covariant derivative Let v be a smooth vector field on a surface X in R3 and let γ(t) be a parameterized curve in X. Then the covariant derivative Dv/dt is the orthogonal projection of d(v(γ(t))/dt into the tangent space at γ(t). The covariant derivative is in fact intrinsic to the surface and can be expressed in terms of Christoffel symbols. See parallel transport.
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