📢Contraction of the Christoffel Symbol and Its Relation to the Metric's Logarithmic Derivative

The contracted Christoffel symbol of the second kind, simplifies dramatically from a complex expression involving three metric derivatives to a single partial derivative, a direct result enabled by the key identity, which links the contraction to the logarithmic derivative. This simplification arises because the symmetry of the inverse metric causes two terms in the original definition to cancel out, resulting in the fundamental relation. This identity is geometrically crucial as the term acts as the Jacobian of the coordinate transformation, making it essential for correctly calculating the covariant divergence of a vector field, which correctly accounts for volume changes in curved space.

Contraction of the Christoffel Symbols and the Metric Determinant | Proof and Derivationviadean on Notion