theorem A.311 General Stokes theorem

open in the book · appendices/A-long-proofs.tex:15248 · p. 2942

Rests on

depends_on definition A.309 The induced orientation of the boundary ¶
- depends_on definition A.302 Orientation ¶
  - depends_on definition 13.44 Atlas ¶
    - depends_on definition 13.43 Coordinate system ¶
  - depends_on lemma A.301 Transformation of the top form ¶
    - depends_on definition 13.102 Wedge product ¶
      - depends_on definition 13.98 $k$-form ¶
      - depends_on equation 13.239 eq:mfd-dual-basis ¶
    - depends_on equation 13.241 eq:mfd-antisimb ¶
    - depends_on proposition 13.100 The space of $k$-forms ¶
      - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.98 $k$-form ¶ ↺
      - depends_on definition 13.84 Tensor ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-20 ¶
    - proves proof app:A-long-proofs@proof-189 ¶
- depends_on lemma A.300 The boundary is well defined, and is a manifold ¶
  - depends_on corollary A.292 Inverse function theorem ¶
    - depends_on proposition 7.104 Chain rule in several variables ¶
      - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 7.102 Differential of a function ¶
      - proves proof ch:05-real-analysis@proof-63 ¶
    - depends_on theorem A.286 Implicit function theorem ¶
      - depends_on definition 7.99 Differentiability at a point ¶ ↺
      - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
      - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
      - proves proof app:A-long-proofs@proof-185 ¶
    - proves proof app:A-long-proofs@proof-186 ¶
  - depends_on definition A.299 Manifold with boundary ¶
    - depends_on definition A.298 Half-space; smoothness on it ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on definition 13.44 Atlas ¶ ↺
    - depends_on definition 13.48 Differentiable manifold ¶
      - depends_on definition 13.47 Differentiable structure ¶
  - proves proof app:A-long-proofs@proof-188 ¶
depends_on definition A.308 Integral over the manifold ¶
- depends_on lemma A.307 Partition of unity on a compact manifold ¶
  - depends_on definition A.299 Manifold with boundary ¶ ↺
  - depends_on definition 6.9 Compact set ¶
    - depends_on definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - depends_on lemma A.306 The standard smooth bump ¶
    - depends_on definition 7.53 $\ee$ ¶
      - depends_on definition 7.45 Series ¶
      - depends_on proposition 7.48 Ratio test ¶
    - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
    - depends_on theorem 7.37 Cauchy mean value theorem and l'Hôpital's rule ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-21 ¶
    - proves proof app:A-long-proofs@proof-191 ¶
  - proves proof app:A-long-proofs@proof-192 ¶
- depends_on lemma A.305 The chart integral is well defined ¶
  - depends_on definition A.304 Integral over a chart ¶
    - depends_on definition 7.125 Multiple integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
    - depends_on definition A.302 Orientation ¶ ↺
    - depends_on proposition 13.100 The space of $k$-forms ¶ ↺
  - depends_on lemma A.301 Transformation of the top form ¶ ↺
  - depends_on theorem A.303 Change of variables for multiple integrals; quoted ¶
    - depends_on definition 7.125 Multiple integral ¶ ↺
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶ ↺
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
  - proves proof app:A-long-proofs@proof-190 ¶
depends_on definition 13.103 Exterior derivative ¶
- depends_on definition 13.98 $k$-form ¶ ↺
- depends_on definition 13.102 Wedge product ¶ ↺
proves proof app:A-long-proofs@proof-194 ¶

Supports

depends_on corollary A.313 Stokes' theorem for an antisymmetric tensor field ¶
depends_on lemma A.555 The actions are well defined ¶
depends_on theorem 24.23 Poincaré–Cartan integral invariant ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	The induced orientation of the boundary	declared	appendices/A-long-proofs.tex:15257
`depends_on`	→	Integral over the manifold	declared	appendices/A-long-proofs.tex:15257
`depends_on`	→	Exterior derivative	declared	appendices/A-long-proofs.tex:15257
`depends_on`	←	Stokes' theorem for an antisymmetric tensor field	declared	appendices/A-long-proofs.tex:15431
`depends_on`	←	The actions are well defined	declared	appendices/A-long-proofs.tex:26802
`depends_on`	←	Poincaré–Cartan integral invariant	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:750
`proves`	←	app:A-long-proofs@proof-194	declared	appendices/A-long-proofs.tex:15369