The Total Derivative in $R^{m} \to R^{n}$

About this page

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 4 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 4 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

Set Theory

Topology

2 · Logical flowchart

3 · Items

Definition

For a differentiable scalar field, Dᵥf(a)=⟨∇ f(a),v⟩ and the unit direction of steepest ascent is the normalized gradient

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

The Total Derivative in $R^{m} \to R^{n}$

3 · Items

A linear map L:ℝᵐ→ℝⁿ in Euclidean coordinates

Every Euclidean linear map has a unique matrix and satisfies ‖Lh‖₂≤ K‖h‖₂ for some K≥0

A convex subset of ℝᵐ contains every line segment between two of its points

Directional derivatives and partial derivatives of a map U⊆ℝᵐ→ℝⁿ

The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder

The total derivative at a point is unique

Total differentiability gives a local O(‖h‖₂) increment bound and therefore continuity

The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case

A total derivative computes every directional derivative, and its matrix is the Jacobian

For a differentiable scalar field, Dᵥf(a)=⟨∇ f(a),v⟩ and the unit direction of steepest ascent is the normalized gradient

Sums and scalar multiples of totally differentiable maps are totally differentiable with the expected derivatives

The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)

Small coordinate-by-coordinate increments stay inside a Euclidean ball and telescope the total increment

If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative

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

A totally differentiable map with zero derivative on a convex open set is constant

Dimension, openness, norm, Jacobian, and the native Euclidean linear-map agreement seam

The Total Derivative in Rm→RnRm→Rn

3 · Items

A linear map L:ℝᵐ→ℝⁿ in Euclidean coordinates

Every Euclidean linear map has a unique matrix and satisfies ‖Lh‖₂≤ K‖h‖₂ for some K≥0

A convex subset of ℝᵐ contains every line segment between two of its points

Directional derivatives and partial derivatives of a map U⊆ℝᵐ→ℝⁿ

The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder

The total derivative at a point is unique

Total differentiability gives a local O(‖h‖₂) increment bound and therefore continuity

The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case

A total derivative computes every directional derivative, and its matrix is the Jacobian

For a differentiable scalar field, Dᵥf(a)=⟨∇ f(a),v⟩ and the unit direction of steepest ascent is the normalized gradient

Sums and scalar multiples of totally differentiable maps are totally differentiable with the expected derivatives

The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)

Small coordinate-by-coordinate increments stay inside a Euclidean ball and telescope the total increment

If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative

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

A totally differentiable map with zero derivative on a convex open set is constant

Dimension, openness, norm, Jacobian, and the native Euclidean linear-map agreement seam

The Total Derivative in $R^{m} \to R^{n}$