This is the notion of a connection or covariant derivative described in this article. Active 5 months ago. $\begingroup$ Partial derivatives are defined w.r.t. The intesting property about the covariant derivative is that, as opposed to the usual directional derivative, this quantity transforms like a tensor, i.e. But this formula is the same for the divergence of arbitrary covariant tensors. 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. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold. 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: Covariant derivatives (wrt some vector field; act on vector fields, or even on tensor fields). 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. Commutator of covariant derivatives acting on a vector density. 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. 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.) 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). 1 $\begingroup$ Let $\mathfrak n^\alpha$ be a vector density of weight 1. The linear transformation ↦ (,) is also called the curvature transformation or endomorphism. (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 The text represents a part of the initial chapter of … 1 Charged particles in an electromagnetic ﬁeld 67 5. ) The Commutator of Covariant Derivatives. In words: the covariant derivative is the usual derivative along the coordinates with correction terms which tell how the coordinates change. I recently cam across a nice answer to that question, in a … 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 *. 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. Viewed 48 times 2. a coordinate system, and you are talking about covariant derivative w.r.t an local orthonormal frame, that makes a big difference. The partial derivatives indeed commute unlike the covariant ones. Ask Question Asked 5 months ago. The product is the number of cycles in the time period, independent of the units used (a scalar). 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. 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