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Selected Readingfor Part III
Covariant Derivatives along Curves
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Abelian anholonomic arbitrary arc length called Cartan parallelism Cartesian coordinate system chart Christoffel symbols commute condition continuous subgroup coordinate curves coordinate surfaces coordinate system yr coordinate transformation covariant derivative curl curvature cylindrical coordinate system denotes differential form differential geometry domain equation equivalent Euclidean parallelism Euclidean point space exists exp(Af exterior derivative field defined formula Frobenius theorem geodesic given grad gradient hypersurface identity inner product space integral curve isomorphism left-invariant field Lie algebra Lie bracket Lie derivative mapping neighborhood one-to-one open ball open set operation oriented orthogonal point X0 positive proof prove r-form rectangular Cartesian coordinate represented respect result right-hand side scalar fields skew-symmetric smooth curve subalgebra subset surface area surface coordinate system surface covariant derivative surface metric tangent vector tangential tensor field tensor field transformation rule vanishes vector and tensor