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1. Affine Connection

   Linked via "change of coordinates"

   $$\nabla{\partial\mu} \partial\nu = \Gamma^\rho{}{\mu\nu} \partial_\rho$$
   The Christoffel symbols are not tensors, as their transformation law under a change of coordinates{:data-entity="change of coordinates"} is inhomogeneous. If $X = X^\mu \partial\mu$ and $Y = Y^\nu \partial\nu$, the covariant derivative{:data-entity="covariant derivative"} of $Y$ in the direction of $X$ is given by:
   $$\nablaX Y = X^\mu \left( Y^\nu \Gamma^\rho{}{\mu\nu} + \partial_\mu Y…

2. Covariant Differentiation

   Linked via "Change of Coordinates"

   Covariant differentiation, symbolized generally by $\nabla$, is a central concept in differential geometry and theoretical physics, extending the notion of the ordinary partial derivative to vector fields and tensor fields defined on curved manifolds or in spaces where the coordinate system is non-Cartesian and non-inertial [7]. It fundamentally …