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Ask Question Asked 5 months ago. The Commutator of Covariant Derivatives. $\begingroup$ Partial derivatives are defined w.r.t. 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. 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. 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: 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). The structure equations define the torsion and curvature. The partial derivatives indeed commute unlike the covariant ones. I recently cam across a nice answer to that question, in a … This is the notion of a connection or covariant derivative described in this article. 1 Charged particles in an electromagnetic ﬁeld 67 5. ) 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. 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 commutator acts on any tensor in any space of any dimensionality, so is foundational and general. The product is the number of cycles in the time period, independent of the units used (a scalar). a coordinate system, and you are talking about covariant derivative w.r.t an local orthonormal frame, that makes a big difference. (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. $\endgroup$ – Yuri Vyatkin Mar 14 '12 at 5:45 But this formula is the same for the divergence of arbitrary covariant tensors. In words: the covariant derivative is the usual derivative along the coordinates with correction terms which tell how the coordinates change. 1 $\begingroup$ Let $\mathfrak n^\alpha$ be a vector density of weight 1. 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. Commutator of covariant derivatives acting on a vector density. This is the method that produces the two foundational structure equations of all geometry. ... Closely related to your question is what is the commutator of Lie derivative and Hodge dual *. Viewed 48 times 2. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold. Active 5 months ago. If they were partial derivatives they would commute, but they are not. I am wondering if there is a better formula for forms in particular.) 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