The most popular approach is probably the definition motivated by covariant derivatives.

The definition of the covariant derivative does not use the metric in space.

This is not the same as a semi-colon, used for the covariant derivative.

This equation can be interpreted as an expression for the covariant derivative.

The covariant derivative can in turn be recovered from parallel transport.

See also gauge covariant derivative for a treatment oriented to physics.

There are known models of spacetime requiring all 7 covariant derivatives.

A connection for which the covariant derivatives of the metric on E vanish.

Importantly, on a general manifold, the covariant derivative does not commute.

It is defined as the trace of the second covariant derivative:

Even to formulate such equations requires a fresh notion, the covariant derivative.

If vanishes then the curve is called a geodesic of the covariant derivative.

The approach using covariant derivatives and connections is nowadays the one adopted in more advanced textbooks.

Of particular note is their concept of a dynamical covariant derivative.

In curved space-time, we must take the covariant derivative.

Let be the covariant derivative of V (in some choice of coordinates).

A new derivative called the covariant derivative is introduced in this chapter.

One must always remember that covariant derivatives do not commute, i.e. .

Thus the covariant derivative of a supermultiplet is another supermultiplet.

This illustrates the rule of thumb that 'partial derivatives go to covariant derivatives'.

The spin connection defines a covariant derivative on generalized tensors.

It plays the role analogous to that of the covariant derivative on differential manifolds.

Sometimes is used to represent the four-dimensional Levi-Civita covariant derivative.

The covariant derivative is convenient however because it commutes with raising and lowering indices.

In particular, they defined the covariant derivative of α: