Multivariable Differential Calculus

(i) Base. This setting extends Functions of Class Cᵏ: Notation and Measurable Functions and the Lebesgue Integral: Standing Notation: their conventions and notation are in force.

(ii) First-order derivatives. Background: The Mean Value Theorem and Inequality on Convex Open Sets, besides the first-order results carried by the base.

(iii) Higher derivatives. The Hessian matrix $D^{2}f(x)$ and the Hessian function $D^{2}f$ of a function $f:U\to\mathbb{R}$ of class $C^{2}$ on $U$ are as in The Hessian Matrix of a Function of Class C² (i) and The Hessian Matrix of a Function of Class C² (ii). Background: Basic Properties of Cᵏ Functions, Restriction of a Cᵏ Function to an Open Subset, Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, Composites of Cᵏ Functions Are of Class Cᵏ, Euclidean Space Is Open in Itself, Equality of Mixed Partial Derivatives, and Symmetry and Continuity of the Hessian, Derivatives of a Function Along a Line Segment in ℝⁿ, Second-Order Taylor Expansion, Multivariate Taylor Expansion with Uniform Second-Order Remainder and First- and Second-Order Conditions at a Local Extremum.

(iv) Diffeomorphisms, inverse and implicit functions. For $k\in\mathbb{N}$, $C^{k}$ and smooth diffeomorphisms between open subsets of $\mathbb{R}^{n}$, and local $C^{k}$ and local smooth diffeomorphisms, are as in Cᵏ Diffeomorphisms Between Open Subsets of ℝⁿ (i), Local Cᵏ Diffeomorphisms (i) and Local Cᵏ Diffeomorphisms (ii). Background: The Inverse Function Theorem, The Inverse Function Theorem for Smooth Functions, The Implicit Function Theorem and Lagrange Multipliers.

(v) Integrals depending on a parameter. Background: Differentiation under the Integral Sign.