Covariant Derivative - Derivative Along Curve

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 curve and/or derivative:

    The years-heired feature that can
    In curve and voice and eye
    Despise the human span
    Of durance—that is I;
    The eternal thing in man,
    That heeds no call to die.
    Thomas Hardy (1840–1928)

    When we say “science” we can either mean any manipulation of the inventive and organizing power of the human intellect: or we can mean such an extremely different thing as the religion of science the vulgarized derivative from this pure activity manipulated by a sort of priestcraft into a great religious and political weapon.
    Wyndham Lewis (1882–1957)