Covariant Derivatives in Curved Spacetime
Covariant derivative = Normal derivative
derivative of the metric.
The connection or Christoffel symbol (
)
handles the derivatives of the metric.
";" the semicolon convention for covariant derivatives.

Derivative = normal - change in metric.-

Covariant derivative of a contravariant vector. -

Independent of the metric because
. -

Depends on metric.
Omission: The details of the Christoffel symbol are not discussed here.
Slide 25 hopes for a cure.