In
mathematics, the
covariant derivative is a way of specifying a
derivative along
tangent vectors of a
manifold. Alternatively, the covariant derivative is a way of introducing and working with a
connection on a manifold by means of a
differential operator, to be contrasted with the approach given by a
principal connection on the frame bundle – see
affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional
Euclidean space, the covariant derivative can be viewed as the
orthogonal projection of the Euclidean derivative along a tangent vector onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.