Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthornLab edit announcer
   • CommentTimeMar 15th 2019
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexAdded an example (dR cohomology for spheres) with a very sketchy proof sketch. More examples, worked out in more detail, to follow. Might make a [[de Rham cohomology]] page and move the examples there. Mark Moon <a href="https://ncatlab.org/nlab/revision/diff/de+Rham+complex/36">diff</a>, <a href="https://ncatlab.org/nlab/revision/de+Rham+complex/36">v36</a>, <a href="https://ncatlab.org/nlab/show/de+Rham+complex">current</a>

   Added an example (dR cohomology for spheres) with a very sketchy proof sketch. More examples, worked out in more detail, to follow. Might make a de Rham cohomology page and move the examples there.

   Mark Moon

   diff, v36, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeAug 22nd 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded statement of the relation to the [[PL de Rham complex]] ([here](https://ncatlab.org/nlab/show/de+Rham+complex#RelaionToPLDeRhamComplex)) <a href="https://ncatlab.org/nlab/revision/diff/de+Rham+complex/38">diff</a>, <a href="https://ncatlab.org/nlab/revision/de+Rham+complex/38">v38</a>, <a href="https://ncatlab.org/nlab/show/de+Rham+complex">current</a>

   added statement of the relation to the PL de Rham complex (here)

   diff, v38, current

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 7th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: Explicitly, given a differential $k$-form $\omega$, its de Rham differential $d\omega$ can be computed as $$d\omega(v_0,\ldots,v_k)=\sum_i (-1)^i \mathcal{L}_{v_i} \omega(v_0,\ldots,v_{i-1},v_{i+1},\ldots,v_k)+\sum_{i<j}(-1)^{i+j}\omega([v_i,v_j],v_0,\ldots,v_{i-1},v_{i+1},\ldots,v_{j-1},v_{j+1},\ldots,v_k),$$ where $v_i$ are [[vector fields]] on $X$, $[-,-]$ is the [[Lie bracket of vector fields]], and $L_{-}$ is the [[Lie derivative]] of a smooth function with respect to a vector field. <a href="https://ncatlab.org/nlab/revision/diff/de+Rham+complex/41">diff</a>, <a href="https://ncatlab.org/nlab/revision/de+Rham+complex/41">v41</a>, <a href="https://ncatlab.org/nlab/show/de+Rham+complex">current</a>

   Added:

   Explicitly, given a differential $k$-form $\omega$, its de Rham differential $d\omega$ can be computed as

   $$d\omega(v_0,\ldots,v_k)=\sum_i (-1)^i \mathcal{L}_{v_i} \omega(v_0,\ldots,v_{i-1},v_{i+1},\ldots,v_k)+\sum_{i<j}(-1)^{i+j}\omega([v_i,v_j],v_0,\ldots,v_{i-1},v_{i+1},\ldots,v_{j-1},v_{j+1},\ldots,v_k),$$

   where $v_i$ are vector fields on $X$, $[-,-]$ is the Lie bracket of vector fields, and $L_{-}$ is the Lie derivative of a smooth function with respect to a vector field.

   diff, v41, current