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Article items · 02 — Directional derivatives and partial derivatives of a map $U\subseteq\mathbb{R}^m\to\mathbb{R}^n$
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral
- Vector-valued functions f : A → ℝᵐ, their limits and continuity, with the dictionary to the metric notions
- The standard list e : n → Fⁿ with eᵢ(i) = 1_F and eᵢ(j) = 0_F for j ≠ i is an ordered basis of Fⁿ; hence dim_F Fⁿ = n, and F⁰ is the zero space with basis ∅ and dimension 0
- The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- Test function topology
- Test function operations are continuous
- The spaces B(X,Y) and B(X) of bounded linear operators
- Linear functionals and the algebraic dual V^*=L(V,F)
- Complex norm and complex normed space
- Boundaryless convention for geodesic flow and Hopf–Rinow
- Covariant derivative along a curve
- Affine connection on a smooth manifold
- Bounded, coercive and symmetric sesquilinear forms
- Real and imaginary parts, complex conjugation, and modulus
- Complex Lp classes and Euclidean test-function conventions
- The Axiom of Countable Choice (AC_ω)
- The notation Hᵏ and the reserved zero-boundary symbol
- The space Lᵖ(μ) as the quotient by null functions
- Zero-boundary Sobolev space as a norm closure
- If Y is Banach then B(X,Y) is Banach
- Real and complex inner-product spaces and their induced length
- The induced length is a norm
- Banach space
- Complete metric space: every Cauchy sequence converges in the space
- Strongly continuous semigroup
- A bounded linear operator between normed spaces
- Unbounded linear operators: domain, graph and extension
- Densely defined, closed and closable operators, and cores
- Normed subspace
- Closure properties of measurable functions used by the integral
- The nonnegative Lebesgue integral
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open
- Divergence and curl of a C¹ vector field
- Cᵏ maps and multi-index derivative notation in Euclidean space
- Cᵏ Euclidean maps and diffeomorphisms
- Metric compatible connection on a riemannian vector bundle
- Torsion tensor of an affine connection
- Vector space over a field
- Subgroup
- Group and abelian group
- In any vector space 0_F v = 0_V, λ 0_V = 0_V, (-λ)v = -(λ v), (-1_F)v = -v, and λ v = 0_V forces λ = 0_F or v = 0_V
- Regular distribution from a locally integrable function
- Locally integrable functions embed in distributions
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- Order on the reals
- Nonnegativity of a metric is a consequence of the other axioms, not an axiom
- Complete ordered field (least-upper-bound property)
- Ordered field
- 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
- Finite, countably infinite, countable, uncountable
- Every subset of an at most countable set is at most countable
- The Bessel completion embeds canonically in tempered distributions
- Plancherel theorem
- The Cauchy-sequence reals have the least-upper-bound property
- Weighted Fourier candidate norm on Schwartz space
- The weighted Fourier seminorm separates Schwartz functions
- Completion of a normed space
- The metric completion of a normed space carries a unique compatible Banach-space structure
- A smooth tensor field
- Smooth bundle metrics
- Piecewise c one curve on a manifold
- Pointwise norm and angle from a riemannian metric
- Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces
- Each smooth atlas is contained in a unique maximal smooth atlas
- Weak derivative of a locally integrable function
- Weak differentiation ignores null-set changes
- Uniqueness of a weak derivative as an almost-everywhere class
- The Lᵖ norm descends to the quotient and makes Lᵖ a normed space for 1 ≤ p ≤ ∞
- Complex Holder, Minkowski, and the quotient norm
- A convex subset of ℝᵐ contains every line segment between two of its points
2 · Logical flowchart
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3 · Items
DefinitionLibrary prerequisite
Directional derivatives and partial derivatives of a map U⊆ℝᵐ→ℝⁿ
DefinitionLibrary prerequisite
Distribution
DefinitionLibrary prerequisite
Distributional derivative
DefinitionLibrary prerequisite
The dual space X^* of a normed space and its dual norm
DefinitionLibrary prerequisite
Geodesic of an affine connection
DefinitionLibrary prerequisite
The negative Sobolev space H⁻¹(Ω)
DefinitionLibrary prerequisite
Hilbert space
DefinitionLibrary prerequisite
Infinitesimal generator of a C0-semigroup
DefinitionLibrary prerequisite
Integrable real and complex functions, and their integrals
DefinitionLibrary prerequisite
Integral over a measurable subset
DefinitionLibrary prerequisite
Interior, closure, boundary, exterior, derived set and isolated point in a topological space
DefinitionLibrary prerequisite
The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case
DefinitionLibrary prerequisite
The Laplacian of a C² function and of a C² vector field
DefinitionLibrary prerequisite
Levi civita connection
DefinitionLibrary prerequisite
Linear subspace of a vector space
DefinitionLibrary prerequisite
Locally integrable functions as regular distributions
DefinitionLibrary prerequisite
Open ball, closed ball and sphere in a metric space
DefinitionLibrary prerequisite
Open cover, subcover, compact metric space, and compact subset of a metric space
DefinitionLibrary prerequisite
Multiplication of a distribution by a smooth function
DefinitionLibrary prerequisite
Restriction Bessel-potential spaces on polyhedral domains
DefinitionLibrary prerequisite
The operator norm as the least bound and as the unit-sphere or unit-ball supremum
DefinitionLibrary prerequisite
Evolution triples and the parabolic energy space
DefinitionLibrary prerequisite
Real-order Bessel-potential completion Hˢ
DefinitionLibrary prerequisite
Riemannian metric and riemannian manifold
DefinitionLibrary prerequisite
Riemannian speed and length
DefinitionLibrary prerequisite
Smooth manifolds and their smooth charts
DefinitionLibrary prerequisite
Integer-order Sobolev spaces and their norms
DefinitionLibrary prerequisite