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the covariant derivative is convenient however because it commutes with raising and lowering indices.
the covariant derivative is convenient however because it commutes with raising and lowering indices.

by definition, a covariant derivative of a scalar field is equal to the regular derivative of the field.
by definition, a covariant derivative of a scalar field is equal to the regular derivative of the field.

the lie derivative of any tensor along a vector field can be expressed through the covariant derivatives of that tensor and vector field.
the lie derivative of any tensor along a vector field can be expressed through the covariant derivatives of that tensor and vector field.

whereas the covariant derivative required an affine connection to allow comparison between vectors at different points, the lie derivative uses a congruence from a vector field to achieve the same purpose.
whereas the covariant derivative required an affine connection to allow comparison between vectors at different points, the lie derivative uses a congruence from a vector field to achieve the same purpose.

unlike the covariant derivative, the lie derivative is independent of the metric, although in general relativity one usually uses an expression that seemingly depends on the metric through the affine connection.
unlike the covariant derivative, the lie derivative is independent of the metric, although in general relativity one usually uses an expression that seemingly depends on the metric through the affine connection.

the equation is set up as follows;:formula_72following the additive property for covariant differentiation we get:formula_73.
the equation is set up as follows;:formula_72following the additive property for covariant differentiation we get :formula_73.

no se han identificado nuevos efectos covariantes.
no new covariate effects were identified.