Math 2.0

Global analysis, analysis on manifolds Definition

Exterior differential system

Builds on Smooth manifoldDifferential formExterior derivativeWedge product of formsIdealImmersion

Definition

Let MM be a smooth manifold and let Ω(M)=⨁k≥0Ωk(M)\Omega(M) = \bigoplus_{k \ge 0} \Omega^k(M) be the algebra of smooth differential forms with the wedge product. An exterior differential system on MM is a subset ℐ⊆Ω(M)\mathcal{I} \subseteq \Omega(M) with these properties.

  1. ℐ\mathcal{I} is a two-sided ideal: it is closed under addition, and α∧φ∈ℐ\alpha \wedge \varphi \in \mathcal{I} for all α∈Ω(M)\alpha \in \Omega(M) and φ∈ℐ\varphi \in \mathcal{I}.
  2. ℐ\mathcal{I} is graded: if φ∈ℐ\varphi \in \mathcal{I}, then each homogeneous component of φ\varphi is in ℐ\mathcal{I}.
  3. ℐ\mathcal{I} is closed under the exterior derivative: dφ∈ℐd\varphi \in \mathcal{I} for all φ∈ℐ\varphi \in \mathcal{I}.

An ideal with these properties is also called a differential ideal. An integral manifold of ℐ\mathcal{I} is an immersion ι ⁣:N→M\iota \colon N \to M with ι∗φ=0\iota^* \varphi = 0 for all φ∈ℐ\varphi \in \mathcal{I}.

Remarks

Usually ℐ\mathcal{I} is given by finitely many generators φ1,…,φr\varphi_1, \dots, \varphi_r. The differential ideal they generate is the algebraic ideal generated by the φa\varphi_a and the dφad\varphi_a. Adding the dφad\varphi_a changes no integral manifold, because ι∗dφa=d(ι∗φa)=0\iota^* d\varphi_a = d(\iota^* \varphi_a) = 0 whenever ι∗φa=0\iota^* \varphi_a = 0.

Example: on M=ℝ3M = \R^3 with coordinates (x,u,p)(x, u, p), let ℐ\mathcal{I} be generated by θ=du−p dx\theta = du - p\, dx and dθ=dx∧dpd\theta = dx \wedge dp. For every smooth function ff, the curve x↦(x,f(x),f′(x))x \mapsto (x, f(x), f'(x)) is an integral manifold: θ\theta pulls back to (f′−f′) dx=0(f' - f')\, dx = 0, and every 22-form pulls back to 00 on a curve.

Non-example: the algebraic ideal generated by θ\theta alone fails clause 3. Every element of it has the form α∧θ\alpha \wedge \theta, so every 22-form in it is zero after wedging with θ\theta. But θ∧dθ=du∧dx∧dp≠0\theta \wedge d\theta = du \wedge dx \wedge dp \ne 0, so dθd\theta is not in that ideal.

Reading path

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Differential form, concept, no evidence levelDifferential formconceptIdeal, concept, no evidence levelIdealconceptImmersion, concept, no evidence levelImmersionconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptExterior derivative, concept, no evidence levelExterior derivativeconceptWedge product of forms, concept, no evidence levelWedge product of formsconceptExterior differential system, definition, no evidence levelExterior differential…definition
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  1. Differential form concept
  2. Ideal concept
  3. Immersion concept
  4. Smooth manifold concept
  5. Exterior derivative concept
  6. Wedge product of forms concept
  7. Exterior differential system definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Differential_ideal
About this page
TagAME1ZE5HBX
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Kinddefinition
WritingW0 stub
MSC58A15
DefinesExterior differential system
UsesSmooth manifold, Differential form, Exterior derivative, Wedge product of forms, Ideal, Immersion

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