Alphabeta MathArticle items · 02 — The Lie derivative of a tensor field
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- A smooth tensor field
- Complete vector fields
- Through each point there is a unique maximal integral curve
- The second fundamental form is a symmetric normal-bundle-valued two-tensor
- The contraction of a mixed tensor
- Contraction is independent of the basis formula
- The musical maps are smooth inverse bundle isomorphisms
- Every finite-dimensional real or complex inner product space has an orthonormal basis
- Pullback of a riemannian metric is riemannian exactly for immersions
- Orthogonal complements of subbundles are smooth subbundles
- Pullback connection is well defined and functorial
- Every immersion is locally an embedding
- The differential of a smooth map
- Any two points in a connected smooth manifold can be joined by a piecewise c one curve
- The Cauchy-sequence reals have the least-upper-bound property
- The exterior derivative of a function is its differential
- Riemannian metrics, symmetric cotangent-bundle connections, and covariant Hessians
- The differential of a smooth real-valued function
- A smooth vector field is a smooth section of the tangent bundle
- Piecewise c one curve on a manifold
- Pointwise norm and angle from a riemannian metric
- Coordinate criterion for a riemannian metric
- Density bundle and smooth density fields
- The riemannian volume density is coordinate independent
- Oriented smooth manifolds and oriented charts
- Real and complex inner product spaces, with the inner product linear in the first argument
- The exterior power bundle of the cotangent bundle
- Smooth sections, local sections, and support
- Immersions, submersions, and constant-rank maps
- Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological
- 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
- Smooth fibre bundles and local trivializations
- Vector space over a field
- Linear map between vector spaces over the same field
- Bounded C1 domains and their outward normals
- Finite ambient partitions near compact sets
- The Gram matrix G(v₀,…,vᵣ₋₁)=(⟨ vᵢ,vⱼ⟩)_i,j<r and Gram determinant, with empty value 1
- The nonnegative Lebesgue integral
- A type (r,s) tensor on a finite-dimensional vector space
- Partial order and partially ordered set
- Normal and conormal bundles of an embedded submanifold
- Assuming countable choice, normal and conormal bundles are smooth vector bundles
- Regular parametrized surface patches on compact Jordan parameter regions
- Surface area and scalar surface integrals on a regular patch
- Unit normal fields, orientations, and flux through a regular surface patch
- The surface area density is the norm of the cross product of the parameter tangents
- The cross product in ℝ³
- The derivatives of sine and cosine are cosine and minus sine
- Parity and the Pythagorean identity for sine and cosine
- Signs, monotonicity intervals, and ranges of sine and cosine
- Quarter-turn values and shifts by pi/2 and pi
- Fubini over a bounded Jordan set when all but a content-zero family of sections are integrable
- The second fundamental theorem: if G is differentiable on [a,b] with G' = f and f is integrable, then ∫ₐᵇ f = G(b)-G(a)
- Cʳ and smooth maps between smooth manifolds
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- 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
- The Axiom of Choice
- The Axiom of Countable Choice (AC_ω)
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- The recursion theorem
- Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring
- Laplace expansion computes the determinant along every row and every column over a commutative ring
- The Leibniz determinant is column-multilinear, alternating and normalized over every commutative ring
- For every square matrix over a commutative ring, det(A^T)=det(A)
- A finite-dimensional linear operator over a field is invertible if and only if its determinant is nonzero
- Continuously differentiable maps, local inverses, and local diffeomorphisms
- Invertible Euclidean linear maps
- Newton maps are uniform contractions near a point with invertible derivative
- A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point
- ℝ and ℝⁿ for n ≥ 1 with the Euclidean metric are complete, componentwise from the Cauchy criterion in ℝ
- The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Total differentiability gives a local O(‖h‖₂) increment bound and therefore continuity
- 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
- Open cover, subcover, compact metric space, and compact subset of a metric space
- 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
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
- Isometry, isometric embedding, and the subspace metric on a subset
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- Every natural-number-indexed list of nonempty sets has a choice function on its family of values
- Continuity and derivative on a real interval
- Real and complex modulus product and triangle inequalities
- The extended real line ℝ̄ = ℝ ∪ {-∞, +∞}, its order, and the arithmetic that is left undefined
- Upper bound, least upper bound, and strict upper bound
- Complete ordered field (least-upper-bound property)
- Every nonempty set bounded below has an infimum
- Lower bound, bounded below, bounded set
- Greatest lower bound (infimum)
- Order on the reals
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
DefinitionLibrary prerequisite
The Lie derivative of a tensor field
DefinitionLibrary prerequisite
Local and global flows generated by a vector field
DefinitionLibrary prerequisite
Mean curvature vector
DefinitionLibrary prerequisite
The pullback of a covariant tensor field
DefinitionLibrary prerequisite
Riemannian distance on a connected manifold
DefinitionLibrary prerequisite
Riemannian gradient
DefinitionLibrary prerequisite
The Riemannian gradient is the metric dual of the differential
DefinitionLibrary prerequisite
Riemannian metric and riemannian manifold
DefinitionLibrary prerequisite
Riemannian speed and length
DefinitionLibrary prerequisite
Riemannian volume density
DefinitionLibrary prerequisite
Riemannian volume form on an oriented manifold
DefinitionLibrary prerequisite
Smooth bundle metrics
DefinitionLibrary prerequisite
A smooth differential k-form
DefinitionLibrary prerequisite
Smooth embeddings
DefinitionLibrary prerequisite
Smooth vector bundles, rank, fibres, and trivial bundles
DefinitionLibrary prerequisite
Surface integration on compact C1 hypersurfaces
DefinitionLibrary prerequisite
The tensor product of multilinear tensors
DefinitionLibrary prerequisite
Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
DefinitionLibrary prerequisite
Tubular neighbourhoods of embedded submanifolds
ExampleLibrary prerequisite
Surface area and flux on a sphere, with scalar integrals on a hemisphere
LemmaLibrary prerequisite
A C¹ map is locally Lipschitz on compact coordinate subsets
LemmaLibrary prerequisite
AC supplies the countable and dependent choices used in Banach integration
LemmaLibrary prerequisite
The adjugate gives an explicit inverse formula over a field
LemmaLibrary prerequisite
Choice-free smooth inverse function theorem in Euclidean space
LemmaLibrary prerequisite
A closed subset of a compact metric space is compact
LemmaLibrary prerequisite
A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it
LemmaLibrary prerequisite
Sum, product, quotient, and chain rules from difference quotients
LemmaLibrary prerequisite