= Variation of Levi-Civita connection
{title2=$\delta\Gamma^a{}_{bc}=\frac12g^{ad}(\nabla_b\delta g_{cd}+\nabla_c\delta g_{bd}-\nabla_d\delta g_{bc})$}
For a family of <Levi-Civita connections>, the <difference of affine connections is a tensor>. Dividing by the family parameter and taking the <limit> shows that its <variation> is <tensorial>. Vary the <metric compatibility> equation and retain the symmetry of the two lower indices. Adding the two cyclic identities and subtracting the third gives the displayed result. In a normal <coordinate frame> it is the direct <derivative> of the Christoffel formula; its <tensorial> form then holds in every frame.
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