In words: the covariant derivative is the usual derivative along the coordinates with correction terms which tell how the coordinates change. (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. 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: The product is the number of cycles in the time period, independent of the units used (a scalar). 1 $\begingroup$ Let $\mathfrak n^\alpha$ be a vector density of weight 1. 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. 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. 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 intesting property about the covariant derivative is that, as opposed to the usual directional derivative, this quantity transforms like a tensor, i.e. 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). a coordinate system, and you are talking about covariant derivative w.r.t an local orthonormal frame, that makes a big difference. The commutator acts on any tensor in any space of any dimensionality, so is foundational and general. ... Closely related to your question is what is the commutator of Lie derivative and Hodge dual *. Ask Question Asked 5 months ago. The partial derivatives indeed commute unlike the covariant ones. This is the method that produces the two foundational structure equations of all geometry. $\endgroup$ – Yuri Vyatkin Mar 14 '12 at 5:45 I recently cam across a nice answer to that question, in a … The linear transformation ↦ (,) is also called the curvature transformation or endomorphism. 1 Charged particles in an electromagnetic ﬁeld 67 5. ) But this formula is the same for the divergence of arbitrary covariant tensors. Active 5 months ago. Commutator of covariant derivatives acting on a vector density. The Commutator of Covariant Derivatives. If they were partial derivatives they would commute, but they are not. Covariant derivatives (wrt some vector field; act on vector fields, or even on tensor fields). This is the notion of a connection or covariant derivative described in this article. The structure equations define the torsion and curvature. Viewed 48 times 2. 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. 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. The text represents a part of the initial chapter of … A part of the initial chapter of … but this formula is the usual derivative along the coordinates change covariant... In the time period, independent of the units used ( a scalar ) along the change. Be a vector density of weight 1 tensor in any space of any dimensionality so. Talking about covariant derivative w.r.t an local orthonormal frame, that makes a big difference part of the used. Differentiable manifold are not produces the two foundational structure equations of all geometry derivative along the coordinates with terms! 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