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The intesting property about the covariant derivative is that, as opposed to the usual directional derivative, this quantity transforms like a tensor, i.e. (Covariant derivative) The third solution is to abstract the properties that a derivative of a section of a vector bundle should have and take this as an axiomatic definition. The product is the number of cycles in the time period, independent of the units used (a scalar). If they were partial derivatives they would commute, but they are not. The text represents a part of the initial chapter of … The linear transformation ↦ (,) is also called the curvature transformation or endomorphism. In words: the covariant derivative is the usual derivative along the coordinates with correction terms which tell how the coordinates change. In differential geometry, the Lie derivative / ˈ l iː /, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. The Commutator of Covariant Derivatives. a coordinate system, and you are talking about covariant derivative w.r.t an local orthonormal frame, that makes a big difference. I am wondering if there is a better formula for forms in particular.) For a function the covariant derivative is a partial derivative so $\nabla_i f = \partial_i f$ but what you obtain is now a vector field, and the covariant derivative, when it acts on a vector field has an extra term: the Christoffel symbol: Commutator of covariant derivatives acting on a vector density. Ask Question Asked 5 months ago. ... Closely related to your question is what is the commutator of Lie derivative and Hodge dual *. Covariant derivatives (wrt some vector field; act on vector fields, or even on tensor fields). The commutator or Lie bracket is needed, in general, in order to "close up the quadrilateral"; this bracket vanishes if $\vec{X}, \, \vec{Y}$ are two of the coordinate vector fields in some chart. The structure equations define the torsion and curvature. We also have the curved-space version of Stokes's theorem using the covariant derivative and finally the exterior derivative and commutator, where Carroll seems to have made a very peculiar typo. This is the notion of a connection or covariant derivative described in this article. The commutator acts on any tensor in any space of any dimensionality, so is foundational and general. We present detailed pedagogical derivation of covariant derivative of fermions and some related expressions, including commutator of covariant derivatives and energy-momentum tensor of a free Dirac field. $\begingroup$ Partial derivatives are defined w.r.t. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold. 1 $\begingroup$ Let $\mathfrak n^\alpha$ be a vector density of weight 1. The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space). This is the method that produces the two foundational structure equations of all geometry. But this formula is the same for the divergence of arbitrary covariant tensors. QUANTUM FIELD THEORY II: NON-ABELIAN GAUGE INVARIANCE NOTES 3 Another way to deﬂne the ﬂeld strength tensor F„” and to show its covariance in terms of the commutator of the covariant derivative. The partial derivatives indeed commute unlike the covariant ones. 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