Alphabeta MathMixed Partials, Taylor Formulae, and Extrema
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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✓ 16 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 16 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Cᵏ maps and multi-index derivative notation in Euclidean space
Definition
The Hessian matrix and critical points of a scalar field
Lemma
A rectangular second difference equals a mixed partial times the side lengths
Theorem
Peano's mixed-partial theorem from continuity of one mixed partial
Theorem
Young's theorem: total differentiability of the first partials forces equality of mixed partials
Theorem
Clairaut--Schwarz theorem for continuous second partial derivatives
Corollary
The Hessian of a C² scalar field is symmetric
Theorem
Continuous mixed partials of order k are invariant under permutations
Lemma
Repeated derivatives along a line expand by the multinomial formula
Definition
The multivariable Taylor polynomial in multi-index notation
Theorem
Multivariable Taylor formula with a Lagrange remainder along a line segment
Corollary
Multivariable Taylor formula with o(‖h‖ᵏ) remainder
Corollary
Second-order Taylor expansion f(a+h)=f(a)+∇ f(a)· h+1/2h^TH_f(a)h+o(‖h‖²)
Definition
Local and strict local extrema for scalar fields on Euclidean open sets
Theorem
Fermat's theorem: an interior differentiable local extremum has zero gradient
Definition
Positive definite, negative definite, semidefinite, and indefinite quadratic forms
Lemma
A definite quadratic form has a uniform signed bound on the Euclidean unit sphere
Theorem
The multivariable second-derivative test by definiteness of the Hessian
Corollary
The two-variable Hessian determinant test
Theorem
A constrained local extremum annihilates every velocity of a differentiable parametrization
Corollary