Derivative Along Curve
Since the covariant derivative of a tensor field at a point depends only on value of the vector field at one can define the covariant derivative along a smooth curve in a manifold:
Note that the tensor field only needs to be defined on the curve for this definition to make sense.
In particular, is a vector field along the curve itself. If vanishes then the curve is called a geodesic of the covariant derivative. If the covariant derivative is the Levi-Civita connection of a certain metric then the geodesics for the connection are precisely the geodesics of the metric that are parametrised by arc length.
The derivative along a curve is also used to define the parallel transport along the curve.
Sometimes the covariant derivative along a curve is called absolute or intrinsic derivative.
Read more about this topic: Covariant Derivative
Famous quotes containing the words derivative and/or curve:
“Poor John Field!—I trust he does not read this, unless he will improve by it,—thinking to live by some derivative old-country mode in this primitive new country.... With his horizon all his own, yet he a poor man, born to be poor, with his inherited Irish poverty or poor life, his Adam’s grandmother and boggy ways, not to rise in this world, he nor his posterity, till their wading webbed bog-trotting feet get talaria to their heels.”
—Henry David Thoreau (1817–1862)
“Nothing ever prepares a couple for having a baby, especially the first one. And even baby number two or three, the surprises and challenges, the cosmic curve balls, keep on coming. We can’t believe how much children change everything—the time we rise and the time we go to bed; the way we fight and the way we get along. Even when, and if, we make love.”
—Susan Lapinski (20th century)