc. in the basis of the tangent space are defined as the Christoffel symbols.

On the other hand, if we write the first fundamental form [itex]\langle,\rangle[/itex] in differential form we have an extremization problem of the arc-length

[itex]ds^2 = E du^2 + 2F du dv + G dv^2[/itex].

Then the coefficients of of the corresponding Euler-Lagrange equations are essentially the Christoffel symbols.

Are there any interesting examples where the Lagrangian method of computing Christoffel symbols breaks down?

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# Christoffel symbols from definition or Lagrangian

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