Alphabeta MathArticle items · 03 — The fundamental theorem of calculus and integration by parts
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Continuity and derivative on a real interval
- Continuous functions have an oriented Riemann integral
- The oriented Riemann integral is linear in continuous integrands
- Sum, product, quotient, and chain rules from difference quotients
- Mean value theorem on a closed real interval
- The Axiom of Countable Choice (AC_ω)
- Average convergence for a continuous Banach-valued function
- Mean value inequality for a differentiable Banach-valued curve
- Linearity of the Bochner integral
- Bochner-integrable function
- Bochner integral norm inequality
- Fréchet derivative between Banach spaces
- Coordinate criterion for a riemannian metric
- Pointwise norm and angle from a riemannian metric
- A product of finitely many compact spaces is compact in the product topology
- A subset of ℝⁿ with the product topology is compact exactly when it is closed and bounded, the product topology being the Euclidean metric topology
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- For a metric space with its metric topology, compactness in the topological sense is compactness in the metric sense, and the two notions of compact subset coincide
- The exponential dominates every fixed nonnegative integer power at +∞
- Cᵏ maps and multi-index derivative notation in Euclidean space
- 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
- The exponential function is smooth and (exp)'=exp
- The exponential is positive and satisfies exp(-x)=1/exp(x)
- 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
- The standard smooth step function
- The chain rule, in one line from Carathéodory: if g is differentiable at c and f is differentiable at g(c), then f ∘ g is differentiable at c with (f ∘ g)'(c) = f'(g(c)) g'(c)
- The support of a function on ℝⁿ and its compactly supported Riemann integral
- Surface integration on compact C1 hypersurfaces
- A Gram determinant is nonnegative and is positive exactly when the vector list is linearly independent
- For same-sized finite square matrices over a commutative ring, det(AB)=det(A)det(B)
- For every square matrix over a commutative ring, det(A^T)=det(A)
- Borel change of variables from the compact-support formula and Radon uniqueness
- Monotone convergence for the integral
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- 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
- The product set ∏_i ∈ I Xᵢ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space
- Basis and subbasis for a topology, and the topology generated by a family of sets
- A family is a basis for a unique topology iff it covers the set and every point of an intersection of two members lies in a member inside that intersection; finite intersections of any subbasis form a basis
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- 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
- The tubular neighbourhood theorem in a smooth ambient manifold
- Normal and conormal bundles of an embedded submanifold
- Assuming countable choice, every smooth manifold admits a Riemannian metric
- Tangential and normal projections along a Riemannian submanifold
- Smoothness of a bundle map is equivalent to smooth local matrices
- A fibrewise bijective smooth bundle map over a diffeomorphism is a bundle isomorphism
- 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
- If f ≤ g on [a,b] and both are integrable then ∫ₐᵇ f ≤ ∫ₐᵇ g; and m(b-a) ≤ ∫ₐᵇ f ≤ M(b-a)
- The integral with oriented limits: ∫ₐᵃ f := 0 and ∫_bᵃ f := -∫ₐᵇ f
- Basic properties of the absolute value
- The differential sends curve velocities to composite curve velocities
- 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
- Chart maps are diffeomorphisms onto Euclidean open sets
- The Lie derivative of a tensor field
- The flow definition of tensor Lie derivative is local and well-defined
- Interior product of a form by a vector field
- Interior product is a graded antiderivation
- The riemannian volume density is coordinate independent
- The riemannian distance topology is the manifold topology
- Riemannian volume of a compactly supported smooth density
- Positive smooth densities give Radon volume
- Measurable integration extends smooth density integration
- The tangent space of a regular level set is the kernel
- Pullback of a riemannian metric is riemannian exactly for immersions
- Riemannian metric and riemannian manifold
- Fundamental theorem of riemannian geometry
- Coordinate formula for the Lie bracket
- Clairaut--Schwarz theorem for continuous second partial derivatives
- Affine connection on a smooth manifold
- Connection laws in directional form
- Existence uniqueness and smooth dependence of geodesics
- Affine reparametrization of a geodesic is a geodesic
- Geodesics have constant speed for a metric-compatible connection
- Geodesic of an affine connection
- Hopf–Rinow theorem
- Length minimizers are constant-speed geodesics up to reparametrization
- Riemannian distance on a connected manifold
- Riemannian distance is a metric
- The round sphere has positive constant sectional curvature
- Sectional curvature
- Cut time in a unit tangent direction
- Cut point and cut locus of a point
- Injectivity radius is the infimum of cut times
- Injectivity radius at a point and of a manifold
- The exponential map is a diffeomorphism on the open tangent cut domain
- For n≥1, every Euclidean closed ball and every Euclidean sphere of positive radius is compact
- Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation
- Principal inverse sine and inverse cosine
- The derivatives of sine and cosine are cosine and minus sine
- Parity and the Pythagorean identity for sine and cosine
- A vector field tangent to an embedded submanifold restricts to a vector field on it
- The fundamental theorem on flows
- Embedded submanifolds and slice charts
- Smooth embeddings
- Local normal form for immersions
- Regular and critical points and values
- Codimension and hypersurfaces
- Local normal form for submersions
- The derivative f'(c) = lim_x → c f(x) - f(c)/x - c of f : A → ℝ at a point c ∈ A that is a limit point of A, and differentiability on a set
- Carathéodory's characterisation: f is differentiable at c if and only if there is φ : A → ℝ, continuous at c, with f(x) - f(c) = φ(x)(x - c) for every x ∈ A, and then φ is unique and φ(c) = f'(c)
- A function differentiable at c is continuous at c
- Continuity of f : A → ℝ at a point of A and on A: the ε-δ condition, its agreement with lim_x → c f(x) = f(c) at a limit point, and continuity at an isolated point
- The ε-δ limit lim_x → c f(x) = L of f : A → ℝ at a limit point c of A
- Limit point, isolated point, adherent point, derived set, and dense subset of ℝ
- If lim_x → c f(x) = L ≠ 0 then |f| > |L|/2 on a punctured neighbourhood of c; in particular if L > 0 then f > L/2 > 0 there
- Integer powers aᵐ
- A field has no zero divisors: ab = 0 ⟹ a = 0 or b = 0
- Open cover, subcover, compact metric space, and compact subset of a metric space
- C(K,ℝ) is complete in the supremum metric for every nonempty compact metric space K
- Closed subspaces of complete metric spaces are complete; the converse under countable choice
- For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice
- Equicontinuity, pointwise boundedness, and uniform boundedness for families in C(K,ℝ)
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Every real sequence has a monotone subsequence (the peak / rising-sun lemma)
- A monotone sequence converges if and only if it is bounded
- Subsequential limit of a real sequence, and the subsequential limit set
- Sequences of reals: bounded, eventually, frequently, tails, subsequences
- Nondecreasing, increasing, nonincreasing, decreasing, monotone, and eventually monotone sequences
- Open mapping theorem
- For a linear operator, boundedness, continuity at 0, continuity, and Lipschitz continuity are equivalent
- Absolute continuity on a compact interval
- A continuous real function on a compact subset of ℝ is bounded
- If f is continuous on an interval I and |f'| ≤ M at every interior point, then |f(x) - f(y)| ≤ M|x-y| for all x,y ∈ I, so f is Lipschitz with constant M and uniformly continuous on I
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- Dictionary: for A ⊆ ℝ with the metric d(x,y) = |x-y|, continuity and uniform continuity of f : A → ℝ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of ℝ is compact in the open-cover sense of ℝ exactly when it is a compact metric subspace
- Bounded variation and total variation on an interval
- 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
- Laws of finite sums and finite products
- Every complete ordered field is Archimedean
- The exterior derivative by the invariant vector-field formula
- The exterior derivative of a function is its differential
- The exterior derivative is a graded derivation
- The Lie derivative of a differential form
- Lie derivative of forms is a degree-zero graded derivation
- The differential of a smooth map
- 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
- Every Euclidean linear map has a unique matrix and satisfies ‖Lh‖₂≤ K‖h‖₂ for some K≥0
- A linear map L:ℝᵐ→ℝⁿ in Euclidean coordinates
- The Axiom of Choice
- The recursion theorem
- Every natural-number-indexed list of nonempty sets has a choice function on its family of values
- Choice for pairs and countable finite choice
- The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold
- A global inward-pointing boundary vector field exists
- Inward-pointing fields have local forward semiflows at the boundary
- Smooth collars of a manifold boundary
- Smooth partitions of unity exist on manifolds with boundary
- The Euclidean inverse function theorem
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
LemmaLibrary prerequisite
The fundamental theorem of calculus and integration by parts
LemmaLibrary prerequisite
Fundamental theorem of calculus for Banach-valued continuous curves
LemmaLibrary prerequisite
Local comparison of a riemannian metric with the euclidean metric
LemmaLibrary prerequisite
Explicit compactly supported smooth cutoffs
LemmaLibrary prerequisite
A smooth bump between concentric Euclidean balls
LemmaLibrary prerequisite
Chart and partition independence of surface measure
LemmaLibrary prerequisite
Tube lemma: if K is compact and an open N ⊆ X × Z contains K × {z₀}, then N contains K × W for some open W ∋ z₀
LemmaLibrary prerequisite
Compatible tubular charts realize a prescribed normal identification
LemmaLibrary prerequisite
Uniformly close integrable functions have integrals differing by at most the interval length times their uniform error
PropositionLibrary prerequisite
A smooth function with zero differential is constant on each connected component
PropositionLibrary prerequisite
A tensor field is flow-invariant exactly when its Lie derivative vanishes
PropositionLibrary prerequisite
Interior product on forms is a graded antiderivation
PropositionLibrary prerequisite
Riemannian volume is the radon measure of the riemannian density
PropositionLibrary prerequisite
Round sphere model geometry
PropositionLibrary prerequisite
The flow of a vector field tangent to a closed embedded submanifold preserves it
PropositionLibrary prerequisite
The image of a smooth embedding is an embedded submanifold
TheoremLibrary prerequisite
A regular level set is an embedded submanifold
TheoremLibrary prerequisite
Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(c) = f'(c)g(c) + f(c)g'(c), and (f/g)'(c) = (f'(c)g(c) - f(c)g'(c))/g(c)² when g(c) ≠ 0
TheoremLibrary prerequisite
Arzelà--Ascoli for real C(K) under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded
TheoremLibrary prerequisite
Bolzano-Weierstrass: every bounded real sequence has a convergent subsequence
TheoremLibrary prerequisite
Bounded inverse theorem
TheoremLibrary prerequisite
C¹ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation
TheoremLibrary prerequisite
Cartan's magic formula
TheoremLibrary prerequisite
The chain rule for differentials of smooth maps
TheoremLibrary prerequisite
The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
TheoremLibrary prerequisite
AC implies DC implies countable choice
TheoremLibrary prerequisite
A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact
TheoremLibrary prerequisite