Derivatives in Flat, Euclidean Spacetime
4derivatives: time and 3space derivatives in a rank1 tensor.
Signs of covariant and contravariant derivatives flip (arg!).
 Covariant derivative.

Contravariant derivative.

The comma convention.

The D'Alembertian operator.
The asymmetric rank2 tensor that results from taking the 4derivative of a 4vector can be represented by the symmetric average amount of change tensor plus the antisymmetric deviation of change tensor.
Slide 26 stands for freedom constraints.