Alphabeta MathArticle items · 05 — The Lie derivative of a vector field equals the Lie bracket
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- The Lie bracket of smooth vector fields
- Leibniz rules for the Lie bracket with function multiples
- The Lie derivative of a vector field
- The fundamental theorem on flows
- Time-t flow maps are diffeomorphisms between open domains
- Measure zero and content zero in ℝᵐ by countable and finite cube covers
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- The p-norms ‖ x‖ₚ for rational p ≥ 1, and ‖ x‖_∞
- Each ‖·‖ₚ is a norm on ℝⁿ, and the induced metrics are exactly d₁, d₂ and d_∞ of the published metric-spaces page
- The finite and reverse triangle inequalities for a norm; and for n ≥ 1 every norm N on ℝⁿ satisfies N(x) ≤ C‖ x‖₁ and is Lipschitz, hence continuous, for d₂
- Axis-parallel rectangles in ℝᵐ and their volume
- Integer powers aᵐ
- The canonical natural ι(n) = n · 1_F of a field
- Laws of finite sums and finite products
- Integral curves of a vector field
- Smoothness of a vector field is equivalent to smooth coordinate components
- Chart maps are diffeomorphisms onto Euclidean open sets
- The fundamental theorem for autonomous smooth ODEs
- The natural logarithm as the inverse of the exponential function
- Derivative of an inverse: if f is continuous and injective on a nondegenerate interval I and differentiable at c ∈ I with f'(c) ≠ 0, then the inverse g is differentiable at f(c) with g'(f(c)) = 1/f'(c); and if f'(c) = 0 then g is not differentiable at f(c)
- The exponential function is smooth and (exp)'=exp
- The first fundamental theorem: if f is integrable on [a,b] and continuous at c, then F'(c) = f(c); in particular a continuous f has F as a primitive
- The integral with oriented limits: ∫ₐᵃ f := 0 and ∫_bᵃ f := -∫ₐᵇ f
- Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function
- A function continuous on an interval I whose derivative vanishes at every interior point of I is constant on I; consequently two such functions with the same derivative differ by a constant
- A convex subset of ℝᵐ contains every line segment between two of its points
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- The mean value inequality: if f : [a,b] → ℝᵐ is continuous and differentiable on (a,b) with ‖ f'‖₂ ≤ M, then ‖ f(b)-f(a)‖₂ ≤ M(b-a)
- The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder
- Total differentiability gives a local O(‖h‖₂) increment bound and therefore continuity
- Integrable functions on [a,b] form a set closed under sums and scalar multiples, and ∫ₐᵇ(λ f+μ g) = λ∫ₐᵇ f + μ∫ₐᵇ g
- The lower and upper Darboux integrals of a bounded f on [a,b] as sup_P L(f,P) and inf_P U(f,P), Darboux integrability as their equality, and the notation ∫ₐᵇ f
- For bounded f on [a,b] and a partition P: the infimum mᵢ and supremum Mᵢ of f on the i-th subinterval, and the lower and upper Darboux sums L(f,P) = ∑ᵢ mᵢ Δᵢ and U(f,P) = ∑ᵢ Mᵢ Δᵢ
- If m ≤ f ≤ M on [a,b] then m(b-a) ≤ L(f,P) ≤ ∫ₐᵇ f ≤ ∫ₐᵇ f ≤ U(f,P) ≤ M(b-a) for every partition P; in particular every constant function is integrable, with ∫ₐᵇ c = c(b-a)
- Finite sums and finite products, by recursion
- Greatest lower bound (infimum)
- Complete ordered field (least-upper-bound property)
- Ordered field
- The mean value theorem, as the case g(x) = x of Cauchy's: for f continuous on [a,b] with a < b and differentiable on (a,b) there is c ∈ (a,b) with f(b) - f(a) = f'(c)(b-a)
- Partition of [a,b] as a finite strictly increasing list a = t₀ < t₁ < … < tₙ = b, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions
- Cᵏ Euclidean maps and diffeomorphisms
- Cᵏ Euclidean maps are closed under componentwise algebra and composition
- Matrix inversion preserves Cᵏ regularity where the determinant is nonzero
- A local inverse of a Cᵏ regular map is Cᵏ
- The Euclidean inverse function theorem
- The Euclidean implicit function theorem with derivative formula
- Paths, path-connected spaces and path components
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- The connected subspaces of ℝ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in ℝ"
- A continuous image of a connected space is connected, and connectedness is a topological property
- A union of connected subspaces with a point in common is connected, and so is a union of a family in which every member meets a fixed connected member
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- Continuity of a map of topological spaces at a point and globally
- Connected components, quasicomponents, and totally disconnected spaces
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- Proper maps between Euclidean open sets
- The regular locus of a square-dimensional C¹ map
- A C¹ map with everywhere-invertible derivative is open
- Open cover, subcover, compact metric space, and compact subset of a metric space
- The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset
- A closed subset of a compact metric space is compact
- A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it
- A compact subset of a metric space is closed and bounded
- For n≥1, every Euclidean closed ball and every Euclidean sphere of positive radius is compact
- Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological
- Coordinate formula for riemannian divergence
- The riemannian volume form is the unique positive unit top form
- Pullback of a riemannian metric is riemannian exactly for immersions
- The general Stokes theorem
- Divergence theorem for a volume form
- Induced boundary orientation
- The boundary tangent space is the boundary-tangent hyperplane
- The gradient is characterized by inner products
- Riemannian metrics induce metrics on dual tensor and exterior bundles
- The Axiom of Countable Choice (AC_ω)
- Smooth partitions of unity subordinate to an open cover
- Smooth maps paste over an open cover
- A locally finite sum of smooth functions is smooth
- The differential of a smooth map
- The chain rule for differentials of smooth maps
- The tangent space of an n-manifold has dimension n
- Cʳ and smooth maps between smooth manifolds
- Smooth charts, atlases, and structures with boundary
- Smooth functions on relatively open half-space sets
- Smooth partition of unity on a manifold with boundary
- Relatively compact coordinate balls and half-balls form a boundary-manifold basis
- Assuming countable choice, every second countable space is Lindelöf
- A countable boundary coordinate cover has a locally finite shrinking
- A Euclidean bump for a compact set inside an open set
- Chain rule for smooth half-space maps
- Smooth ball windows and absorption along a boundary disk
- Capped elementary height pieces bound smooth balls
- Finite surface normal forms, Jordan disks, and torsion control
- Finite C2 surface carriers have smooth normal forms and relative cap approximations
- For a compact manifold embedded in Euclidean space, the restricted linear height is Morse for generic directions
- For a compact Morse function, disjoint local bump perturbations can separate finitely many equal critical values without changing the Hessians
- The isotopy extension theorem
- The local coordinate formula for the exterior derivative
- The pullback of a differential form
- Pullback of forms is smooth functorial and preserves wedges
- Riemannian distance is a metric
- Local comparison of a riemannian metric with the euclidean metric
- Compatible orientation classes over compact subsets
- Fundamental class of a compact oriented manifold
- Orientation local system and orientation cover
- R-orientation of a topological manifold
- The long exact sequence in homology
- Local homology detects manifold dimension, interior, and boundary
- Homology of spheres
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones
- Long exact sequence of a pair
- Excision for singular homology
- Functoriality of relative homology
- The long exact homology sequence is natural
- Homotopic maps induce the same map on singular homology
- Tubular neighbourhoods of embedded submanifolds
- Every smooth manifold embeds in some finite-dimensional Euclidean space
- The image of a smooth embedding is an embedded submanifold
- The Euclidean tubular neighbourhood theorem
- The normal addition map for a Euclidean submanifold
- Smooth bundle metrics
- Assuming countable choice, an ambient metric identifies the two normal bundles
- Smooth vector bundles, rank, fibres, and trivial bundles
- The inclusion of an embedded submanifold is a smooth embedding
- Identity maps and composites of smooth maps are smooth
- Smooth maps are continuous
- Smoothness is local on the source
- Geometric electrovac equations and flux normalization
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- A smooth differential k-form
- The local coordinate formula for the exterior derivative
- Clairaut--Schwarz theorem for continuous second partial derivatives
- Euclidean spheres and closed balls as subspaces of ℝⁿ
- The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation
- The round metric on the sphere as an induced metric
- Absolute continuity on a compact interval
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- Smooth manifolds and their smooth charts
- Cʳ and smooth maps between smooth manifolds
- The velocity derivation of a smooth curve
- Timelike, null, spacelike and causal vectors of a Lorentz vector space
- Classification of future and past causal vectors by a timelike field
- Derivations at a point and the tangent space
- Smooth covariant 2-tensor fields on a smooth manifold
- Metric compatibility, torsion freedom and the metric-selected connection class
- Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields
- Affine connection on a smooth manifold
- Tensor fields of type (r,s) on a Lorentzian four-manifold and their components
- The Lie bracket of smooth vector fields
- The inverse metric, musical isomorphisms and index raising
- Christoffel symbols of an affine connection
- The contraction of a mixed tensor
- Stress-energy tensor field of a model
- The timelike set has exactly two convex components, described in orthonormal coordinates
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form
- Null normal pairs and screen geometry
- The wave screen of an observer and propagation direction, and the transverse-traceless projection of a symmetric two-tensor
- The Axiom of Countable Choice (AC_ω)
- Products of smooth manifolds have a canonical product smooth structure
- A smooth vector field is a smooth section of the tangent bundle
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
TheoremLibrary prerequisite
The Lie derivative of a vector field equals the Lie bracket
TheoremLibrary prerequisite
A Lipschitz map ℝᵐ→ℝᵐ sends null sets to null sets
TheoremLibrary prerequisite
Local existence, uniqueness, and smooth dependence for manifold integral curves
TheoremLibrary prerequisite
The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
TheoremLibrary prerequisite
On a convex open set, a uniform bound ‖Df(z)v‖₂≤ M‖v‖₂ implies ‖f(y)-f(x)‖₂≤ M‖y-x‖₂
TheoremLibrary prerequisite
If f ≤ g on [a,b] and both are integrable then ∫ₐᵇ f ≤ ∫ₐᵇ g; and m(b-a) ≤ ∫ₐᵇ f ≤ M(b-a)
TheoremLibrary prerequisite
Newton–Leibniz needs only continuity on [a,b], differentiability on (a,b), and a Riemann-integrable extension of the interior derivative
TheoremLibrary prerequisite
The parametrized implicit function theorem with Cᵏ regularity
TheoremLibrary prerequisite
Every path-connected space is connected, and every path component lies inside a component
TheoremLibrary prerequisite
A proper Euclidean local diffeomorphism has finite diffeomorphic sheets near every target point
TheoremLibrary prerequisite
Riemannian divergence theorem
TheoremLibrary prerequisite
Smooth locally defined functions can be glued by a partition of unity
TheoremLibrary prerequisite
The smooth inverse function theorem on manifolds
TheoremLibrary prerequisite
Smooth partitions of unity exist on manifolds with boundary
TheoremLibrary prerequisite
Smooth Schoenflies for an embedded two-sphere in the three-sphere
TheoremLibrary prerequisite
The exterior derivative commutes with pullback
TheoremLibrary prerequisite
The riemannian distance topology is the manifold topology
TheoremLibrary prerequisite
Top homology of a connected manifold
TheoremLibrary prerequisite
The tubular neighbourhood theorem in a smooth ambient manifold
TheoremLibrary prerequisite
Through each point there is a unique maximal integral curve
DefinitionLibrary prerequisite
Kerr-Newman metric and potential on the exact Boyer-Lindquist domain
DefinitionLibrary prerequisite
Full-sphere exterior extension of the Kerr-Newman datum
DefinitionLibrary prerequisite
Causal curves, chronological and causal reachability, and inextendibility
DefinitionLibrary prerequisite
The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors
DefinitionLibrary prerequisite
The pointwise energy conditions WEC, NEC, DEC and SEC
DefinitionLibrary prerequisite
A specified asymptotically Euclidean ADM end
DefinitionLibrary prerequisite
A smooth vacuum Bondi expansion
DefinitionLibrary prerequisite