The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder

Depends on

• A linear map $L:\mathbb{R}^m\to\mathbb{R}^n$ in Euclidean coordinates
• A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
• Vector-valued functions $f : A \to \mathbb{R}^m$, their limits and continuity, with the dictionary to the metric notions
• 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

Used by

• The subdifferential of a differentiable convex function is its gradient singleton Corollary
• A locally constant step map on the disconnected open set ℝ∖{0} has zero total derivative but is not globally Lipschitz Counterexample
• Zero derivative need not give constancy on a disconnected open set Counterexample
• Bounded C1 domains and their outward normals Definition
• Continuously differentiable maps, local inverses, and local diffeomorphisms Definition
• Holomorphic functions on an open subset of ℂᵐ Definition
• Orientation-preserving conformality for a real-differentiable complex map at a point Definition
• Wirtinger operators in ℂᵐ Definition
• r² sin(1/r) is differentiable at the origin with a discontinuous gradient Example
• xy sin(1/(x²+y²)) is differentiable at the origin with unbounded partial derivatives nearby Example
• Choice-free smooth inverse function theorem in Euclidean space Lemma
• Smooth orientation sign is the local integral homology multiplier Lemma
• The total derivative at a point is unique Lemma
• Compatibility of smooth atlases is an equivalence relation, and smooth Euclidean maps compose Proposition
• Dimension, openness, norm, Jacobian, and the native Euclidean linear-map agreement seam Remark
• A conjugate difference quotient characterizes antiholomorphic maps Theorem
• A differentiable map on a connected open Euclidean set has zero derivative exactly when it is constant Theorem
• A total derivative computes every directional derivative, and its matrix is the Jacobian Theorem
• Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with ∂_z̄f=0, or with the Cauchy–Riemann equations Theorem
• For C¹ functions, holomorphy, complex linearity of the real derivative, and the Cauchy–Riemann system agree Theorem
• If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative Theorem
• 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 Theorem
• Sums and scalar multiples of totally differentiable maps are totally differentiable with the expected derivatives Theorem
• The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a) Theorem
• The Euclidean implicit function theorem with derivative formula Theorem
• The Euclidean inverse function theorem Theorem
• Total differentiability gives a local O(‖h‖₂) increment bound and therefore continuity Theorem
• Young's theorem: total differentiability of the first partials forces equality of mixed partials Theorem

Dependency tree · two levels

32 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• J. Lebl, Basic Analysis I, §8.3 (standard reference, not scraped)